# Mode Settings

This chapter describes commands that set modes in the Calculator. They do not affect the contents of the stack, although they may change the appearance or interpretation of the stack's contents.

## General Mode Commands

You can save all of the current mode settings in your .emacs' file with the m m (calc-save-modes) command. This will cause Emacs to reestablish these modes each time it starts up. The modes saved in the file include everything controlled by the m and d prefix keys, the current precision and binary word size, whether or not the trail is displayed, the current height of the Calc window, and more. The current interface (used when you type M-# M-#) is also saved. If there were already saved mode settings in the file, they are replaced. Otherwise, the new mode information is appended to the end of the file.

The m R (calc-mode-record-mode) command tells Calc to record the new mode settings (as if by pressing m m) every time a mode setting changes. If Embedded Mode is enabled, other options are available; see section Mode Settings in Embedded Mode.

The m F (calc-settings-file-name) command allows you to choose a different place than your .emacs' file for m m, Z P, and similar commands to save permanent information. You are prompted for a file name. All Calc modes are then reset to their default values, then settings from the file you named are loaded if this file exists, and this file becomes the one that Calc will use in the future for commands like m m. The default settings file name is ~/.emacs'. You can see the current file name by giving a blank response to the m F prompt. See also the discussion of the calc-settings-file variable; see section Installation.

If the file name you give contains the string .emacs' anywhere inside it, m F will not automatically load the new file. This is because you are presumably switching to your ~/.emacs' file, which may contain other things you don't want to reread. You can give a numeric prefix argument of 1 to m F to force it to read the file no matter what its name. Conversely, an argument of -1 tells m F not to read the new file. An argument of 2 or -2 tells m F not to reset the modes to their defaults beforehand, which is useful if you intend your new file to have a variant of the modes present in the file you were using before.

The m x (calc-always-load-extensions) command enables a mode in which the first use of Calc loads the entire program, including all extensions modules. Otherwise, the extensions modules will not be loaded until the various advanced Calc features are used. Since this mode only has effect when Calc is first loaded, m x is usually followed by m m to make the mode-setting permanent. To load all of Calc just once, rather than always in the future, you can press M-# L.

The m S (calc-shift-prefix) command enables a mode in which all of Calc's letter prefix keys may be typed shifted as well as unshifted. If you are typing, say, a S (calc-solve-for) quite often you might find it easier to turn this mode on so that you can type A S instead. When this mode is enabled, the commands that used to be on those single shifted letters (e.g., A (calc-abs)) can now be invoked by pressing the shifted letter twice: A A. Note that the v prefix key always works both shifted and unshifted, and the z and Z prefix keys are always distinct. Also, the h prefix is not affected by this mode. Press m S again to disable shifted-prefix mode.

## Precision

The p (calc-precision) command controls the precision to which floating-point calculations are carried. The precision must be at least 3 digits and may be arbitrarily high, within the limits of memory and time. This affects only floats: Integer and rational calculations are always carried out with as many digits as necessary.

The p key prompts for the current precision. If you wish you can instead give the precision as a numeric prefix argument.

Many internal calculations are carried to one or two digits higher precision than normal. Results are rounded down afterward to the current precision. Unless a special display mode has been selected, floats are always displayed with their full stored precision, i.e., what you see is what you get. Reducing the current precision does not round values already on the stack, but those values will be rounded down before being used in any calculation. The c 0 through c 9 commands (see section Conversions) can be used to round an existing value to a new precision.

It is important to distinguish the concepts of precision and accuracy. In the normal usage of these words, the number 123.4567 has a precision of 7 digits but an accuracy of 4 digits. The precision is the total number of digits not counting leading or trailing zeros (regardless of the position of the decimal point). The accuracy is simply the number of digits after the decimal point (again not counting trailing zeros). In Calc you control the precision, not the accuracy of computations. If you were to set the accuracy instead, then calculations like exp(100)' would generate many more digits than you would typically need, while exp(-100)' would probably round to zero! In Calc, both these computations give you exactly 12 (or the requested number of) significant digits.

The only Calc features that deal with accuracy instead of precision are fixed-point display mode for floats (d f; see section Float Formats), and the rounding functions like floor and round (see section Integer Truncation). Also, c 0 through c 9 deal with both precision and accuracy depending on the magnitudes of the numbers involved.

If you need to work with a particular fixed accuracy (say, dollars and cents with two digits after the decimal point), one solution is to work with integers and an "implied" decimal point. For example, $8.99 divided by 6 would be entered 899 RET 6 /, yielding 149.833 (actually$1.49833 with our implied decimal point); pressing R would round this to 150 cents, i.e., $1.50. See section Floats, for still more on floating-point precision and related issues. ## Inverse and Hyperbolic Flags There is no single-key equivalent to the calc-arcsin function. Instead, you must first press I (calc-inverse) to set the Inverse Flag, then press S (calc-sin). The I key actually toggles the Inverse Flag. When this flag is set, the word Inv' appears in the mode line. Likewise, the H key (calc-hyperbolic) sets or clears the Hyperbolic Flag, which transforms calc-sin into calc-sinh. If both of these flags are set at once, the effect will be calc-arcsinh. (The Hyperbolic flag is also used by some non-trigonometric commands; for example H L computes a base-10, instead of base-e, logarithm.) Command names like calc-arcsin are provided for completeness, and may be executed with x or M-x. Their effect is simply to toggle the Inverse and/or Hyperbolic flags and then execute the corresponding base command (calc-sin in this case). The Inverse and Hyperbolic flags apply only to the next Calculator command, after which they are automatically cleared. (They are also cleared if the next keystroke is not a Calc command.) Digits you type after I or H (or K) are treated as prefix arguments for the next command, not as numeric entries. The same is true of C-u, but not of the minus sign (K - means to subtract and keep arguments). The third Calc prefix flag, K (keep-arguments), is discussed elsewhere. See section Keep Arguments. ## Calculation Modes The commands in this section are two-key sequences beginning with the m prefix. (That's the letter m, not the META key.) The m a' (calc-algebraic-mode) command is described elsewhere (see section Algebraic Entry). ### Angular Modes The Calculator supports three notations for angles: radians, degrees, and degrees-minutes-seconds. When a number is presented to a function like sin that requires an angle, the current angular mode is used to interpret the number as either radians or degrees. If an HMS form is presented to sin, it is always interpreted as degrees-minutes-seconds. Functions that compute angles produce a number in radians, a number in degrees, or an HMS form depending on the current angular mode. If the result is a complex number and the current mode is HMS, the number is instead expressed in degrees. (Complex-number calculations would normally be done in radians mode, though. Complex numbers are converted to degrees by calculating the complex result in radians and then multiplying by 180 over @c{$\pi$} pi.) The m r (calc-radians-mode), m d (calc-degrees-mode), and m h (calc-hms-mode) commands control the angular mode. The current angular mode is displayed on the Emacs mode line. The default angular mode is degrees. ### Polar Mode The Calculator normally "prefers" rectangular complex numbers in the sense that rectangular form is used when the proper form can not be decided from the input. This might happen by multiplying a rectangular number by a polar one, by taking the square root of a negative real number, or by entering ( 2 SPC 3 ). The m p (calc-polar-mode) command toggles complex-number preference between rectangular and polar forms. In polar mode, all of the above example situations would produce polar complex numbers. ### Fraction Mode Division of two integers normally yields a floating-point number if the result cannot be expressed as an integer. In some cases you would rather get an exact fractional answer. One way to accomplish this is to multiply fractions instead: 6 RET 1:4 * produces 3:2 even though 6 RET 4 / produces 1.5. To set the Calculator to produce fractional results for normal integer divisions, use the m f (calc-frac-mode) command. For example, 8/4 produces 2 in either mode, but 6/4 produces 3:2 in Fraction Mode, 1.5 in Float Mode. At any time you can use c f (calc-float) to convert a fraction to a float, or c F (calc-fraction) to convert a float to a fraction. See section Conversions. ### Infinite Mode The Calculator normally treats results like 1 / 0 as errors; formulas like this are left in unsimplified form. But Calc can be put into a mode where such calculations instead produce "infinite" results. The m i (calc-infinite-mode) command turns this mode on and off. When the mode is off, infinities do not arise except in calculations that already had infinities as inputs. (One exception is that infinite open intervals like [0 .. inf)' can be generated; however, intervals closed at infinity ([0 .. inf]') will not be generated when infinite mode is off.) With infinite mode turned on, 1 / 0' will generate uinf, an undirected infinity. See section Infinities, for a discussion of the difference between inf and uinf. Also, 0 / 0 evaluates to nan, the "indeterminate" symbol. Various other functions can also return infinities in this mode; for example, ln(0) = -inf', and gamma(-7) = uinf'. Once again, note that exp(inf) = inf' regardless of infinite mode because this calculation has infinity as an input. The m i command with a numeric prefix argument of zero, i.e., C-u 0 m i, turns on a "positive infinite mode" in which zero is treated as positive instead of being directionless. Thus, 1 / 0 = inf' and -1 / 0 = -inf' in this mode. Note that zero never actually has a sign in Calc; there are no separate representations for +0 and -0. Positive infinite mode merely changes the interpretation given to the single symbol, 0'. One consequence of this is that, while you might expect 1 / -0 = -inf', actually 1 / -0' is equivalent to 1 / 0', which is equal to positive inf. ### Symbolic Mode Calculations are normally performed numerically wherever possible. For example, the calc-sqrt command, or sqrt function in an algebraic expression, produces a numeric answer if the argument is a number or a symbolic expression if the argument is an expression: 2 Q pushes 1.4142 but ' x+1 RET Q pushes sqrt(x+1)'. In symbolic mode, controlled by the m s (calc-symbolic-mode) command, functions which would produce inexact, irrational results are left in symbolic form. Thus 16 Q pushes 4, but 2 Q pushes sqrt(2)'. The shift-N (calc-eval-num) command evaluates numerically the expression at the top of the stack, by temporarily disabling calc-symbolic-mode and executing = (calc-evaluate). Given a numeric prefix argument, it also sets the floating-point precision to the specified value for the duration of the command. To evaluate a formula numerically without expanding the variables it contains, you can use the key sequence m s a v m s (this uses calc-alg-evaluate, which resimplifies but doesn't evaluate variables.) ### Matrix and Scalar Modes Calc sometimes makes assumptions during algebraic manipulation that are awkward or incorrect when vectors and matrices are involved. Calc has two modes, matrix mode and scalar mode, which modify its behavior around vectors in useful ways. Press m v (calc-matrix-mode) once to enter matrix mode. In this mode, all objects are assumed to be matrices unless provably otherwise. One major effect is that Calc will no longer consider multiplication to be commutative. (Recall that in matrix arithmetic, A*B' is not the same as B*A'.) This assumption affects rewrite rules and algebraic simplification. Another effect of this mode is that calculations that would normally produce constants like 0 and 1 (e.g., a - a and a / a, respectively) will now produce function calls that represent "generic" zero or identity matrices: idn(0)', idn(1)'. The idn function idn(a,n)' returns a times an nxn identity matrix; if n is omitted, it doesn't know what dimension to use and so the idn call remains in symbolic form. However, if this generic identity matrix is later combined with a matrix whose size is known, it will be converted into a true identity matrix of the appropriate size. On the other hand, if it is combined with a scalar (as in idn(1) + 2'), Calc will assume it really was a scalar after all and produce, e.g., 3. Press m v a second time to get scalar mode. Here, objects are assumed not to be vectors or matrices unless provably so. For example, normally adding a variable to a vector, as in [x, y, z] + a', will leave the sum in symbolic form because as far as Calc knows, a' could represent either a number or another 3-vector. In scalar mode, a' is assumed to be a non-vector, and the addition is evaluated to [x+a, y+a, z+a]'. Press m v a third time to return to the normal mode of operation. If you press m v with a numeric prefix argument n, you get a special "dimensioned matrix mode" in which matrices of unknown size are assumed to be nxn square matrices. Then, the function call idn(1)' will expand into an actual matrix rather than representing a "generic" matrix. Of course these modes are approximations to the true state of affairs, which is probably that some quantities will be matrices and others will be scalars. One solution is to "declare" certain variables or functions to be scalar-valued. See section Declarations, to see how to make declarations in Calc. There is nothing stopping you from declaring a variable to be scalar and then storing a matrix in it; however, if you do, the results you get from Calc may not be valid. Suppose you let Calc get the result [x+a, y+a, z+a]' shown above, and then stored [1, 2, 3]' in a'. The result would not be the same as for [x, y, z] + [1, 2, 3]', but that's because you have broken your earlier promise to Calc that a' would be scalar. Another way to mix scalars and matrices is to use selections (see section Selecting Sub-Formulas). Use matrix mode when operating on your formula normally; then, to apply scalar mode to a certain part of the formula without affecting the rest just select that part, change into scalar mode and press = to resimplify the part under this mode, then change back to matrix mode before deselecting. ### Automatic Recomputation The evaluates-to operator, =>', has the special property that any =>' formulas on the stack are recomputed whenever variable values or mode settings that might affect them are changed. See section The Evaluates-To Operator. The m C (calc-auto-recompute) command turns this automatic recomputation on and off. If you turn it off, Calc will not update =>' operators on the stack (nor those in the attached Embedded Mode buffer, if there is one). They will not be updated unless you explicitly do so by pressing = or until you press m C to turn recomputation back on. (While automatic recomputation is off, you can think of m C m C as a command to update all =>' operators while leaving recomputation off.) To update =>' operators in an Embedded buffer while automatic recomputation is off, use M-# u. See section Embedded Mode. ### Working Messages Since the Calculator is written entirely in Emacs Lisp, which is not designed for heavy numerical work, many operations are quite slow. The Calculator normally displays the message Working...' in the echo area during any command that may be slow. In addition, iterative operations such as square roots and trigonometric functions display the intermediate result at each step. Both of these types of messages can be disabled if you find them distracting. Type m w (calc-working) with a numeric prefix of 0 to disable all "working" messages. Use a numeric prefix of 1 to enable only the plain Working...' message. Use a numeric prefix of 2 to see intermediate results as well. With no numeric prefix this displays the current mode. While it may seem that the "working" messages will slow Calc down considerably, experiments have shown that their impact is actually quite small. But if your terminal is slow you may find that it helps to turn the messages off. ## Simplification Modes The current simplification mode controls how numbers and formulas are "normalized" when being taken from or pushed onto the stack. Some normalizations are unavoidable, such as rounding floating-point results to the current precision, and reducing fractions to simplest form. Others, such as simplifying a formula like a+a (or 2+3), are done by default but can be turned off when necessary. When you press a key like + when 2 and 3 are on the stack, Calc pops these numbers, normalizes them, creates the formula 2+3, normalizes it, and pushes the result. Of course the standard rules for normalizing 2+3 will produce the result 5. Simplification mode commands consist of the lower-case m prefix key followed by a shifted letter. The m O (calc-no-simplify-mode) command turns off all optional simplifications. These would leave a formula like 2+3 alone. In fact, nothing except simple numbers are ever affected by normalization in this mode. The m N (calc-num-simplify-mode) command turns off simplification of any formulas except those for which all arguments are constants. For example, 1+2 is simplified to 3, and a+(2-2) is simplified to a+0 but no further, since one argument of the sum is not a constant. Unfortunately, (a+2)-2 is not simplified because the top-level -' operator's arguments are not both constant numbers (one of them is the formula a+2). A constant is a number or other numeric object (such as a constant error form or modulo form), or a vector all of whose elements are constant. The m D (calc-default-simplify-mode) command restores the default simplifications for all formulas. This includes many easy and fast algebraic simplifications such as a+0 to a, and a + 2 a to 3 a, as well as evaluating functions like deriv(x^2, x) to 2 x. The m B (calc-bin-simplify-mode) mode applies the default simplifications to a result and then, if the result is an integer, uses the b c (calc-clip) command to clip the integer according to the current binary word size. See section Binary Number Functions. Real numbers are rounded to the nearest integer and then clipped; other kinds of results (after the default simplifications) are left alone. The m A (calc-alg-simplify-mode) mode does algebraic simplification; it applies all the default simplifications, and also the more powerful (and slower) simplifications made by a s (calc-simplify). See section Algebraic Simplifications. The m E (calc-ext-simplify-mode) mode does "extended" algebraic simplification, as by the a e (calc-simplify-extended) command. See section "Unsafe" Simplifications. The m U (calc-units-simplify-mode) mode does units simplification; it applies the command u s (calc-simplify-units), which in turn is a superset of a s. In this mode, variable names which are identifiable as unit names (like mm' for "millimeters") are simplified with their unit definitions in mind. A common technique is to set the simplification mode down to the lowest amount of simplification you will allow to be applied automatically, then use manual commands like a s and c c (calc-clean) to perform higher types of simplifications on demand. See section Programming with Formulas, for another sample use of no-simplification mode. ## Declarations A declaration is a statement you make that promises you will use a certain variable or function in a restricted way. This may give Calc the freedom to do things that it couldn't do if it had to take the fully general situation into account. ### Declaration Basics The s d (calc-declare-variable) command is the easiest way to make a declaration for a variable. This command prompts for the variable name, then prompts for the declaration. The default at the declaration prompt is the previous declaration, if any. You can edit this declaration, or press C-k to erase it and type a new declaration. (Or, erase it and press RET to clear the declaration, effectively "undeclaring" the variable.) A declaration is in general a vector of type symbols and range values. If there is only one type symbol or range value, you can write it directly rather than enclosing it in a vector. For example, s d foo RET real RET declares foo to be a real number, and s d bar RET [int, const, [1..6]] RET declares bar to be a constant integer between 1 and 6. (Actually, you can omit the outermost brackets and Calc will provide them for you: s d bar RET int, const, [1..6] RET.) Declarations in Calc are kept in a special variable called Decls. This variable encodes the set of all outstanding declarations in the form of a matrix. Each row has two elements: A variable or vector of variables declared by that row, and the declaration specifier as described above. You can use the s D command to edit this variable if you wish to see all the declarations at once. See section Other Operations on Variables, for a description of this command and the s p command that allows you to save your declarations permanently if you wish. Items being declared can also be function calls. The arguments in the call are ignored; the effect is to say that this function returns values of the declared type for any valid arguments. The s d command declares only variables, so if you wish to make a function declaration you will have to edit the Decls matrix yourself. For example, the declaration matrix [ [ foo, real ] [ [j, k, n], int ] [ f(1,2,3), [0 .. inf) ] ]  declares that foo represents a real number, j, k and n represent integers, and the function f always returns a real number in the interval shown. If there is a declaration for the variable All, then that declaration applies to all variables that are not otherwise declared. It does not apply to function names. For example, using the row [All, real]' says that all your variables are real unless they are explicitly declared without real in some other row. The s d command declares All if you give a blank response to the variable-name prompt. ### Kinds of Declarations The type-specifier part of a declaration (that is, the second prompt in the s d command) can be a type symbol, an interval, or a vector consisting of zero or more type symbols followed by zero or more intervals or numbers that represent the set of possible values for the variable. [ [ a, [1, 2, 3, 4, 5] ] [ b, [1 .. 5] ] [ c, [int, 1 .. 5] ] ]  Here a is declared to contain one of the five integers shown; b is any number in the interval from 1 to 5 (any real number since we haven't specified), and c is any integer in that interval. Thus the declarations for a and c are nearly equivalent (see below). The type-specifier can be the empty vector []' to say that nothing is known about a given variable's value. This is the same as not declaring the variable at all except that it overrides any All declaration which would otherwise apply. The initial value of Decls is the empty vector []'. If Decls has no stored value or if the value stored in it is not valid, it is ignored and there are no declarations as far as Calc is concerned. (The s d command will replace such a malformed value with a fresh empty matrix, []', before recording the new declaration.) Unrecognized type symbols are ignored. The following type symbols describe what sorts of numbers will be stored in a variable: int Integers. numint Numerical integers. (Integers or integer-valued floats.) frac Fractions. (Rational numbers which are not integers.) rat Rational numbers. (Either integers or fractions.) float Floating-point numbers. real Real numbers. (Integers, fractions, or floats. Actually, intervals and error forms with real components also count as reals here.) pos Positive real numbers. (Strictly greater than zero.) nonneg Nonnegative real numbers. (Greater than or equal to zero.) number Numbers. (Real or complex.) Calc uses this information to determine when certain simplifications of formulas are safe. For example, (x^y)^z' cannot be simplified to x^(y z)' in general; for example, ((-3)^2)^1:2' is 3, but (-3)^(2*1:2) = (-3)^1' is -3. However, this simplification is safe if z is known to be an integer, or if x is known to be a nonnegative real number. If you have given declarations that allow Calc to deduce either of these facts, Calc will perform this simplification of the formula. Calc can apply a certain amount of logic when using declarations. For example, (x^y)^(2n+1)' will be simplified if n has been declared int; Calc knows that an integer times an integer, plus an integer, must always be an integer. (In fact, Calc would simplify (-x)^(2n+1)' to -(x^(2n+1))' since it is able to determine that 2n+1' must be an odd integer.) Similarly, (abs(x)^y)^z' will be simplified to abs(x)^(y z)' because Calc knows that the abs function always returns a nonnegative real. If you had a myabs function that also had this property, you could get Calc to recognize it by adding the row [myabs(), nonneg]' to the Decls matrix. One instance of this simplification is sqrt(x^2)' (since the sqrt function is effectively a one-half power). Normally Calc leaves this formula alone. After the command s d x RET real RET, however, it can simplify the formula to abs(x)'. And after s d x RET nonneg RET, Calc can simplify this formula all the way to x'. If there are any intervals or real numbers in the type specifier, they comprise the set of possible values that the variable or function being declared can have. In particular, the type symbol real is effectively the same as the range [-inf .. inf]' (note that infinity is included in the range of possible values); pos is the same as (0 .. inf]', and nonneg is the same as [0 .. inf]'. Saying [real, [-5 .. 5]]' is redundant because the fact that the variable is real can be deduced just from the interval, but [int, [-5 .. 5]]' and [rat, [-5 .. 5]]' are useful combinations. Note that the vector of intervals or numbers is in the same format used by Calc's set-manipulation commands. See section Set Operations using Vectors. The type specifier [1, 2, 3]' is equivalent to [numint, 1, 2, 3]', not to [int, 1, 2, 3]'. In other words, the range of possible values means only that the variable's value must be numerically equal to a number in that range, but not that it must be equal in type as well. Calc's set operations act the same way; in(2, [1., 2., 3.])' and in(1.5, [1:2, 3:2, 5:2])' both report "true." If you use a conflicting combination of type specifiers, the results are unpredictable. An example is [pos, [0 .. 5]]', where the interval does not lie in the range described by the type symbol. "Real" declarations mostly affect simplifications involving powers like the one described above. Another case where they are used is in the a P command which returns a list of all roots of a polynomial; if the variable has been declared real, only the real roots (if any) will be included in the list. "Integer" declarations are used for simplifications which are valid only when certain values are integers (such as (x^y)^z' shown above). Another command that makes use of declarations is a s, when simplifying equations and inequalities. It will cancel x from both sides of a x = b x' only if it is sure x is non-zero, say, because it has a pos declaration. To declare specifically that x is real and non-zero, use [[-inf .. 0), (0 .. inf]]'. (There is no way in the current notation to say that x is nonzero but not necessarily real.) The a e command does "unsafe" simplifications, including cancelling x' from the equation when x' is not known to be nonzero. Another set of type symbols distinguish between scalars and vectors. scalar The value is not a vector. vector The value is a vector. matrix The value is a matrix (a rectangular vector of vectors). These type symbols can be combined with the other type symbols described above; [int, matrix]' describes an object which is a matrix of integers. Scalar/vector declarations are used to determine whether certain algebraic operations are safe. For example, [a, b, c] + x' is normally not simplified to [a + x, b + x, c + x]', but it will be if x has been declared scalar. On the other hand, multiplication is usually assumed to be commutative, but the terms in x y' will never be exchanged if both x and y are known to be vectors or matrices. (Calc currently never distinguishes between vector and matrix declarations.) See section Matrix and Scalar Modes, for a discussion of "matrix mode" and "scalar mode," which are similar to declaring [All, matrix]' or [All, scalar]' but much more convenient. One more type symbol that is recognized is used with the H a d command for taking total derivatives of a formula. See section Calculus. const The value is a constant with respect to other variables. Calc does not check the declarations for a variable when you store a value in it. However, storing -3.5 in a variable that has been declared pos, int, or matrix may have unexpected effects; Calc may evaluate sqrt(x^2)' to 3.5 if it substitutes the value first, or to -3.5 if x was declared pos and the formula sqrt(x^2)' is simplified to x' before the value is substituted. Before using a variable for a new purpose, it is best to use s d or s D to check to make sure you don't still have an old declaration for the variable that will conflict with its new meaning. ### Functions for Declarations Calc has a set of functions for accessing the current declarations in a convenient manner. These functions return 1 if the argument can be shown to have the specified property, or 0 if the argument can be shown not to have that property; otherwise they are left unevaluated. These functions are suitable for use with rewrite rules (see section Conditional Rewrite Rules) or programming constructs (see section Conditionals in Keyboard Macros). They can be entered only using algebraic notation. See section Logical Operations, for functions that perform other tests not related to declarations. For example, dint(17)' returns 1 because 17 is an integer, as do dint(n)' and dint(2 n - 3)' if n has been declared int, but dint(2.5)' and dint(n + 0.5)' return 0. Calc consults knowledge of its own built-in functions as well as your own declarations: dint(floor(x))' returns 1. The dint function checks if its argument is an integer. The dnatnum function checks if its argument is a natural number, i.e., a nonnegative integer. The dnumint function checks if its argument is numerically an integer, i.e., either an integer or an integer-valued float. Note that these and the other data type functions also accept vectors or matrices composed of suitable elements, and that real infinities inf' and -inf' are considered to be integers for the purposes of these functions. The drat function checks if its argument is rational, i.e., an integer or fraction. Infinities count as rational, but intervals and error forms do not. The dreal function checks if its argument is real. This includes integers, fractions, floats, real error forms, and intervals. The dimag function checks if its argument is imaginary, i.e., is mathematically equal to a real number times i. The dpos function checks for positive (but nonzero) reals. The dneg function checks for negative reals. The dnonneg function checks for nonnegative reals, i.e., reals greater than or equal to zero. Note that the a s command can simplify an expression like x > 0 to 1 or 0 using dpos, and that a s is effectively applied to all conditions in rewrite rules, so the actual functions dpos, dneg, and dnonneg are rarely necessary. The dnonzero function checks that its argument is nonzero. This includes all nonzero real or complex numbers, all intervals that do not include zero, all nonzero modulo forms, vectors all of whose elements are nonzero, and variables or formulas whose values can be deduced to be nonzero. It does not include error forms, since they represent values which could be anything including zero. (This is also the set of objects considered "true" in conditional contexts.) The deven function returns 1 if its argument is known to be an even integer (or integer-valued float); it returns 0 if its argument is known not to be even (because it is known to be odd or a non-integer). The a s command uses this to simplify a test of the form x % 2 = 0'. There is also an analogous dodd function. The drange function returns a set (an interval or a vector of intervals and/or numbers; see section Set Operations using Vectors) that describes the set of possible values of its argument. If the argument is a variable or a function with a declaration, the range is copied from the declaration. Otherwise, the possible signs of the expression are determined using a method similar to dpos, etc., and a suitable set like [0 .. inf]' is returned. If the expression is not provably real, the drange function remains unevaluated. The dscalar function returns 1 if its argument is provably scalar, or 0 if its argument is provably non-scalar. It is left unevaluated if this cannot be determined. (If matrix mode or scalar mode are in effect, this function returns 1 or 0, respectively, if it has no other information.) When Calc interprets a condition (say, in a rewrite rule) it considers an unevaluated formula to be "false." Thus, dscalar(a)' is "true" only if a is provably scalar, and !dscalar(a)' is "true" only if a is provably non-scalar; both are "false" if there is insufficient information to tell. ## Display Modes The commands in this section are two-key sequences beginning with the d prefix. The d l (calc-line-numbering) and d b (calc-line-breaking) commands are described elsewhere; see section Stack Basics and see section Normal Language Modes, respectively. Display formats for vectors and matrices are also covered elsewhere; see section Vector and Matrix Display Formats. One thing all display modes have in common is their treatment of the H prefix. This prefix causes any mode command that would normally refresh the stack to leave the stack display alone. The word "Dirty" will appear in the mode line when Calc thinks the stack display may not reflect the latest mode settings. The d RET (calc-refresh-top) command reformats the top stack entry according to all the current modes. Positive prefix arguments reformat the top n entries; negative prefix arguments reformat the specified entry, and a prefix of zero is equivalent to d SPC (calc-refresh), which reformats the entire stack. For example, H d s M-2 d RET changes to scientific notation but reformats only the top two stack entries in the new mode. The I prefix has another effect on the display modes. The mode is set only temporarily; the top stack entry is reformatted according to that mode, then the original mode setting is restored. In other words, I d s is equivalent to H d s d RET H d (old mode). ### Radix Modes Calc normally displays numbers in decimal (base-10 or radix-10) notation. Calc can actually display in any radix from two (binary) to 36. When the radix is above 10, the letters A to Z are used as digits. When entering such a number, letter keys are interpreted as potential digits rather than terminating numeric entry mode. The key sequences d 2, d 8, d 6, and d 0 select binary, octal, hexadecimal, and decimal as the current display radix, respectively. Numbers can always be entered in any radix, though the current radix is used as a default if you press # without any initial digits. A number entered without a # is always interpreted as decimal. To set the radix generally, use d r (calc-radix) and enter an integer from 2 to 36. You can specify the radix as a numeric prefix argument; otherwise you will be prompted for it. Integers normally are displayed with however many digits are necessary to represent the integer and no more. The d z (calc-leading-zeros) command causes integers to be padded out with leading zeros according to the current binary word size. (See section Binary Number Functions, for a discussion of word size.) If the absolute value of the word size is w, all integers are displayed with at least enough digits to represent @c{$2^w-1} (2^w)-1 in the current radix. (Larger integers will still be displayed in their entirety.) ### Grouping Digits Long numbers can be hard to read if they have too many digits. For example, the factorial of 30 is 33 digits long! Press d g (calc-group-digits) to enable grouping mode, in which digits are displayed in clumps of 3 or 4 (depending on the current radix) separated by commas. The d g command toggles grouping on and off. With a numerix prefix of 0, this command displays the current state of the grouping flag; with an argument of minus one it disables grouping; with a positive argument N it enables grouping on every N digits. For floating-point numbers, grouping normally occurs only before the decimal point. A negative prefix argument -N enables grouping every N digits both before and after the decimal point. The d , (calc-group-char) command allows you to choose any character as the grouping separator. The default is the comma character. If you find it difficult to read vectors of large integers grouped with commas, you may wish to use spaces or some other character instead. This command takes the next character you type, whatever it is, and uses it as the digit separator. As a special case, d , \ selects \,' (TeX's thin-space symbol) as the digit separator. Please note that grouped numbers will not generally be parsed correctly if re-read in textual form, say by the use of M-# y and M-# g. (See section Kill and Yank Functions, for details on these commands.) One exception is the \,' separator, which doesn't interfere with parsing because it is ignored by TeX language mode. ### Float Formats Floating-point quantities are normally displayed in standard decimal form, with scientific notation used if the exponent is especially high or low. All significant digits are normally displayed. The commands in this section allow you to choose among several alternative display formats for floats. The d n (calc-normal-notation) command selects the normal display format. All significant figures in a number are displayed. With a positive numeric prefix, numbers are rounded if necessary to that number of significant digits. With a negative numerix prefix, the specified number of significant digits less than the current precision is used. (Thus C-u -2 d n displays 10 digits if the current precision is 12.) The d f (calc-fix-notation) command selects fixed-point notation. The numeric argument is the number of digits after the decimal point, zero or more. This format will relax into scientific notation if a nonzero number would otherwise have been rounded all the way to zero. Specifying a negative number of digits is the same as for a positive number, except that small nonzero numbers will be rounded to zero rather than switching to scientific notation. The d s (calc-sci-notation) command selects scientific notation. A positive argument sets the number of significant figures displayed, of which one will be before and the rest after the decimal point. A negative argument works the same as for d n format. The default is to display all significant digits. The d e (calc-eng-notation) command selects engineering notation. This is similar to scientific notation except that the exponent is rounded down to a multiple of three, with from one to three digits before the decimal point. An optional numeric prefix sets the number of significant digits to display, as for d s. It is important to distinguish between the current precision and the current display format. After the commands C-u 10 p and C-u 6 d n the Calculator computes all results to ten significant figures but displays only six. (In fact, intermediate calculations are often carried to one or two more significant figures, but values placed on the stack will be rounded down to ten figures.) Numbers are never actually rounded to the display precision for storage, except by commands like C-k and M-# y which operate on the actual displayed text in the Calculator buffer. The d . (calc-point-char) command selects the character used as a decimal point. Normally this is a period; users in some countries may wish to change this to a comma. Note that this is only a display style; on entry, periods must always be used to denote floating-point numbers, and commas to separate elements in a list. ### Complex Formats There are three supported notations for complex numbers in rectangular form. The default is as a pair of real numbers enclosed in parentheses and separated by a comma: (a,b)'. The d c (calc-complex-notation) command selects this style. The other notations are d i (calc-i-notation), in which numbers are displayed in a+bi' form, and d j (calc-j-notation) which displays the form a+bj' preferred in some disciplines. Complex numbers are normally entered in (a,b)' format. If you enter 2+3i' as an algebraic formula, it will be stored as the formula 2 + 3 * i'. However, if you use = to evaluate this formula and you have not changed the variable i', the i' will be interpreted as (0,1)' and the formula will be simplified to (2,3)'. Other commands (like calc-sin) will not interpret the formula 2 + 3 * i' as a complex number. See section Variables, under "special constants." ### Fraction Formats Display of fractional numbers is controlled by the d o (calc-over-notation) command. By default, a number like eight thirds is displayed in the form 8:3'. The d o command prompts for a one- or two-character format. If you give one character, that character is used as the fraction separator. Common separators are :' and /'. (During input of numbers, the : key must be used regardless of the display format; in particular, the / is used for RPN-style division, not for entering fractions.) If you give two characters, fractions use "integer-plus-fractional-part" notation. For example, the format +/' would display eight thirds as 2+2/3'. If two colons are present in a number being entered, the number is interpreted in this form (so that the entries 2:2:3 and 8:3 are equivalent). It is also possible to follow the one- or two-character format with a number. For example: :10' or +/3'. In this case, Calc adjusts all fractions that are displayed to have the specified denominator, if possible. Otherwise it adjusts the denominator to be a multiple of the specified value. For example, in :6' mode the fraction 1:6 will be unaffected, but 2:3 will be displayed as 4:6, 1:2 will be displayed as 3:6, and 1:8 will be displayed as 3:24. Integers are also affected by this mode: 3 is displayed as 18:6. Note that the format :1' writes fractions the same as :', but it writes integers as n:1. The fraction format does not affect the way fractions or integers are stored, only the way they appear on the screen. The fraction format never affects floats. ### HMS Formats The d h (calc-hms-notation) command controls the display of HMS (hours-minutes-seconds) forms. It prompts for a string which consists basically of an "hours" marker, optional punctuation, a "minutes" marker, more optional punctuation, and a "seconds" marker. Punctuation is zero or more spaces, commas, or semicolons. The hours marker is one or more non-punctuation characters. The minutes and seconds markers must be single non-punctuation characters. The default HMS format is @ ' "', producing HMS values of the form 23@ 30' 15.75"'. The format deg, ms' would display this same value as 23deg, 30m15.75s'. During numeric entry, the h or o keys are recognized as synonyms for @ regardless of display format. The m and s keys are recognized as synonyms for ' and ", respectively, but only if an @ (or h or o) has already been typed; otherwise, they have their usual meanings (m- prefix and s- prefix). Thus, 5 ", 0 @ 5 ", and 0 h 5 s are some of the ways to enter the quantity "five seconds." The ' key is recognized as "minutes" only if @ (or h or o) has already been pressed; otherwise it means to switch to algebraic entry. ### Date Formats The d d (calc-date-notation) command controls the display of date forms (see section Date Forms). It prompts for a string which contains letters that represent the various parts of a date and time. To show which parts should be omitted when the form represents a pure date with no time, parts of the string can be enclosed in < >' marks. If you don't include < >' markers in the format, Calc guesses at which parts, if any, should be omitted when formatting pure dates. The default format is: <H:mm:SSpp >Www Mmm D, YYYY'. An example string in this format is 3:32pm Wed Jan 9, 1991'. If you enter a blank format string, this default format is reestablished. Calc uses < >' notation for nameless functions as well as for dates. See section Specifying Operators. To avoid confusion with nameless functions, your date formats should avoid using the #' character. #### Date Formatting Codes When displaying a date, the current date format is used. All characters except for letters and <' and >' are copied literally when dates are formatted. The portion between < >' markers is omitted for pure dates, or included for date/time forms. Letters are interpreted according to the table below. When dates are read in during algebraic entry, Calc first tries to match the input string to the current format either with or without the time part. The punctuation characters (including spaces) must match exactly; letter fields must correspond to suitable text in the input. If this doesn't work, Calc checks if the input is a simple number; if so, the number is interpreted as a number of days since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and flexible algorithm which is described in the next section. Weekday names are ignored during reading. Two-digit year numbers are interpreted as lying in the range from 1941 to 2039. Years outside that range are always entered and displayed in full. Year numbers with a leading +' sign are always interpreted exactly, allowing the entry and display of the years 1 through 99 AD. Here is a complete list of the formatting codes for dates: Y Year: "91" for 1991, "7" for 2007, "+23" for 23 AD. YY Year: "91" for 1991, "07" for 2007, "+23" for 23 AD. BY Year: "91" for 1991, " 7" for 2007, "+23" for 23 AD. YYY Year: "1991" for 1991, "23" for 23 AD. YYYY Year: "1991" for 1991, "+23" for 23 AD. aa Year: "ad" or blank. AA Year: "AD" or blank. aaa Year: "ad " or blank. (Note trailing space.) AAA Year: "AD " or blank. aaaa Year: "a.d." or blank. AAAA Year: "A.D." or blank. bb Year: "bc" or blank. BB Year: "BC" or blank. bbb Year: " bc" or blank. (Note leading space.) BBB Year: " BC" or blank. bbbb Year: "b.c." or blank. BBBB Year: "B.C." or blank. M Month: "8" for August. MM Month: "08" for August. BM Month: " 8" for August. MMM Month: "AUG" for August. Mmm Month: "Aug" for August. mmm Month: "aug" for August. MMMM Month: "AUGUST" for August. Mmmm Month: "August" for August. D Day: "7" for 7th day of month. DD Day: "07" for 7th day of month. BD Day: " 7" for 7th day of month. W Weekday: "0" for Sunday, "6" for Saturday. WWW Weekday: "SUN" for Sunday. Www Weekday: "Sun" for Sunday. www Weekday: "sun" for Sunday. WWWW Weekday: "SUNDAY" for Sunday. Wwww Weekday: "Sunday" for Sunday. d Day of year: "34" for Feb. 3. ddd Day of year: "034" for Feb. 3. bdd Day of year: " 34" for Feb. 3. h Hour: "5" for 5 AM; "17" for 5 PM. hh Hour: "05" for 5 AM; "17" for 5 PM. bh Hour: " 5" for 5 AM; "17" for 5 PM. H Hour: "5" for 5 AM and 5 PM. HH Hour: "05" for 5 AM and 5 PM. BH Hour: " 5" for 5 AM and 5 PM. p AM/PM: "a" or "p". P AM/PM: "A" or "P". pp AM/PM: "am" or "pm". PP AM/PM: "AM" or "PM". pppp AM/PM: "a.m." or "p.m.". PPPP AM/PM: "A.M." or "P.M.". m Minutes: "7" for 7. mm Minutes: "07" for 7. bm Minutes: " 7" for 7. s Seconds: "7" for 7; "7.23" for 7.23. ss Seconds: "07" for 7; "07.23" for 7.23. bs Seconds: " 7" for 7; " 7.23" for 7.23. SS Optional seconds: "07" for 7; blank for 0. BS Optional seconds: " 7" for 7; blank for 0. N Numeric date/time: "726842.25" for 6:00am Wed Jan 9, 1991. n Numeric date: "726842" for any time on Wed Jan 9, 1991. J Julian date/time: "2448265.75" for 6:00am Wed Jan 9, 1991. j Julian date: "2448266" for any time on Wed Jan 9, 1991. U Unix time: "663400800" for 6:00am Wed Jan 9, 1991. X Brackets suppression. An "X" at the front of the format causes the surrounding < >' delimiters to be omitted when formatting dates. Note that the brackets are still required for algebraic entry. If "SS" or "BS" (optional seconds) is preceded by a colon, the colon is also omitted if the seconds part is zero. If "bb," "bbb" or "bbbb" or their upper-case equivalents appear in the format, then negative year numbers are displayed without a minus sign. Note that "aa" and "bb" are mutually exclusive. Some typical usages would be YYYY AABB'; AAAYYYYBBB'; YYYYBBB'. The formats "YY," "YYYY," "MM," "DD," "ddd," "hh," "HH," "mm," "ss," and "SS" actually match any number of digits during reading unless several of these codes are strung together with no punctuation in between, in which case the input must have exactly as many digits as there are letters in the format. The "j," "J," and "U" formats do not make any time zone adjustment. They effectively use julian(x,0)' and unixtime(x,0)' to make the conversion; see section Date Arithmetic. #### Free-Form Dates When reading a date form during algebraic entry, Calc falls back on the algorithm described here if the input does not exactly match the current date format. This algorithm generally "does the right thing" and you don't have to worry about it, but it is described here in full detail for the curious. Calc does not distinguish between upper- and lower-case letters while interpreting dates. First, the time portion, if present, is located somewhere in the text and then removed. The remaining text is then interpreted as the date. A time is of the form hh:mm:ss', possibly with the seconds part omitted and possibly with an AM/PM indicator added to indicate 12-hour time. If the AM/PM is present, the minutes may also be omitted. The AM/PM part may be any of the words am', pm', noon', or midnight'; each of these may be abbreviated to one letter, and the alternate forms a.m.', p.m.', and mid' are also understood. Obviously noon' and midnight' are allowed only on 12:00:00. The words noon', mid', and midnight' are also recognized with no number attached. If there is no AM/PM indicator, the time is interpreted in 24-hour format. To read the date portion, all words and numbers are isolated from the string; other characters are ignored. All words must be either month names or day-of-week names (the latter of which are ignored). Names can be written in full or as three-letter abbreviations. Large numbers, or numbers with +' or -' signs, are interpreted as years. If one of the other numbers is greater than 12, then that must be the day and the remaining number in the input is therefore the month. Otherwise, Calc assumes the month, day and year are in the same order that they appear in the current date format. If the year is omitted, the current year is taken from the system clock. If there are too many or too few numbers, or any unrecognizable words, then the input is rejected. If there are any large numbers (of five digits or more) other than the year, they are ignored on the assumption that they are something like Julian dates that were included along with the traditional date components when the date was formatted. One of the words ad', a.d.', bc', or b.c.' may optionally be used; the latter two are equivalent to a minus sign on the year value. If you always enter a four-digit year, and use a name instead of a number for the month, there is no danger of ambiguity. #### Standard Date Formats There are actually ten standard date formats, numbered 0 through 9. Entering a blank line at the d d command's prompt gives you format number 1, Calc's usual format. You can enter any digit to select the other formats. To create your own standard date formats, give a numeric prefix argument from 0 to 9 to the d d command. The format you enter will be recorded as the new standard format of that number, as well as becoming the new current date format. You can save your formats permanently with the m m command (see section Mode Settings). 0 N' (Numerical format) 1 <H:mm:SSpp >Www Mmm D, YYYY' (American format) 2 D Mmm YYYY<, h:mm:SS>' (European format) 3 Www Mmm BD< hh:mm:ss> YYYY' (Unix written date format) 4 M/D/Y< H:mm:SSpp>' (American slashed format) 5 D.M.Y< h:mm:SS>' (European dotted format) 6 M-D-Y< H:mm:SSpp>' (American dashed format) 7 D-M-Y< h:mm:SS>' (European dashed format) 8 j<, h:mm:ss>' (Julian day plus time) 9 YYddd< hh:mm:ss>' (Year-day format) ### Truncating the Stack The d t (calc-truncate-stack) command moves the .' line that marks the top-of-stack up or down in the Calculator buffer. The number right above that line is considered to the be at the top of the stack. Any numbers below that line are "hidden" from all stack operations. This is similar to the Emacs "narrowing" feature, except that the values below the .' are visible, just temporarily frozen. This feature allows you to keep several independent calculations running at once in different parts of the stack, or to apply a certain command to an element buried deep in the stack. Pressing d t by itself moves the .' to the line the cursor is on. Thus, this line and all those below it become hidden. To un-hide these lines, move down to the end of the buffer and press d t. With a positive numeric prefix argument n, d t hides the bottom n values in the buffer. With a negative argument, it hides all but the top n values. With an argument of zero, it hides zero values, i.e., moves the .' all the way down to the bottom. The d [ (calc-truncate-up) and d ] (calc-truncate-down) commands move the .' up or down one line at a time (or several lines with a prefix argument). ### Justification Values on the stack are normally left-justified in the window. You can control this arrangement by typing d < (calc-left-justify), d > (calc-right-justify), or d = (calc-center-justify). For example, in right-justification mode, stack entries are displayed flush-right against the right edge of the window. If you change the width of the Calculator window you may have to type d SPC (calc-refresh) to re-align right-justified or centered text. Right-justification is especially useful together with fixed-point notation (see d f; calc-fix-notation). With these modes together, the decimal points on numbers will always line up. With a numeric prefix argument, the justification commands give you a little extra control over the display. The argument specifies the horizontal "origin" of a display line. It is also possible to specify a maximum line width using the d b command (see section Normal Language Modes). For reference, the precise rules for formatting and breaking lines are given below. Notice that the interaction between origin and line width is slightly different in each justification mode. In left-justified mode, the line is indented by a number of spaces given by the origin (default zero). If the result is longer than the maximum line width, if given, or too wide to fit in the Calc window otherwise, then it is broken into lines which will fit; each broken line is indented to the origin. In right-justified mode, lines are shifted right so that the rightmost character is just before the origin, or just before the current window width if no origin was specified. If the line is too long for this, then it is broken; the current line width is used, if specified, or else the origin is used as a width if that is specified, or else the line is broken to fit in the window. In centering mode, the origin is the column number of the center of each stack entry. If a line width is specified, lines will not be allowed to go past that width; Calc will either indent less or break the lines if necessary. If no origin is specified, half the line width or Calc window width is used. Note that, in each case, if line numbering is enabled the display is indented an additional four spaces to make room for the line number. The width of the line number is taken into account when positioning according to the current Calc window width, but not when positioning by explicit origins and widths. In the latter case, the display is formatted as specified, and then uniformly shifted over four spaces to fit the line numbers. ### Labels The d { (calc-left-label) command prompts for a string, then displays that string to the left of every stack entry. If the entries are left-justified (see section Justification), then they will appear immediately after the label (unless you specified an origin greater than the length of the label). If the entries are centered or right-justified, the label appears on the far left and does not affect the horizontal position of the stack entry. Give a blank string (with d { RET) to turn the label off. The d } (calc-right-label) command similarly adds a label on the righthand side. It does not affect positioning of the stack entries unless they are right-justified. Also, if both a line width and an origin are given in right-justified mode, the stack entry is justified to the origin and the righthand label is justified to the line width. One application of labels would be to add equation numbers to formulas you are manipulating in Calc and then copying into a document (possibly using Embedded Mode). The equations would typically be centered, and the equation numbers would be on the left or right as you prefer. ## Language Modes The commands in this section change Calc to use a different notation for entry and display of formulas, corresponding to the conventions of some other common language such as Pascal or TeX. Objects displayed on the stack or yanked from the Calculator to an editing buffer will be formatted in the current language; objects entered in algebraic entry or yanked from another buffer will be interpreted according to the current language. The current language has no effect on things written to or read from the trail buffer, nor does it affect numeric entry. Only algebraic entry is affected. You can make even algebraic entry ignore the current language and use the standard notation by giving a numeric prefix, e.g., C-u '. For example, suppose the formula 2*a[1] + atan(a[2])' occurs in a C program; elsewhere in the program you need the derivatives of this formula with respect to a[1]' and a[2]'. First, type d C to switch to C notation. Now use C-u M-# g to grab the formula into the Calculator, a d a[1] RET to differentiate with respect to the first variable, and M-# y to yank the formula for the derivative back into your C program. Press U to undo the differentiation and repeat with a d a[2] RET for the other derivative. Without being switched into C mode first, Calc would have misinterpreted the brackets in a[1]' and a[2]', would not have known that atan was equivalent to Calc's built-in arctan function, and would have written the formula back with notations (like implicit multiplication) which would not have been legal for a C program. As another example, suppose you are maintaining a C program and a TeX document, each of which needs a copy of the same formula. You can grab the formula from the program in C mode, switch to TeX mode, and yank the formula into the document in TeX math-mode format. Language modes are selected by typing the letter d followed by a shifted letter key. ### Normal Language Modes The d N (calc-normal-language) command selects the usual notation for Calc formulas, as described in the rest of this manual. Matrices are displayed in a multi-line tabular format, but all other objects are written in linear form, as they would be typed from the keyboard. The d O (calc-flat-language) command selects a language identical with the normal one, except that matrices are written in one-line form along with everything else. In some applications this form may be more suitable for yanking data into other buffers. Even in one-line mode, long formulas or vectors will still be split across multiple lines if they exceed the width of the Calculator window. The d b (calc-line-breaking) command turns this line-breaking feature on and off. (It works independently of the current language.) If you give a numeric prefix argument of five or greater to the d b command, that argument will specify the line width used when breaking long lines. The d B (calc-big-language) command selects a language which uses textual approximations to various mathematical notations, such as powers, quotients, and square roots:  ____________ | a + 1 2 | ----- + c \| b  in place of sqrt((a+1)/b + c^2)'. Subscripts like a_i' are displayed as actual subscripts in "big" mode. Double subscripts, a_i_j' (subscr(subscr(a, i), j)') are displayed as a' with subscripts separated by commas: i, j'. They must still be entered in the usual underscore notation. One slight ambiguity of Big notation is that  3 - - 4  can represent either the negative rational number -3:4, or the actual expression -(3/4)'; but the latter formula would normally never be displayed because it would immediately be evaluated to -3:4 or -0.75, so this ambiguity is not a problem in typical use. Non-decimal numbers are displayed with subscripts. Thus there is no way to tell the difference between 16#C2' and C2_16', though generally you will know which interpretation is correct. Logarithms log(x,b)' and log10(x)' also use subscripts in Big mode. In Big mode, stack entries often take up several lines. To aid readability, stack entries are separated by a blank line in this mode. You may find it useful to expand the Calc window's height using C-x ^ (enlarge-window) or to make the Calc window the only one on the screen with C-x 1 (delete-other-windows). Long lines are currently not rearranged to fit the window width in Big mode, so you may need to use the < and > keys to scroll across a wide formula. For really big formulas, you may even need to use { and } to scroll up and down. The d U (calc-unformatted-language) command altogether disables the use of operator notation in formulas. In this mode, the formula shown above would be displayed: sqrt(add(div(add(a, 1), b), pow(c, 2)))  These four modes differ only in display format, not in the format expected for algebraic entry. The standard Calc operators work in all four modes, and unformatted notation works in any language mode (except that Mathematica mode expects square brackets instead of parentheses). ### C, FORTRAN, and Pascal Modes The d C (calc-c-language) command selects the conventions of the C language for display and entry of formulas. This differs from the normal language mode in a variety of (mostly minor) ways. In particular, C language operators and operator precedences are used in place of Calc's usual ones. For example, a^b' means xor(a,b)' in C mode; a value raised to a power is written as a function call, pow(a,b)'. In C mode, vectors and matrices use curly braces instead of brackets. Octal and hexadecimal values are written with leading 0' or 0x' rather than using the #' symbol. Array subscripting is translated into subscr calls, so that a[i]' in C mode is the same as a_i' in normal mode. Assignments turn into the assign function, which Calc normally displays using the :=' symbol. The variables var-pi and var-e would be displayed pi' and e' in normal mode, but in C mode they are displayed as M_PI' and M_E', corresponding to the names of constants typically provided in the <math.h>' header. Functions whose names are different in C are translated automatically for entry and display purposes. For example, entering asin(x)' will push the formula arcsin(x)' onto the stack; this formula will be displayed as asin(x)' as long as C mode is in effect. The d P (calc-pascal-language) command selects Pascal conventions. Like C mode, Pascal mode interprets array brackets and uses a different table of operators. Hexadecimal numbers are entered and displayed with a preceding dollar sign. (Thus the regular meaning of2 during algebraic entry does not work in Pascal mode, though $(and$$, etc.) not followed by digits works the same as always.) No special provisions are made for other non-decimal numbers, vectors, and so on, since there is no universally accepted standard way of handling these in Pascal. The d F (calc-fortran-language) command selects FORTRAN conventions. Various function names are transformed into FORTRAN equivalents. Vectors are written as /1, 2, 3/', and may be entered this way or using square brackets. Since FORTRAN uses round parentheses for both function calls and array subscripts, Calc displays both in the same way; a(i)' is interpreted as a function call upon reading, and subscripts must be entered as subscr(a, i)'. Also, if the variable a has been declared to have type vector or matrix then a(i)' will be parsed as a subscript. (See section Declarations.) Usually it doesn't matter, though; if you enter the subscript expression a(i)' and Calc interprets it as a function call, you'll never know the difference unless you switch to another language mode or replace a with an actual vector (or unless a happens to be the name of a built-in function!). Underscores are allowed in variable and function names in all of these language modes. The underscore here is equivalent to the #' in normal mode, or to hyphens in the underlying Emacs Lisp variable names. FORTRAN and Pascal modes normally do not adjust the case of letters in formulas. Most built-in Calc names use lower-case letters. If you use a positive numeric prefix argument with d P or d F, these modes will use upper-case letters exclusively for display, and will convert to lower-case on input. With a negative prefix, these modes convert to lower-case for display and input. ### TeX Language Mode The d T (calc-tex-language) command selects the conventions of "math mode" in the TeX typesetting language, by Donald Knuth. Formulas are entered and displayed in TeX notation, as in \sin\left( a \over b \right)'. Math formulas are usually enclosed by $ $' signs in TeX; these should be omitted when interfacing with Calc. To Calc, the $' sign has the same meaning it always does in algebraic formulas (a reference to an existing entry on the stack).

Complex numbers are displayed as in 3 + 4i'. Fractions and quotients are written using \over; binomial coefficients are written with \choose. Interval forms are written with \ldots, and error forms are written with \pm. Absolute values are written as in |x + 1|', and the floor and ceiling functions are written with \lfloor, \rfloor, etc. The words \left and \right are ignored when reading formulas in TeX mode. Both inf and uinf are written as \infty; when read, \infty always translates to inf.

Function calls are written the usual way, with the function name followed by the arguments in parentheses. However, functions for which TeX has special names (like \sin) will use curly braces instead of parentheses for very simple arguments. During input, curly braces and parentheses work equally well for grouping, but when the document is formatted the curly braces will be invisible. Thus the printed result is sin 2x but @c{$\sin(2 + x)$} sin(2 + x).

Function and variable names not treated specially by TeX are simply written out as-is, which will cause them to come out in italic letters in the printed document. If you invoke d T with a positive numeric prefix argument, names of more than one character will instead be written \hbox{name}'. The \hbox{ }' notation is ignored during reading. If you use a negative prefix argument, such function names are written \name', and function names that begin with \ during reading have the \ removed. (Note that in this mode, long variable names are still written with \hbox. However, you can always make an actual variable name like \bar in any TeX mode.)

During reading, text of the form \matrix{ ... }' is replaced by [ ... ]'. The same also applies to \pmatrix and \bmatrix. The symbol &' is interpreted as a comma, and the symbols \cr' and \\' are interpreted as semicolons. During output, matrices are displayed in \matrix{ a & b \\ c & d}' format; you may need to edit this afterwards to change \matrix to \pmatrix or \\ to \cr.

Accents like \tilde and \bar translate into function calls internally (tilde(x)', bar(x)'). The \underline sequence is treated as an accent. The \vec accent corresponds to the function name Vec, because vec is the name of a built-in Calc function. The following table shows the accents in Calc, TeX, and eqn (described in the next section):

@begingroup @let@calcindexershow=@calcindexernoshow @let@calcindexersh=@calcindexernoshow @endgroup

Calc      TeX           eqn
----      --           ---
acute     \acute
bar       \bar          bar
breve     \breve
check     \check
dot       \dot          dot
dotdot    \ddot         dotdot
grave     \grave
hat       \hat          hat
Prime                   prime
tilde     \tilde        tilde
under     \underline    under
Vec       \vec          vec


The =>' (evaluates-to) operator appears as a \to symbol: {a \to b}'. TeX defines \to as an alias for \rightarrow. However, if the =>' is the top-level expression being formatted, a slightly different notation is used: \evalto a \to b'. The \evalto word is ignored by Calc's input routines, and is undefined in TeX. You will typically want to include one of the following definitions at the top of a TeX file that uses \evalto:

\def\evalto{}
\def\evalto#1\to{}


The first definition formats evaluates-to operators in the usual way. The second causes only the b part to appear in the printed document; the a part and the arrow are hidden. Another definition you may wish to use is \let\to=\Rightarrow' which causes \to to appear more like Calc's =>' symbol. See section The Evaluates-To Operator, for a discussion of evalto.

The complete set of TeX control sequences that are ignored during reading is:

\hbox  \mbox  \text  \left  \right
\displaystyle  \textstyle  \dsize  \tsize
\scriptstyle  \scriptscriptstyle  \ssize  \ssize
\rm  \bf  \it  \sl  \roman  \bold  \italic  \slanted
\cal  \mit  \Cal  \Bbb  \frak  \goth
\evalto


Note that, because these symbols are ignored, reading a TeX formula into Calc and writing it back out may lose spacing and font information.

Also, the "discretionary multiplication sign" \*' is read the same as *'.

Here are some examples of how various Calc formulas are formatted in TeX:

sin(a^2 / b_i)
\sin\left( {a^2 \over b_i} \right)


[(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
[3 + 4i, {3 \over 4}, 3 \pm 4, [3 \ldots \infty)]


[abs(a), abs(a / b), floor(a), ceil(a / b)]
[|a|, \left| a \over b \right|,
\lfloor a \rfloor, \left\lceil a \over b \right\rceil]


[sin(a), sin(2 a), sin(2 + a), sin(a / b)]
[\sin{a}, \sin{2 a}, \sin(2 + a),
\sin\left( {a \over b} \right)]


First with plain d T, then with C-u d T, then finally with C-u - d T (using the example definition \def\foo#1{\tilde F(#1)}':


[f(a), foo(bar), sin(pi)]
[f(a), foo(bar), \sin{\pi}]
[f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}]
[f(a), \foo{\hbox{bar}}, \sin{\pi}]


First with \def\evalto{}', then with \def\evalto#1\to{}':


2 + 3 => 5
\evalto 2 + 3 \to 5


First with standard \to, then with \let\to\Rightarrow':


[2 + 3 => 5, a / 2 => (b + c) / 2]
[{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}]


Matrices normally, then changing \matrix to \pmatrix:


[ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
\matrix{ {a \over b} & 0 \\ 0 & 2^{(x + 1)} }
\pmatrix{ {a \over b} & 0 \\ 0 & 2^{(x + 1)} }


### Eqn Language Mode

Eqn is another popular formatter for math formulas. It is designed for use with the TROFF text formatter, and comes standard with many versions of Unix. The d E (calc-eqn-language) command selects eqn notation.

The eqn language's main idiosyncrasy is that whitespace plays a significant part in the parsing of the language. For example, sqrt x+1 + y' treats x+1' as the argument of the sqrt operator. Eqn also understands more conventional grouping using curly braces: sqrt{x+1} + y'. Braces are required only when the argument contains spaces.

In Calc's eqn mode, however, curly braces are required to delimit arguments of operators like sqrt. The first of the above examples would treat only the x' as the argument of sqrt, and in fact sin x+1' would be interpreted as sin * x + 1', because sin is not a special operator in the eqn language. If you always surround the argument with curly braces, Calc will never misunderstand.

Calc also understands parentheses as grouping characters. Another peculiarity of eqn's syntax makes it advisable to separate words with spaces from any surrounding characters that aren't curly braces, so Calc writes sin ( x + y )' in eqn mode. (The spaces around sin are important to make eqn recognize that sin should be typeset in a roman font, and the spaces around x and y are a good idea just in case the eqn document has defined special meanings for these names, too.)

Powers and subscripts are written with the sub and sup operators, respectively. Note that the caret symbol ^' is treated the same as a space in eqn mode, as is the ~' symbol (these are used to introduce spaces of various widths into the typeset output of eqn).

As in TeX mode, Calc's formatter omits parentheses around the arguments of functions like ln and sin if they are "simple-looking"; in this case Calc surrounds the argument with braces, separated by a ~' from the function name: sin~{x}'.

Font change codes (like roman x') and positioning codes (like ~' and down n x') are ignored by the eqn reader. Also ignored are the words left, right, mark, and lineup. Quotation marks in eqn mode input are treated the same as curly braces: sqrt "1+x"' is equivalent to sqrt {1+x}'; this is only an approximation to the true meaning of quotes in eqn, but it is good enough for most uses.

Accent codes (x dot') are handled by treating them as function calls (dot(x)') internally. See section TeX Language Mode for a table of these accent functions. The prime accent is treated specially if it occurs on a variable or function name: f prime prime ( x prime )' is stored internally as f''(x')'. For example, taking the derivative of f(2 x)' with a d x will produce 2 f'(2 x)', which eqn mode will display as 2 f prime ( 2 x )'.

Assignments are written with the <-' (left-arrow) symbol, and evalto operators are written with ->' or evalto ... ->' (see section TeX Language Mode, for a discussion of this). The regular Calc symbols :=' and =>' are also recognized for these operators during reading.

Vectors in eqn mode use regular Calc square brackets, but matrices are formatted as matrix { ccol { a above b } ... }'. The words lcol and rcol are recognized as synonyms for ccol during input, and are generated instead of ccol if the matrix justification mode so specifies.

### Mathematica Language Mode

The d M (calc-mathematica-language) command selects the conventions of Mathematica, a powerful and popular mathematical tool from Wolfram Research, Inc. Notable differences in Mathematica mode are that the names of built-in functions are capitalized, and function calls use square brackets instead of parentheses. Thus the Calc formula sin(2 x)' is entered and displayed Sin[2 x]' in Mathematica mode.

Vectors and matrices use curly braces in Mathematica. Complex numbers are written 3 + 4 I'. The standard special constants in Calc are written Pi, E, I, GoldenRatio, EulerGamma, Infinity, ComplexInfinity, and Indeterminate in Mathematica mode. Non-decimal numbers are written, e.g., 16^^7fff'. Floating-point numbers in scientific notation are written 1.23*10.^3'. Subscripts use double square brackets: a[[i]]'.

### Maple Language Mode

The d W (calc-maple-language) command selects the conventions of Maple, another mathematical tool from the University of Waterloo.

Maple's language is much like C. Underscores are allowed in symbol names; square brackets are used for subscripts; explicit *'s for multiplications are required. Use either ^' or **' to denote powers.

Maple uses square brackets for lists and curly braces for sets. Calc interprets both notations as vectors, and displays vectors with square brackets. This means Maple sets will be converted to lists when they pass through Calc. As a special case, matrices are written as calls to the function matrix, given a list of lists as the argument, and can be read in this form or with all-capitals MATRIX.

The Maple interval notation 2 .. 3' has no surrounding brackets; Calc reads 2 .. 3' as the closed interval [2 .. 3]', and writes any kind of interval as 2 .. 3'. This means you cannot see the difference between an open and a closed interval while in Maple display mode.

Maple writes complex numbers as 3 + 4*I'. Its special constants are Pi, E, I, and infinity (all three of inf, uinf, and nan display as infinity). Floating-point numbers are written 1.23*10.^3'.

Among things not currently handled by Calc's Maple mode are the various quote symbols, procedures and functional operators, and inert (&') operators.

### Compositions

There are several composition functions which allow you to get displays in a variety of formats similar to those in Big language mode. Most of these functions do not evaluate to anything; they are placeholders which are left in symbolic form by Calc's evaluator but are recognized by Calc's display formatting routines.

Two of these, string and bstring, are described elsewhere. See section Strings. For example, string("ABC")' is displayed as ABC'. When viewed on the stack it will be indistinguishable from the variable ABC, but internally it will be stored as string([65, 66, 67])' and can still be manipulated this way; for example, the selection and vector commands j 1 v v j u would select the vector portion of this object and reverse the elements, then deselect to reveal a string whose characters had been reversed.

The composition functions do the same thing in all language modes (although their components will of course be formatted in the current language mode). The one exception is Unformatted mode (d U), which does not give the composition functions any special treatment. The functions are discussed here because of their relationship to the language modes.

#### Composition Basics

Compositions are generally formed by stacking formulas together horizontally or vertically in various ways. Those formulas are themselves compositions. TeX users will find this analogous to TeX's "boxes." Each multi-line composition has a baseline; horizontal compositions use the baselines to decide how formulas should be positioned relative to one another. For example, in the Big mode formula

          2
a + b
17 + ------
c


the second term of the sum is four lines tall and has line three as its baseline. Thus when the term is combined with 17, line three is placed on the same level as the baseline of 17.

Another important composition concept is precedence. This is an integer that represents the binding strength of various operators. For example, *' has higher precedence (195) than +' (180), which means that (a * b) + c' will be formatted without the parentheses, but a * (b + c)' will keep the parentheses.

The operator table used by normal and Big language modes has the following precedences:

_     1200   (subscripts)
%     1100   (as in n%)
-     1000   (as in -n)
!     1000   (as in !n)
mod    400
+/-    300
!!     210    (as in n!!)
!      210    (as in n!)
^      200
*      195    (or implicit multiplication)
/ % \  190
+ -    180    (as in a+b)
|      170
< =    160    (and other relations)
&&     110
||     100
? :     90
!!!     85
&&&     80
|||     75
:=      50
::      45
=>      40


The general rule is that if an operator with precedence n occurs as an argument to an operator with precedence m, then the argument is enclosed in parentheses if n < m. Top-level expressions and expressions which are function arguments, vector components, etc., are formatted with precedence zero (so that they normally never get additional parentheses).

For binary left-associative operators like +', the righthand argument is actually formatted with one-higher precedence than shown in the table. This makes sure (a + b) + c' omits the parentheses, but the unnatural form a + (b + c)' keeps its parentheses. Right-associative operators like ^' format the lefthand argument with one-higher precedence.

The cprec function formats an expression with an arbitrary precedence. For example, cprec(abc, 185)' will combine into sums and products as follows: 7 + abc', 7 (abc)' (because this cprec form has higher precedence than addition, but lower precedence than multiplication).

A final composition issue is line breaking. Calc uses two different strategies for "flat" and "non-flat" compositions. A non-flat composition is anything that appears on multiple lines (not counting line breaking). Examples would be matrices and Big mode powers and quotients. Non-flat compositions are displayed exactly as specified. If they come out wider than the current window, you must use horizontal scrolling (< and >) to view them.

Flat compositions, on the other hand, will be broken across several lines if they are too wide to fit the window. Certain points in a composition are noted internally as break points. Calc's general strategy is to fill each line as much as possible, then to move down to the next line starting at the first break point that didn't fit. However, the line breaker understands the hierarchical structure of formulas. It will not break an "inner" formula if it can use an earlier break point from an "outer" formula instead. For example, a vector of sums might be formatted as:

[ a + b + c, d + e + f,
g + h + i, j + k + l, m ]


If the m' can fit, then so, it seems, could the g'. But Calc prefers to break at the comma since the comma is part of a "more outer" formula. Calc would break at a plus sign only if it had to, say, if the very first sum in the vector had itself been too large to fit.

Of the composition functions described below, only choriz generates break points. The bstring function (see section Strings) also generates breakable items: A break point is added after every space (or group of spaces) except for spaces at the very beginning or end of the string.

Composition functions themselves count as levels in the formula hierarchy, so a choriz that is a component of a larger choriz will be less likely to be broken. As a special case, if a bstring occurs as a component of a choriz or choriz-like object (such as a vector or a list of arguments in a function call), then the break points in that bstring will be on the same level as the break points of the surrounding object.

#### Horizontal Compositions

The choriz function takes a vector of objects and composes them horizontally. For example, choriz([17, a b/c, d])' formats as 17a b / cd' in normal language mode, or as

  a b
17--d
c


in Big language mode. This is actually one case of the general function choriz(vec, sep, prec)', where either or both of sep and prec may be omitted. Prec gives the precedence to use when formatting each of the components of vec. The default precedence is the precedence from the surrounding environment.

Sep is a string (i.e., a vector of character codes as might be entered with " " notation) which should separate components of the composition. Also, if sep is given, the line breaker will allow lines to be broken after each occurrence of sep. If sep is omitted, the composition will not be breakable (unless any of its component compositions are breakable).

For example, 2 choriz([a, b c, d = e], " + ", 180)' is formatted as 2 a + b c + (d = e)'. To get the choriz to have precedence 180 "outwards" as well as "inwards," enclose it in a cprec form: 2 cprec(choriz(...), 180)' formats as 2 (a + b c + (d = e))'.

The baseline of a horizontal composition is the same as the baselines of the component compositions, which are all aligned.

#### Vertical Compositions

The cvert function makes a vertical composition. Each component of the vector is centered in a column. The baseline of the result is by default the top line of the resulting composition. For example, f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))' formats in Big mode as

f( a ,  2    )
bb   a  + 1
ccc     2
b


There are several special composition functions that work only as components of a vertical composition. The cbase function controls the baseline of the vertical composition; the baseline will be the same as the baseline of whatever component is enclosed in cbase. Thus f(cvert([a, cbase(bb), ccc]), cvert([a^2 + 1, cbase(b^2)]))' displays as

        2
a  + 1
a      2
f(bb ,   b   )
ccc


There are also ctbase and cbbase functions which make the baseline of the vertical composition equal to the top or bottom line (rather than the baseline) of that component. Thus cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) + cvert([cbbase(a / b)])' gives

        a
a       -
- + a + b
b   -
b


There should be only one cbase, ctbase, or cbbase function in a given vertical composition. These functions can also be written with no arguments: ctbase()' is a zero-height object which means the baseline is the top line of the following item, and cbbase()' means the baseline is the bottom line of the preceding item.

The crule function builds a "rule," or horizontal line, across a vertical composition. By itself crule()' uses -' characters to build the rule. You can specify any other character, e.g., crule("=")'. The argument must be a character code or vector of exactly one character code. It is repeated to match the width of the widest item in the stack. For example, a quotient with a thick line is cvert([a + 1, cbase(crule("=")), b^2])':

a + 1
=====
2
b


Finally, the functions clvert and crvert act exactly like cvert except that the items are left- or right-justified in the stack. Thus clvert([a, bb, ccc]) + crvert([a, bb, ccc])' gives:

a   +   a
bb     bb
ccc   ccc


Like choriz, the vertical compositions accept a second argument which gives the precedence to use when formatting the components. Vertical compositions do not support separator strings.

#### Other Compositions

The csup function builds a superscripted expression. For example, csup(a, b)' looks the same as a^b' does in Big language mode. This is essentially a horizontal composition of a' and b', where b' is shifted up so that its bottom line is one above the baseline.

Likewise, the csub function builds a subscripted expression. This shifts b' down so that its top line is one below the bottom line of a' (note that this is not quite analogous to csup). Other arrangements can be obtained by using choriz and cvert directly.

The cflat function formats its argument in "flat" mode, as obtained by d O', if the current language mode is normal or Big. It has no effect in other language modes. For example, a^(b/c)' is formatted by Big mode like csup(a, cflat(b/c))' to improve its readability.

The cspace function creates horizontal space. For example, cspace(4)' is effectively the same as string(" ")'. A second string (i.e., vector of characters) argument is repeated instead of the space character. For example, cspace(4, "ab")' looks like abababab'. If the second argument is not a string, it is formatted in the normal way and then several copies of that are composed together: cspace(4, a^2)' yields

 2 2 2 2
a a a a


If the number argument is zero, this is a zero-width object.

The cvspace function creates vertical space, or a vertical stack of copies of a certain string or formatted object. The baseline is the center line of the resulting stack. A numerical argument of zero will produce an object which contributes zero height if used in a vertical composition.

There are also ctspace and cbspace functions which create vertical space with the baseline the same as the baseline of the top or bottom copy, respectively, of the second argument. Thus cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)' displays as:

        a
-
a       b
-   a   a
b + - + -
a   b   b
-   a
b   -
b


The functions in this section are actual functions; they compose their arguments according to the current language and other display modes, then return a certain measurement of the composition as an integer.

The cwidth function measures the width, in characters, of a composition. For example, cwidth(a + b)' is 5, and cwidth(a / b)' is 5 in normal mode, 1 in Big mode, and 11 in TeX mode (for {a \over b}'). The argument may involve the composition functions described in this section.

The cheight function measures the height of a composition. This is the total number of lines in the argument's printed form.

The functions cascent and cdescent measure the amount of the height that is above (and including) the baseline, or below the baseline, respectively. Thus cascent(x) + cdescent(x)' always equals cheight(x)'. For a one-line formula like a + b', cascent returns 1 and cdescent returns 0. For a / b' in Big mode, cascent returns 2 and cdescent returns 1. The only formula for which cascent will return zero is cvspace(0)' or equivalents.

#### User-Defined Compositions

The Z C (calc-user-define-composition) command lets you define the display format for any algebraic function. You provide a formula containing a certain number of argument variables on the stack. Any time Calc formats a call to the specified function in the current language mode and with that number of arguments, Calc effectively replaces the function call with that formula with the arguments replaced.

Calc builds the default argument list by sorting all the variable names that appear in the formula into alphabetical order. You can edit this argument list before pressing RET if you wish. Any variables in the formula that do not appear in the argument list will be displayed literally; any arguments that do not appear in the formula will not affect the display at all.

You can define formats for built-in functions, for functions you have defined with Z F (see section Programming with Formulas), or for functions which have no definitions but are being used as purely syntactic objects. You can define different formats for each language mode, and for each number of arguments, using a succession of Z C commands. When Calc formats a function call, it first searches for a format defined for the current language mode (and number of arguments); if there is none, it uses the format defined for the Normal language mode. If neither format exists, Calc uses its built-in standard format for that function (usually just func(args)').

If you execute Z C with the number 0 on the stack instead of a formula, any defined formats for the function in the current language mode will be removed. The function will revert to its standard format.

For example, the default format for the binomial coefficient function choose(n, m)' in the Big language mode is

 n
( )
m


You might prefer the notation,

 C
n m


To define this notation, first make sure you are in Big mode, then put the formula

choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])


on the stack and type Z C. Answer the first prompt with choose. The second prompt will be the default argument list of (C m n)'. Edit this list to be (n m)' and press RET. Now, try it out: For example, turn simplification off with m O and enter choose(a,b) + choose(7,3)' as an algebraic entry.

 C  +  C
a b   7 3


As another example, let's define the usual notation for Stirling numbers of the first kind, stir1(n, m)'. This is just like the regular format for binomial coefficients but with square brackets instead of parentheses.

choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])


Now type Z C stir1 RET, edit the argument list to (n m)', and type RET.

The formula provided to Z C usually will involve composition functions, but it doesn't have to. Putting the formula a + b + c' onto the stack and typing Z C foo RET RET would define the function foo(x,y,z)' to display like x + y + z'. This "sum" will act exactly like a real sum for all formatting purposes (it will be parenthesized the same, and so on). However it will be computationally unrelated to a sum. For example, the formula 2 * foo(1, 2, 3)' will display as 2 (1 + 2 + 3)'. Operator precedences have caused the "sum" to be written in parentheses, but the arguments have not actually been summed. (Generally a display format like this would be undesirable, since it can easily be confused with a real sum.)

The special function eval can be used inside a Z C composition formula to cause all or part of the formula to be evaluated at display time. For example, if the formula is a + eval(b + c)', then foo(1, 2, 3)' will be displayed as 1 + 5'. Evaluation will use the default simplifications, regardless of the current simplification mode. There are also evalsimp and evalextsimp which simplify as if by a s and a e (respectively). Note that these "functions" operate only in the context of composition formulas (and also in rewrite rules, where they serve a similar purpose; see section Rewrite Rules). On the stack, a call to eval will be left in symbolic form.

It is not a good idea to use eval except as a last resort. It can cause the display of formulas to be extremely slow. For example, while eval(a + b)' might seem quite fast and simple, there are several situations where it could be slow. For example, a' and/or b' could be polar complex numbers, in which case doing the sum requires trigonometry. Or, a' could be the factorial fact(100)' which is unevaluated because you have typed m O; eval will evaluate it anyway to produce a large, unwieldy integer.

You can save your display formats permanently using the Z P command (see section Creating User Keys).

### Syntax Tables

Syntax tables do for input what compositions do for output: They allow you to teach custom notations to Calc's formula parser. Calc keeps a separate syntax table for each language mode.

(Note that the Calc "syntax tables" discussed here are completely unrelated to the syntax tables described in the Emacs manual.)

The Z S (calc-edit-user-syntax) command edits the syntax table for the current language mode. If you want your syntax to work in any language, define it in the normal language mode. Type M-# M-# to finish editing the syntax table, or M-# x to cancel the edit. The m m command saves all the syntax tables along with the other mode settings; see section General Mode Commands.

#### Syntax Table Basics

Parsing is the process of converting a raw string of characters, such as you would type in during algebraic entry, into a Calc formula. Calc's parser works in two stages. First, the input is broken down into tokens, such as words, numbers, and punctuation symbols like +', :=', and +/-'. Space between tokens is ignored (except when it serves to separate adjacent words). Next, the parser matches this string of tokens against various built-in syntactic patterns, such as "an expression followed by +' followed by another expression" or "a name followed by (', zero or more expressions separated by commas, and )'."

A syntax table is a list of user-defined syntax rules, which allow you to specify new patterns to define your own favorite input notations. Calc's parser always checks the syntax table for the current language mode, then the table for the normal language mode, before it uses its built-in rules to parse an algebraic formula you have entered. Each syntax rule should go on its own line; it consists of a pattern, a :=' symbol, and a Calc formula with an optional condition. (Syntax rules resemble algebraic rewrite rules, but the notation for patterns is completely different.)

A syntax pattern is a list of tokens, separated by spaces. Except for a few special symbols, tokens in syntax patterns are matched literally, from left to right. For example, the rule,

foo ( ) := 2+3


would cause Calc to parse the formula 4+foo()*5' as if it were 4+(2+3)*5'. Notice that the parentheses were written as two separate tokens in the rule. As a result, the rule works for both foo()' and foo ( )'. If we had written the rule as foo () := 2+3', then Calc would treat ()' as a single, indivisible token, so that foo( )' would not be recognized by the rule. (It would be parsed as a regular zero-argument function call instead.) In fact, this rule would also make trouble for the rest of Calc's parser: An unrelated formula like bar()' would now be tokenized into bar ()' instead of bar ( )', so that the standard parser for function calls would no longer recognize it!

While it is possible to make a token with a mixture of letters and punctuation symbols, this is not recommended. It is better to break it into several tokens, as we did with foo()' above.

The symbol #' in a syntax pattern matches any Calc expression. On the righthand side, the things that matched the #'s can be referred to as #1', #2', and so on (where #1' matches the leftmost #' in the pattern). For example, these rules match a user-defined function, prefix operator, infix operator, and postfix operator, respectively:

foo ( # ) := myfunc(#1)
foo # := myprefix(#1)
# foo # := myinfix(#1,#2)
# foo := mypostfix(#1)


Thus foo(3)' will parse as myfunc(3)', and 2+3 foo' will parse as mypostfix(2+3)'.

It is important to write the first two rules in the order shown, because Calc tries rules in order from first to last. If the pattern foo #' came first, it would match anything that could match the foo ( # )' rule, since an expression in parentheses is itself a valid expression. Thus the foo ( # )' rule would never get to match anything. Likewise, the last two rules must be written in the order shown or else 3 foo 4' will be parsed as mypostfix(3) * 4'. (Of course, the best way to avoid these ambiguities is not to use the same symbol in more than one way at the same time! In case you're not convinced, try the following exercise: How will the above rules parse the input foo(3,4)', if at all? Work it out for yourself, then try it in Calc and see.)

Calc is quite flexible about what sorts of patterns are allowed. The only rule is that every pattern must begin with a literal token (like foo' in the first two patterns above), or with a #' followed by a literal token (as in the last two patterns). After that, any mixture is allowed, although putting two #'s in a row will not be very useful since two expressions with nothing between them will be parsed as one expression that uses implicit multiplication.

As a more practical example, Maple uses the notation sum(a(i), i=1..10)' for sums, which Calc's Maple mode doesn't recognize at present. To handle this syntax, we simply add the rule,

sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)


to the Maple mode syntax table. As another example, C mode can't read assignment operators like ++' and *='. We can define these operators quite easily:

# *= # := muleq(#1,#2)
# ++ := postinc(#1)
++ # := preinc(#1)


To complete the job, we would use corresponding composition functions and Z C to cause these functions to display in their respective Maple and C notations. (Note that the C example ignores issues of operator precedence, which are discussed in the next section.)

You can enclose any token in quotes to prevent its usual interpretation in syntax patterns:

# ":=" # := becomes(#1,#2)


Quotes also allow you to include spaces in a token, although once again it is generally better to use two tokens than one token with an embedded space. To include an actual quotation mark in a quoted token, precede it with a backslash. (This also works to include backslashes in tokens.)

# "bad token" # "/\"\\" # := silly(#1,#2,#3)


This will parse 3 bad token 4 /"\ 5' to silly(3,4,5)'.

The token # has a predefined meaning in Calc's formula parser; it is not legal to use "#"' in a syntax rule. However, longer tokens that include the #' character are allowed. Also, while "$"' and "\""' are allowed as tokens, their presence in the syntax table will prevent those characters from working in their usual ways (referring to stack entries and quoting strings, respectively). Finally, the notation %%' anywhere in a syntax table causes the rest of the line to be ignored as a comment. #### Precedence Different operators are generally assigned different precedences. By default, an operator defined by a rule like # foo # := foo(#1,#2)  will have an extremely low precedence, so that 2*3+4 foo 5 == 6' will be parsed as (2*3+4) foo (5 == 6)'. To change the precedence of an operator, use the notation #/p' in place of #', where p is an integer precedence level. For example, 185 lies between the precedences for +' and *', so if we change this rule to #/185 foo #/186 := foo(#1,#2)  then 2+3 foo 4*5' will be parsed as 2+(3 foo (4*5))'. Also, because we've given the righthand expression slightly higher precedence, our new operator will be left-associative: 1 foo 2 foo 3' will be parsed as (1 foo 2) foo 3'. By raising the precedence of the lefthand expression instead, we can create a right-associative operator. See section Composition Basics, for a table of precedences of the standard Calc operators. For the precedences of operators in other language modes, look in the Calc source file calc-lang.el'. #### Advanced Syntax Patterns To match a function with a variable number of arguments, you could write foo ( # ) := myfunc(#1) foo ( # , # ) := myfunc(#1,#2) foo ( # , # , # ) := myfunc(#1,#2,#3)  but this isn't very elegant. To match variable numbers of items, Calc uses some notations inspired regular expressions and the "extended BNF" style used by some language designers. foo ( { # }*, ) := apply(myfunc,#1)  The token {' introduces a repeated or optional portion. One of the three tokens }*', }+', or }?' ends the portion. These will match zero or more, one or more, or zero or one copies of the enclosed pattern, respectively. In addition, }*' and }+' can be followed by a separator token (with no space in between, as shown above). Thus { # }*,' matches nothing, or one expression, or several expressions separated by commas. A complete { ... }' item matches as a vector of the items that matched inside it. For example, the above rule will match foo(1,2,3)' to get apply(myfunc,[1,2,3])'. The Calc apply function takes a function name and a vector of arguments and builds a call to the function with those arguments, so the net result is the formula myfunc(1,2,3)'. If the body of a { ... }' contains several #'s (or nested { ... }' constructs), then the items will be strung together into the resulting vector. If the body does not contain anything but literal tokens, the result will always be an empty vector. foo ( { # , # }+, ) := bar(#1) foo ( { { # }*, }*; ) := matrix(#1)  will parse foo(1,2,3,4)' as bar([1,2,3,4])', and foo(1,2;3,4)' as matrix([[1,2],[3,4]])'. Also, after some thought it's easy to see how this pair of rules will parse foo(1,2,3)' as matrix([[1,2,3]])', since the first rule will only match an even number of arguments. The rule foo ( # { , # , # }? ) := bar(#1,#2)  will parse foo(2,3,4)' as bar(2,[3,4])', and foo(2)' as bar(2,[])'. The notation { ... }?.' (note the trailing period) works just the same as regular { ... }?', except that it does not count as an argument; the following two rules are equivalent: foo ( # , { also }? # ) := bar(#1,#3) foo ( # , { also }?. # ) := bar(#1,#2)  Note that in the first case the optional text counts as #2', which will always be an empty vector, but in the second case no empty vector is produced. Another variant is { ... }?$', which means the body is optional only at the end of the input formula. All built-in syntax rules in Calc use this for closing delimiters, so that during algebraic entry you can type [sqrt(2), sqrt(3 RET, omitting the closing parenthesis and bracket. Calc does this automatically for trailing )', ]', and >' tokens in syntax rules, but you can use { ... }?$' explicitly to get this effect with any token (such as "}"' or end'). Like { ... }?.', this notation does not count as an argument. Conversely, you can use quotes, as in ")"', to prevent a closing-delimiter token from being automatically treated as optional. Calc's parser does not have full backtracking, which means some patterns will not work as you might expect: foo ( { # , }? # , # ) := bar(#1,#2,#3)  Here we are trying to make the first argument optional, so that foo(2,3)' parses as bar([],2,3)'. Unfortunately, Calc first tries to match 2,' against the optional part of the pattern, finds a match, and so goes ahead to match the rest of the pattern. Later on it will fail to match the second comma, but it doesn't know how to go back and try the other alternative at that point. One way to get around this would be to use two rules: foo ( # , # , # ) := bar([#1],#2,#3) foo ( # , # ) := bar([],#1,#2)  More precisely, when Calc wants to match an optional or repeated part of a pattern, it scans forward attempting to match that part. If it reaches the end of the optional part without failing, it "finalizes" its choice and proceeds. If it fails, though, it backs up and tries the other alternative. Thus Calc has "partial" backtracking. A fully backtracking parser would go on to make sure the rest of the pattern matched before finalizing the choice. #### Conditional Syntax Rules It is possible to attach a condition to a syntax rule. For example, the rules foo ( # ) := ifoo(#1) :: integer(#1) foo ( # ) := gfoo(#1)  will parse foo(3)' as ifoo(3)', but will parse foo(3.5)' and foo(x)' as calls to gfoo. Any number of conditions may be attached; all must be true for the rule to succeed. A condition is "true" if it evaluates to a nonzero number. See section Logical Operations, for a list of Calc functions like integer that perform logical tests. The exact sequence of events is as follows: When Calc tries a rule, it first matches the pattern as usual. It then substitutes #1', #2', etc., in the conditions, if any. Next, the conditions are simplified and evaluated in order from left to right, as if by the a s algebra command (see section Simplifying Formulas). Each result is true if it is a nonzero number, or an expression that can be proven to be nonzero (see section Declarations). If the results of all conditions are true, the expression (such as ifoo(#1)') has its #'s substituted, and that is the result of the parse. If the result of any condition is false, Calc goes on to try the next rule in the syntax table. Syntax rules also support let conditions, which operate in exactly the same way as they do in algebraic rewrite rules. See section Other Features of Rewrite Rules, for details. A let condition is always true, but as a side effect it defines a variable which can be used in later conditions, and also in the expression after the :=' sign: foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)  The dnumint function tests if a value is numerically an integer, i.e., either a true integer or an integer-valued float. This rule will parse foo with a half-integer argument, like foo(3.5)', to a call like hifoo(4.)'. The lefthand side of a syntax rule let must be a simple variable, not the arbitrary pattern that is allowed in rewrite rules. The matches function is also treated specially in syntax rule conditions (again, in the same way as in rewrite rules). See section Matching Commands. If the matching pattern contains meta-variables, then those meta-variables may be used in later conditions and in the result expression. The arguments to matches are not evaluated in this situation. sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])  This is another way to implement the Maple mode sum notation. In this approach, we allow #2' to equal the whole expression i=1..10'. Then, we use matches to break it apart into its components. If the expression turns out not to match the pattern, the syntax rule will fail. Note that Z S always uses Calc's normal language mode for editing expressions in syntax rules, so we must use regular Calc notation for the interval [b..c]' that will correspond to the Maple mode interval 1..10'. ## The Modes Variable The m g (calc-get-modes) command pushes onto the stack a vector of numbers that describes the various mode settings that are in effect. With a numeric prefix argument, it pushes only the nth mode, i.e., the nth element of this vector. Keyboard macros can use the m g command to modify their behavior based on the current mode settings. The modes vector is also available in the special variable Modes. In other words, m g is like s r Modes RET. It will not work to store into this variable; in fact, if you do, Modes will cease to track the current modes. (The m g command will continue to work, however.) In general, each number in this vector is suitable as a numeric prefix argument to the associated mode-setting command. (Recall that the ~ key takes a number from the stack and gives it as a numeric prefix to the next command.) The elements of the modes vector are as follows: 1. Current precision. Default is 12; associated command is p. 2. Binary word size. Default is 32; associated command is b w. 3. Stack size (not counting the value about to be pushed by m g). This is zero if m g is executed with an empty stack. 4. Number radix. Default is 10; command is d r. 5. Floating-point format. This is the number of digits, plus the constant 0 for normal notation, 10000 for scientific notation, 20000 for engineering notation, or 30000 for fixed-point notation. These codes are acceptable as prefix arguments to the d n command, but note that this may lose information: For example, d s and C-u 12 d s have similar (but not quite identical) effects if the current precision is 12, but they both produce a code of 10012, which will be treated by d n as C-u 12 d s. If the precision then changes, the float format will still be frozen at 12 significant figures. 6. Angular mode. Default is 1 (degrees). Other values are 2 (radians) and 3 (HMS). The m d command accepts these prefixes. 7. Symbolic mode. Value is 0 or 1; default is 0. Command is m s. 8. Fraction mode. Value is 0 or 1; default is 0. Command is m f. 9. Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0. Command is m p. 10. Matrix/scalar mode. Default value is -1. Value is 0 for scalar mode, -2 for matrix mode, or N for @c{$N\times N\$} NxN matrix mode. Command is m v.
11. Simplification mode. Default is 1. Value is -1 for off (m O), 0 for m N, 2 for m B, 3 for m A, 4 for m E, or 5 for m U. The m D command accepts these prefixes.
12. Infinite mode. Default is -1 (off). Value is 1 if the mode is on, or 0 if the mode is on with positive zeros. Command is m i.

For example, the sequence M-1 m g RET 2 + ~ p increases the precision by two, leaving a copy of the old precision on the stack. Later, ~ p will restore the original precision using that stack value. (This sequence might be especially useful inside a keyboard macro.)

As another example, M-3 m g 1 - ~ DEL deletes all but the oldest (bottommost) stack entry.

Yet another example: The HP-48 "round" command rounds a number to the current displayed precision. You could roughly emulate this in Calc with the sequence M-5 m g 10000 % ~ c c. (This would not work for fixed-point mode, but it wouldn't be hard to do a full emulation with the help of the Z [ and Z ] programming commands. See section Conditionals in Keyboard Macros.)

## The Calc Mode Line

This section is a summary of all symbols that can appear on the Calc mode line, the highlighted bar that appears under the Calc stack window (or under an editing window in Embedded Mode).

The basic mode line format is:

--%%-Calc: 12 Deg other modes       (Calculator)


The %%' is the Emacs symbol for "read-only"; it shows that regular Emacs commands are not allowed to edit the stack buffer as if it were text.

The word Calc:' changes to CalcEmbed:' if Embedded Mode is enabled. The words after this describe the various Calc modes that are in effect.

The first mode is always the current precision, an integer. The second mode is always the angular mode, either Deg, Rad, or Hms.

Here is a complete list of the remaining symbols that can appear on the mode line:

Alg
Algebraic mode (m a; see section Algebraic Entry).
Alg[(
Incomplete algebraic mode (C-u m a).
Alg*
Total algebraic mode (m t).
Symb
Symbolic mode (m s; see section Symbolic Mode).
Matrix
Matrix mode (m v; see section Matrix and Scalar Modes).
Matrixn
Dimensioned matrix mode (C-u n m v).
Scalar
Scalar mode (m v; see section Matrix and Scalar Modes).
Polar
Polar complex mode (m p; see section Polar Mode).
Frac
Fraction mode (m f; see section Fraction Mode).
Inf
Infinite mode (m i; see section Infinite Mode).
+Inf
Positive infinite mode (C-u 0 m i).
NoSimp
Default simplifications off (m O; see section Simplification Modes).
NumSimp
Default simplifications for numeric arguments only (m N).
BinSimpw
Binary-integer simplification mode; word size w (m B, b w).
AlgSimp
Algebraic simplification mode (m A).
ExtSimp
Extended algebraic simplification mode (m E).
UnitSimp
Units simplification mode (m U).
Bin
Oct
Current radix is 8 (d 8).
Hex
Current radix is 16 (d 6).
Radixn
Current radix is n (d r).
Zero
Big
Big language mode (d B; see section Normal Language Modes).
Flat
One-line normal language mode (d O).
Unform
Unformatted language mode (d U).
C
C language mode (d C; see section C, FORTRAN, and Pascal Modes).
Pascal
Pascal language mode (d P).
Fortran
FORTRAN language mode (d F).
TeX
TeX language mode (d T; see section TeX Language Mode).
Eqn
Eqn language mode (d E; see section Eqn Language Mode).
Math
Mathematica language mode (d M; see section Mathematica Language Mode).
Maple
Maple language mode (d W; see section Maple Language Mode).
Normn
Normal float mode with n digits (d n; see section Float Formats).
Fixn
Fixed point mode with n digits after the point (d f).
Sci
Scientific notation mode (d s).
Scin
Scientific notation with n digits (d s).
Eng
Engineering notation mode (d e).
Engn
Engineering notation with n digits (d e).
Leftn
Left-justified display indented by n (d <; see section Justification).
Right
Right-justified display (d >).
Rightn
Right-justified display with width n (d >).
Center
Centered display (d =).
Centern
Centered display with center column n (d =).
Widn
Line breaking with width n (d b; see section Normal Language Modes).
Wide
No line breaking (d b).
Break
Selections show deep structure (j b; see section Making Selections).
Save
Record modes in ~/.emacs' (m R; see section General Mode Commands).
Local
Record modes in Embedded buffer (m R).
LocEdit
Record modes as editing-only in Embedded buffer (m R).
LocPerm
Record modes as permanent-only in Embedded buffer (m R).
Global
Record modes as global in Embedded buffer (m R).
Manual
Automatic recomputation turned off (m C; see section Automatic Recomputation).
Graph
GNUPLOT process is alive in background (see section Graphics).
Sel
Top-of-stack has a selection (Embedded only; see section Making Selections).
Dirty
The stack display may not be up-to-date (see section Display Modes).
Inv
"Inverse" prefix was pressed (I; see section Inverse and Hyperbolic Flags).
Hyp
"Hyperbolic" prefix was pressed (H).
Keep
"Keep-arguments" prefix was pressed (K).
Narrow
Stack is truncated (d t; see section Truncating the Stack).

In addition, the symbols Active and ~Active can appear as minor modes on an Embedded buffer's mode line. See section Embedded Mode.