"Though this be madness, yet there is method in 't." - Shakespeare

 

A HYPERGUIDE: RELATIONSHIPS AMONG TOPICS

 

This page demonstrates some features of our proposed "hyperguide" to math. The main focus will be to highlight patterns within the curriculum.

We hope that organizing material in a manner which emphasizes the big picture will make review or assimilation of topics more manageable.

The main point is that broader topics in pre-calculus algebra fit into a basic pattern of "do it again" and "un-do it", and the calculus curriculum

follows a similar pattern after another ingredient is included. Many of the fundamental formulas of algebra and calculus may be understood in

terms of the interactions of each operation obtained from these procedures with others. The streams of geometry and trigonometry begins

in parallel to the algebraic world, until calculus ultimately unifies them through the exponential function in the complex number plane.

Links to and from historical development and scientific motivations, and extensions to linear algebra and multivariable calculus, and

more advanced topics are being developed elsewhere.
 

ALGEBRA: THE NUMBER LINE

ADDITION UNDO IT SUBTRACTION
DO IT AGAIN DO IT AGAIN
MULTIPLICATION UNDO IT DIVISION
DO IT AGAIN DO IT AGAIN
POWER AND EXPONENTIATION  UNDO IT ROOT AND LOGARITHM

 
 

PLANE GEOMETRY AND TRIGONOMETRY: THE NUMBER PLANE

COMPLEX ADDITION UNDO IT COMPLEX SUBTRACTION
DO IT AGAIN DO IT AGAIN
COMPLEX MULTIPLICATION UNDO IT COMPLEX DIVISION
DO IT AGAIN DO IT AGAIN
COMPLEX POWER AND EXPONENTIATION UNDO IT COMPLEX ROOT AND LOGARITHM
PLANE TO LINE
CIRCULAR/POLAR  TO  RECTANGULAR
r exp (it) = r cos t + i r sin t
 LINE TO PLANE
RECTANGULAR TO CIRCULAR/POLAR
TRIGONOMETRIC/CIRCULAR UNDO IT ARC-TRIGONOMETRIC/CIRCULAR

 
 
 

CALCULUS: THE APPROXIMATED VALUES ("LIMITS") DEFINE CUMULATIVE AND INSTANTANEOUS CHANGE

AND UNIFY ALGEBRA AND GEOMETRY THROUGH THE COMPLEX EXPONENTIAL. POWER FUNCTIONS AND

THE BINOMIAL FORMULA EXPAND TO ALL REAL EXPONENTS.

 

SUMMATION/INTEGRATION UNDO IT DIFFERENCING/DIFFERENTIATION
DO IT AGAIN DO IT AGAIN
MULTIPLE INTEGRALS UNDO IT HIGHER DERIVATIVES

 
 
 
 
 


 
 


RELATIONSHIPS INVOLVING ADDITION

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Addition, m+n, sum Subtraction, x=y-n solves x+n=y, difference Natural or whole numbers, solving requires and definesexpansion to negative. Expand to rational, real, complex, matrix, function, etc. 
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
Multiplication, x+x+...+x=nx See other operations (Multiplication, Power, Exponential, etc.) Shifting,shifted functions, f(x)=x+n,  f(x+n)
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Shifting/translation Combined quantity or effect, shift x+y=y+x, (x+y)+z=x+(y+z), etc.

 
 
 
 

RELATIONSHIPS INVOLVING MULTIPLICATION

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Multiplication, mn, m*n, product Division, x=y/n solves nx=y, quotient Natural or whole numbers, solving requires and defines expansion to rational. Expand to real, complex, function, etc.
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
Power/Exponential xx...x=xn x(y+z)=xy+xz, x(y-z)=xy-xz, (see alsoPower, Exponential, etc.) Linear, scaling, scaled functions, f(x)=nx, f(nx)
Linear behavior/models f(x)=mx+b
Polynomials p(x)=a0+ a1x +...+ anxn
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Scaling, zoom, stretching Repeated additive combination or effect, scaling xy=yx (for "numbers"), (xy)z=x(yz)

 
 
 

RELATIONSHIPS INVOLVING SUBTRACTION

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Subtraction Natural or whole numbers.Expand to integers, rational, real, complex, matrix, function, etc. 
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
Division
VISUALIZATIONS INTERPRETATIONS PROPERTIES

 
 
 

RELATIONSHIPS INVOLVING DIVISION

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Division Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc. 
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
Logarithm Inverse linear/scaling f(x)=x/n, f(x/n) n != 0
Rational functions
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Whole part of x/y is the number of times you can subtract y from x with a result greater than 0.

 
 
 

RELATIONSHIPS INVOLVING EXPONENTIALS

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Exponential Logarithm Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc.
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
with addition:            ax+y= ax ay
with multiplication:  axy = (ax)y		 Exponential function f(x)=ax
Exponential behavior/models f(x)=Cax
VISUALIZATIONS INTERPRETATIONS PROPERTIES

 
 
 

RELATIONSHIPS INVOLVING LOGARITHMS

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Logarithm Exponential Natural or whole numbers,Expand torational, real, complex, matrix, function, etc.
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Whole part of logy x is the number of times you can divide y from x with a result greater than 1.

 
 
 

RELATIONSHIPS INVOLVING POWERS

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Power Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc.
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
with addition:    (x+y)n = xn+nxn-1y+...+yn
with multiplication: (xy)n = xnyn
VISUALIZATIONS INTERPRETATIONS PROPERTIES

 
 
 

RELATIONSHIPS INVOLVING ROOTS

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Root Power Natural or whole numbers,Expand to integer, rational, real, complex, matrix, function, etc.
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
VISUALIZATIONS INTERPRETATIONS PROPERTIES

 
 
 

RELATIONSHIPS INVOLVING DIFFERENTIATION

OPERATION  UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Differentiation, f'(x), df/dx, derivative Integration Differentiable functions, expand to distributions
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
Higher derivatives f(n)(x), (d/dx)nf with addition:             (f + g)' = f' + g'
with subtraction:        (f - g)' = f' - g'

with multiplication:     (cf)'=cf', (fg)'=f'g+fg' 
(product rule)            (see integration by parts)
with division:              (f/g)'=(gf'-fg')/g2
(quotient rule)

with composition:        (f o g)'(x)=f'(g(x))g'(x)
(chain rule)                     (see substitution)
with inversion:            f-1 '(x)=1/f'(f-1(x))
 

with power:                 d/dx(xn/n!)=xn-1/ (n-1)!
with exponential:         d/dx(ex)=ex
with circular functions:
                            cos'(x)=-sin(x),sin'(x)=cos(x)
 
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Slope of tangent line Recover instantaneous change from cumulative change, e.g., speedometer from tripmeter

 
 
 

RELATIONSHIPS INVOLVING INTEGRATION

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Integration, integral Differentiation Integrable functions, distributions
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
Multiple integrals with addition:             I(f+g)= I(f)+ I(g)
with subtraction:        I(f-g)= I(f)  - I(g)

with multiplication:   I(cf)=cI(f),I(f'g)=fg-I(fg')
(integration by parts)  (see product rule)

with composition:      I(f(u(x)u'(x))=
(substitution)                    (see chain rule)
 

with power:                 I(xn/n!)=xn+1/ (n+1)!
with exponential:         I(ex)=ex
with circular functions:
                      I(cos(x))=sin(x),I(sin(x))=-cos(x)
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Area between graph and independent variable axis Recover cumulative change from instantaneous change, e.g., tripmeter from speedometer

 
 
 

RELATIONSHIPS INVOLVING TRIGONOMETRIC/CIRCULAR FUNCTIONS

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Horizontal and vertical projection: cosine,sine arccosine, arcsine Angles, Arclength
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
with addition: Trigonometric Addition Formulas
 

with subtraction:
 

with multiplication: cos (2x)=cos^2 x - sin^2 x
                             sin (2x)=2 sin x cos x

cosx cos y = 1/2 (cos (x+y) + cos (x-y))
sin x cos y = 1/2 (sin (x+y) - sin (x-y))
sin nx sin mx =1/2 (cos (x-y) - cos (x+y))

with division:

 tangent, cotangent, secant, cosecant 
VISUALIZATIONS INTERPRETATIONS PROPERTIES
Relationship between uniform circular motion
and the graphs of natural cosine and sine 		 cos (x) = cos (-x), cos (x+2 \pi)= cos (x)
sin (x) = -sin (-x), sin (x+2 \pi)= sin (x)
sin (x) = cos (x- {\pi \over 2})

 
 
 
 

RELATIONSHIPS INVOLVING ARC-TRIGONOMETRIC/CIRCULAR FUNCTIONS

OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
Angles from horizontal or vertical projections, arccosine, arcsine Horizontal and vertical projections  (trigonometric/circular)
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
with addition:

with subtraction:

with multiplication:

with division:
 

 arctangent, arccotangent, arcsecant, arccosecant
VISUALIZATIONS INTERPRETATIONS PROPERTIES

 
 
 
 
 
 
OPERATION UNDO IT (INVERT/SOLVE) OPERATES ON/TECHNIQUES
DO IT AGAIN (ITERATE) INTERACTION WITH EXISTING OPERATIONS RELATED FUNCTIONS
VISUALIZATIONS INTERPRETATIONS PROPERTIES

 
 
 
 

MORE TO COME INCLUDING

DIFFERENCING            SUMMATION

LIMIT                            VARIATION

LINEAR TRANSFORMATIONS

Warnings/Pitfalls, Alternate notation and terminology