In the remainder of this page, and on other pages connected with this
course, you will frequently see words typeset in
red. These are words that make up the vocabulary of our
subject. It is essential that you understand exactly what they mean.
They are usually defined here when they first occur, but if you don't
understand the definition look up the word in the textbook (use the
index) or in any dictionary. Don't skip over a word thinking that you
will figure it out later, or it's not really necessary for
understanding the subject. There is virtually no redundancy in these
pages! (This is in pronounced contrast, for example, to our textbook.)
There is a glossary which provides links to
the places where key terms are defined.
Some things are the way they are in mathematics because there is no
way they could be meaningfully otherwise. For example, even though it
may appear mysterious, the product of two negative real numbers is
positive because assuming or decreeing otherwise would cause us to run
quickly into contradictions. We will discover many such facts as we
work through the course.
By contrast, a (mathematical)
convention is an agreement people have
made (usually all over the world) to do things a certain way. There
is no mathematical reason for the convention, employing it is just a
matter of convenience. We will adopt all the usual conventions, and
while you are free to do things differently in your own work, there is
usually no reason to fight the conventions. It is, however, crucial
that you understand them! (Well known examples for conventions
include writing from left to right, using 10 as the base of our number
system, or driving on the right side of the street.)
The phrase arithmetic precedence refers to the sequence in which
formulas are evaluated.
When entering formulas into WeBWorK (or a calculator, or a
computer program) they are interpreted according to certain
conventions. Not fully appreciating these conventions appears to
be the largest obstacle to using WeBWorK correctly and hassle
free.
To enter a formula into WeBWorK use the following symbols:
Note: Students frequently use words like plussing,
minussing, or timesing. These words are juvenile.
Using them in university mathematics is like referring to your parents
as your mommy and daddy during a job interview. It
is unfortunate that they are taught and used in primary and secondary
schools. Make a habit of using the proper words defined, introduced,
and used in this class.
A missing operator means multiplication
By convention, when an operator is omitted it means multiplication.
For example, 3a means 3*a and 3(4+2) equals 18.
xy means x*y.
(Actually there is an exception to this rule in the form of
mixed numbers
which for our purpose are mostly useless and should be avoided.)
Sequence of Operations
By convention, formulas are evaluated in the following sequence:
- Exponentiation.
- Multiplication and Division.
- Addition and Subtraction
- In the case of operations of the same level of precedence,
evaluation proceeds from left to right.
Actually, in the above list there should be an item 0 preceding all
the others: standard functions, such as logarithms, trigonometric
functions, etc. However, we will
not use such functions in Math 1010. (But you will study them in
great detail in Math 1030, 1050 and 1060, and in Calculus.)
If these conventions were absolute we would be severely stifled,
To prevent this calamity, the conventions can be modified by the use of parentheses:
- Expressions in parentheses are evaluated first.
Let's illustrate these rules by some examples:
- 2+3*4 = 2+12 = 14. Note that the
multiplication is carried out before the addition.
- (2+3)*4 = 5*4 = 20. You use parentheses to
specify that the addition be carried out first.
- 10 - 4 - 2 = 6 - 2 = 4. You carry out the
subtractions working from left to right.
- 10 - (4-2) = 10-2 = 8. You use parentheses
to carry out the rightmost subtraction first, which alters the
answer.
- 12/3+3 = 4+3 = 7. Division comes
first.
- 12/(3+3) = 12/6 = 2. The parentheses
indicate that you want the addition to be carried out first.
- 2^3*3 = 2*2*2*3 = 24. The power is evaluated
first, and the result is multiplied with 3.
- 2^(3*3) = 2^9 = 2*2*2*2*2*2*2*2*2 =512. The
parentheses indicate that the exponent is the product 3*3=9.
- 12/2*3 = 6*3 = 18. Multiplication and
division have the same precedence, and we work from left to right.
- 12/2/3 = 6/3 = 2. Again, we work from left to right.
- 18/3^2 = 18/9=2. The power has a higher
priority than the division, and is evaluated first. Note that 3^2
means 3*3, but if we replace 3^2 with 3*3 we get a different result:
18/3*3 = 6*3 = 18.
- (18/3)^2 = 6^2 = 36. The parentheses indicate
that the base equals 18/3=6.
There is a good chance that you will get confused about precedence
when entering numbers into WeBWorK. Luckily it does not hurt to
use parentheses when they are not needed. In the above examples,
instead of the expression without parentheses you could have also
entered:
2+3*4 = 2+(3*4)
10-4-2 = (10-4)-2
12/3+3 = (12/3)+3
2^3+3 = (2^3)+3
12/2*3 = (12/2)*3
12/2/3 = (12/2)/3
18/3^2 = 18/(3^2)
- In case of doubt, use parentheses!
The rules illustrated above also apply to formulas involving variables (instead of just numbers).
For example, enter
1
--- as 1/(a+b), not as 1/a+b
a+b
or
x^{2r} as x^(2r), not as x^2r
Parentheses can be nested, i.e.,
pairs of matching parentheses can be contained within other pairs of
parentheses. For example,
12-(6-(4-2)) = 12 - (6-2) = 12 - 4 = 8.
To evaluate formulas involving nested pairs of parentheses you
start with the innermost pairs and work your way out.