There is a 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.)
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.)
By convention, formulas are evaluated in the following sequence:
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:
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.
Conventions
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.
Arithmetic Precedence
The phrase arithmetic precedence refers to the sequence in which
formulas are evaluated.
Sequence of Operations
Exponentiation.
Multiplication and Division.
Addition and Subtraction
In the case of operations of the same level of precedence,
evaluation proceeds from left to right.
Expressions in parentheses are evaluated first.
Let's illustrate these rules by some examples:
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)
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
x2r as x^(2r), not as x^2r