Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online


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.)

Arithmetic Precedence

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:

  1. Exponentiation.
  2. Multiplication and Division.
  3. Addition and Subtraction
  4. 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:

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

   ---  as 1/(a+b),   not as     1/a+b

x2r 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.