# Mathematics 1010 online

## Vocabulary

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.

## 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.

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.

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.
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:

• 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
```
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.