1. Number Theory and the Real Number System
CHAPTER 5
Number Theory and the Real Number System

2. The Integers; Order of Operations
5.2
The Integers; Order of Operations

3. Objectives
Define the integers.
Graph integers on a number line.
Use symbols < and >.
Find the absolute value of an integer.
Perform operations with integers.
Use the order of operations agreement.

4. Define the Integers
The set consisting of the natural numbers, 0, and the negatives of the natural numbers is called the set of integers.
Notice the term positive integers is another name for the natural numbers. The positive integers can be written in two ways:
Use a “+” sign. For example, +4 is “positive four”.
Do not write any sign. For example, 4 is also “positive four”.

5. The Number Line
The number line is a graph we use to visualize the set of integers, as well as sets of other numbers. Notice, zero is neither positive nor negative.

6. Example: Graphing Integers on a Number Line3 4
Solution: Place a dot at the correct location for each integer.

7. Use the Symbols < and >
Looking at the graph, 4 and 1 are graphed below. Observe that 4 is to the left of 1 on the number line. This means that -4 is less than -1. Also observe that 1 is to the right of 4 on the number line. This means that 1 is greater then 4.

8. Use the Symbols < and >
The symbols < and > are called inequality symbols. These symbols always point to the lesser of the two real numbers when the inequality statement is true.

9. Example: Using the Symbols < and >
Insert either < or > in the shaded area between the integers to make each statement true:
4 3
1 5
5 2
0 3

10. Example: Using the Symbols < and > continued
4 < 3 (negative 4 is less than 3) because 4 is to the left of 3 on the number line.
1 > 5 (negative 1 is greater than negative 5) because 1 is to the right of 5 on the number line.
5 < 2 ( negative 5 is less than negative 2) because 5 is to the left of 2 on the number line.
0 > 3 (zero is greater than negative 3) because 0 is to the right of 3 on the number line.

11. Use the Symbols < and >
The symbols < and > may be combined with an equal sign, as shown in the following table:

12. Absolute Value
The absolute value of an integer a, denoted by |a|, is the distance from 0 to a on the number line. Because absolute value describes a distance, it is never negative.

13. Example: Finding Absolute Value
Find the absolute value:
|3| b. |5| c. |0|
Solution:
| 3| = 3 because 3 is 3 units away from 0.
|5| = 5 because 5 is 5 units away from 0.
|0| = 0 because 0 is 0 units away from itself.

14. Addition of Integers Rule If the integers have the same sign, Examples
Add their absolute values.
The sign of the sum is the
same sign of the two numbers.
If the integers have different signs,
Subtract the smaller absolute
value from the larger absolute
value.
The sign of the sum is the same as the sign of the number with the larger absolute value.
Examples

15. Study Tip
A good analogy for adding integers is temperatures above and below zero on the thermometer. Think of a thermometer as a number line standing straight up. For example,

16. Additive Inverses
Additive inverses have the same absolute value, but lie on opposite sides of zero on the number line.
When we add additive inverses, the sum is equal to zero.
For example:
18 + (18) = 0
(7) + 7 = 0
In general, the sum of any integer and its additive inverse is 0: a + (a) = 0

17. Subtraction of Integers
For all integers a and b, a – b = a + (b). In words, to subtract b from a, add the additive inverse of b to a. The result of subtraction is called the difference.

18. Example: Subtracting of Integers
Subtract: a. 17 – (–11) b. –18 – (–5) c. –18 – 5

19. Multiplication of Integers
The result of multiplying two or more numbers is called the product of the numbers. Think of multiplication as repeated addition or subtraction that starts at 0. For example,

20. Multiplication of Integers: Rules
The product of two integers with different signs is found by multiplying their absolute
values. The product is negative.
The product of two integers with the same signs is found by multiplying their absolute
values. The product is positive.
The product of 0 and any integer is 0:
Examples
7(5) = 35
(6)(11) = 66
17(0) = 0

21. Multiplication of Integers: Rules
If no number is 0, a product with an odd number of negative factors is found by
multiplying absolute values. The product is negative.
If no number is 0, a product with an even number of negative factors is found by
multiplying absolute values. The product is positive.
Examples

22. Exponential Notation
Because exponents indicate repeated multiplication, rules for multiplying can be used to evaluate exponential expressions.

23. Example: Evaluating Exponential Notation
Evaluate: a. (6)2 b. 62 c. (5)3 d. (2)4 Solution:

24. Division of Integers
The result of dividing the integer a by the nonzero integer b is called the quotient of numbers. We write this quotient as or a / b.
This means that 4(3) = 12.

25. Division of Integers Rules
Rule 1. The quotient of two integers with different signs is found by dividing their absolute values. The quotient is negative. 2. The quotient of two integers with the same sign is found by dividing their absolute values. The quotient is positive. 3. Zero divided by any nonzero integer is zero. 4. Division by 0 is undefined.
Examples

26. Order of Operations Perform all operations within grouping symbols.
Evaluate all exponential expressions.
Do all the multiplications and divisions in the order in which they occur, working from left to right.
Finally, do all additions and subtractions in the order in which they occur, working from left to right.

27. Example: Using the Order of Operations
Simplify 62 – 24 ÷ 22 · Solution: There are no grouping symbols. Thus, we begin by evaluating exponential expressions. 62 – 24 ÷ 22 · = 36 – 24 ÷ 4 · = 36 – 6 · = 36 – = = 19