1. 5.2
The Integers

2. Whole Numbers
The set of whole numbers contains the set of natural numbers and the number 0.
Whole numbers = {0,1,2,3,4,…}

3. Integers
The set of integers consists of 0, the natural numbers, and the negative natural numbers.
Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…}
On a number line, the positive numbers extend to the right from zero; the negative numbers extend to the left from zero.

4. Writing an Inequality
Insert either > or < in the box between the paired numbers to make the statement correct.
a)  1 b)  7
3 <  9 < 7
c) 4 d)
0 >  < 8

5. Subtraction of Integers
a – b = a + (b)
Evaluate:
a) –7 – 3 = –7 + (–3) = –10
b) –7 – (–3) = –7 + 3 = –4

6. Properties Multiplication Property of Zero Division
For any a, b, and c where b  0, means that c • b = a.

7. Rules for Multiplication
The product of two numbers with like signs (positive  positive or negative  negative) is a positive number.
The product of two numbers with unlike signs (positive  negative or negative  positive) is a negative number.

8. Examples Evaluate: a) (3)(4) b) (7)(5) c) 8 • 7 d) (5)(8)
Solution:
a) (3)(4) = 12 b) (7)(5) = 35
c) 8 • 7 = 56 d) (5)(8) = 40

9. Rules for Division
The quotient of two numbers with like signs (positive  positive or negative  negative) is a positive number.
The quotient of two numbers with unlike signs (positive  negative or negative  positive) is a negative number.

10. Example
Evaluate:
a) b)
c) d)
Solution:

11. Homework
P. 227 # 8 – 70 even

12. 5.3
The Rational Numbers

13. The Rational Numbers
The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q  0.
The following are examples of rational numbers:

14. Fractions Fractions are numbers such as:
The numerator is the number above the fraction line.
The denominator is the number below the fraction line.

15. Reducing Fractions
In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor.
Example: Reduce to its lowest terms.
Solution:

16. Mixed Numbers
A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number.
3 ½ is read “three and one half” and means “3 + ½”.

17. Improper Fractions
Rational numbers greater than 1 or less than –1 that are not integers may be written as mixed numbers, or as improper fractions.
An improper fraction is a fraction whose numerator is greater than its denominator An example of an improper fraction is

18. Add product to the numerator.
Example
Convert to an improper fraction.
Multiply the denominator of the fraction in the mixed number by the integer preceding it.
Add product to the numerator.

19. Example Convert to a mixed number. The mixed number is
Divide the numerator by the denominator. Identify the quotient and the remainder
The remainder is the numerator of the fraction in the mixed number.

20. Terminating or Repeating Decimal Numbers
Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number.
Examples of terminating decimal numbers are 0.7, 2.85,
Examples of repeating decimal numbers … which may be written

21. Multiplication of Fractions
Division of Fractions

22. Example: Multiplying Fractions
Evaluate the following. a) b)

23. Example: Dividing Fractions
Evaluate the following. a) b)

24. Addition and Subtraction of Fractions

25. Example: Add or Subtract Fractions

26. Fundamental Law of Rational Numbers
If a, b, and c are integers, with b  0, c  0, then

27. Example:
Evaluate:
Solution:

28. Homework
P. 256 # 14 – 100 EOE