Natural numbers are counting numbers.

2. = {1, 2, 3, 4, 5…} Natural Numbers Natural numbers are counting numbers.

3. = {0, 1, 2, 3, 4, 5…} Whole Numbers Whole numbers are natural numbers and zero.

4. N is a subset of W.

5. = {...−3, −2, −1, 0, 1, 2, 3…} Integers Integers are whole numbers and opposites of naturals.

6. N and W are subsetsof Z.

7. Rational Numbers Rational numbers are integers and all fractions.

8. ab = { } , , a b & b ≠ 0

10. Irrational Numbers Irrational numbers are totally different from rational numbers. The two have nothing in common.

11. Rationals and irrationals are disjoint sets. In other words, they have no common element.

12. p 2, , 5 7, 1.305276... Irrationals

13. Real Numbers Real numbers are rational and irrational.

14. = & irrationals

15. There are an infinite number of rational numbers between each pair of integers. This is called the density of numbers.

16. A rational number is any number that can be written in the form , where a and b are integers and b ≠ 0. ab Rational Numbers

17. ab A rational fraction is in lowest terms if the GCF of a and b is one. Lowest Terms

18. 1218 2 x 63 x 6 23 1218 = = Example 1 Rename in lowest terms. 12 = 2 • 2 • 3 18 = 2 • 3 • 3 GCF = 2 • 3 = 6

19. 2490 2 x 2 x 2 x 3 2 x 3 x 3 x 5 2 x 2 x 2 x 3 2 x 3 x 3 x 5 = = 2490 415 = Example 2 Rename in lowest terms.

20. 57 = 3042 Example Rename in lowest terms.

21. 57 = 3,0004,200 Example Rename in lowest terms.

22. 45 = 7290 Example Rename in lowest terms.

23. A proper fraction is one whose numerator is less than its denominator.

24. If the numerator is greater than or equal to the denominator, the fraction is greater than or equal to one and is called an improper fraction.

25. A mixed number is actually the sum of a whole number and a fraction.

26. Renaming Improper Fractions as Mixed Numbers • Divide the numerator by the denominator. • Write the quotient as the whole number. • Write the remainder over the divisor as a fraction. • If possible, reduce the fraction to lowest terms.

27. 19 7 2 7 19 5 7 = 2 - 14 5 Example 3 Rename as a mixed number.

28. 12 8 1 8 12 4 8 1 1 = 1 2 - 8 4 Example 3 Rename as a mixed number.

29. 78 36 2 = 1 6 Example Rename the improper fraction as a mixed number.

30. 93 8 = −11 − 5 8 Example Rename the improper fraction as a mixed number.

31. 1 3 = 6 6 38 6 383(2) = 3 19 y3z - 18 19 x 2 3 x 2 19 3 1 = = Example 4 Evaluate the expressionwhen y = 38 and z = 2. Write the answer as a mixed number in lowest terms.

32. yz 3x – = 3 5 6 Example Evaluate when x = 2, y = – 3, and z = 5.

33. 3x2z – y = 2 5 5 Example Evaluate when x = 2, y = – 3, and z = 5.

34. (3x)23yz2 4 25 = − Example Evaluate when x = 2, y = – 3, and z = 5.

35. Renaming Mixed Numbers as Improper Fractions • Multiply the whole number by the denominator. • Add the numerator to the product. • Write the sum over the denominator. • If possible, reduce the fraction to lowest terms.

36. 1 5 3 1 5 5(3) + 15 15 + 15 = 3 16 5 = = Example 5 Rename as an improper fraction in lowest terms.

37. 6 8 7 628 31 x 24 x 2 6 8 8(7) + 68 56 + 68 = = = 7 31 4 = = Example 5 Rename as an improper fraction in lowest terms.

38. 9 11 31 11 2 = Example Rename the mixed number as an improper fraction.

39. 64 5 = − Example Rename the mixed number as an improper fraction. 4 5 − 12