1. Rational and Irrational Numbers
CHAPTER 11
Rational and Irrational Numbers

2. 11-1 Properties of Rational Numbers

3. Rational Numbers
A real number that can be expressed as the quotient of two integers.

4. Examples
7 = 7/1
5 2/3 = 17/3
.43 = 43/100
-1 4/5 = -9/5

5. Write as a quotient of integers3
48%
.60
- 2 3/5

6. Which rational number is greater 8/3 or 17/7

7. Rules a/c > b/d if and only if ad > bc.

8. Examples
4/7 ? 3/8
7/9 ? 4/5
8/15 ? 3/4

9. Density Property
Between every pair of different rational numbers there is another rational number

10. Implication
The density property implies that it is possible to find an unlimited or endless number of rational numbers between two given rational numbers.

11. Formula If a < b, then to find the number halfway from a to b use:
a + ½(b – a)

12. Example
Find a rational number between -5/8 and -1/3.

13. 11-2 Decimal Forms of Rational Numbers

14. Forms of Rational Numbers
Any common fraction can be written as a decimal by dividing the numerator by the denominator.

15. Decimal Forms
Terminating
Nonterminating

16. Examples Express each fraction as a terminating or repeating decimal
5/ / /7

17. Rule
For every integer n and every positive integer d, the decimal form of the rational number n/d either terminates or eventually repeats in a block of fewer than d digits.

18. Rule
To express a terminating decimal as a common fraction, express the decimal as a common fraction with a power of 10 as the denominator.

19. Express as a fraction
.38
.425

20. Solutions
.38 = 38/100 or 19/50
.425 = 425/1000= 17/40

21. Express a Repeating Decimal as a fraction
.542
let N = 0.542
Multiply both sides of the equation by a power of 10

22. Continued
Subtract the original equation from the new equation
Solve

23. 11-3 Rational Square Roots
Rational Numbers
11-3 Rational Square Roots

24. Rule
If a2 = b, then a is a square root of b.

25. Terminology Radical sign is 
Radicand is the number beneath the radical sign

26. Product Property of Square Roots
For any nonnegative real numbers a and b:
ab = (a) (b)

27. Quotient Property of Square Roots
For any nonnegative real number a and any positive real number b:
a/b = (a) /(b)

28. Examples
36
100
- 81/1600
0.04

29. 11-4 Irrational Square Roots
Irrational Numbers
11-4 Irrational Square Roots

30. Irrational Numbers
Real number that cannot be expressed in the form a/b where a and b are integers.

31. Property of Completeness
Every decimal number represents a real number, and every real number can be represented as a decimal.

32. Rational or Irrational
17
49
1.21
5 + 2 2

33. Simplify
63
128
50
6108

34. Simplify 63 = 9 7 = 37 128 = 64 2 = 82 50 = 25 5 = 55
6108= 636 3=36 3

35. 11-5 Square Roots of Variable Expressions
Rational Numbers
11-5 Square Roots of Variable Expressions

36. Simplify
196y2
36x8
m2-6m + 9
18a3

37. Solutions 196y2 = ± 18y 36x8 = ± 6x4 m2-6m + 9 = ±(m -3)
18a3 = ± 3a 2a

38. Solve by factoring Get the equation equal to zero Factor
Set each factor equal to zero and solve

39. Examples
9x2 = 64
45r2 – 500 = 0
81y2 – 16= 0

40. 11-6 The Pythagorean Theorem
Irrational Numbers
11-6 The Pythagorean Theorem

41. The Pythagorean Theorem
In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. a2 + b2 = c2

42. Example c a b

43. Example c 8 15

44. Solution
a2 + b2 = c2
= c2
=c2
289 =c2
17 = c

45. Example
The length of one side of a right triangle is 28 cm. The length of the hypotenuse is 53 cm. Find the length of the unknown side.

46. Solution
a2 + b2 = c2
a = 532
a =2809
a2 =2025
a = 45

47. Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest, then the triangle is a right triangle. The right side is opposite the longest side.

48. 11-7 Multiplying, Dividing, and Simplifying Radicals
Radical Expressions
11-7 Multiplying, Dividing, and Simplifying Radicals

49. Rationalization
The process of eliminating a radical from the denominator.

50. Simplest Form
No integral radicand has a perfect-square factor other than 1
No fractions are under a radical sign, and
No radicals are in a denominator

51. Simplify
3/5
7/ 8
3 3/7
9 3/ 24

52. Solution
3/5 = 3 5 /5
7/ 8= 14/4
3 3/7= 22
9 3/ 24 = 9 2/4

53. 11-8 Adding and Subtracting Radicals
Radical Expressions
11-8 Adding and Subtracting Radicals

54. Simplifying Sums or Differences
Express each radical in simplest form.
Use the distributive property to add or subtract radicals with like radicands.

55. Examples
47 + 57
36 - 213
73 - 46 + 248

56. Solution
97
86 - 213
153 -46

57. 11-9 Multiplication of Binomials Containing Radicals
Radical Expressions
11-9 Multiplication of Binomials Containing Radicals

58. Terminology Binomials – variable expressions containing two terms.
Conjugates – binomials that differ only in the sign of one term.

59. Rationalization of Binomials
Use conjugates to rationalize denominators that contain radicals.

60. Simplify
(6 + 11)(6 - 11)
(3 + 5)2
(23 - 57) 2
3/(5 - 27)

61. Solution 25
14 + 65
187 – 2021 7

62. 11-10 Simple Radical Equations
Radical Expressions
11-10 Simple Radical Equations

63. Terminology
Radical equation – an equation that has a variable in the radicand.

64. Examples
d = 1000
x = 3
x = ± 3

65. Solutions
140 = 2(9.8)d
(5x +1) + 2 = 6
(11x2 – 63) -2x = 0

66. End…End…End…End…End...
End