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CHAPTER 11 Rational and Irrational Numbers. Rational Numbers 11-1 Properties of Rational Numbers.

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Presentation on theme: "CHAPTER 11 Rational and Irrational Numbers. Rational Numbers 11-1 Properties of Rational Numbers."— Presentation transcript:

1 CHAPTER 11 Rational and Irrational Numbers

2 Rational Numbers 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/ /5 = -9/5

5 Write as a quotient of integers 3 48% /5

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

7 Rules a/c > b/d if and only if ad > bc. 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 Rational Numbers 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 5/6 7/11 3 2/7

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

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

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

22 Continued Subtract the original equation from the new equation Solve

23 Rational Numbers 11-3 Rational Square Roots

24 Rule If a 2 = 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   81/1600  0.04

29 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   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 Rational Numbers 11-5 Square Roots of Variable Expressions

36 Simplify  196y 2  36x 8  m 2 -6m + 9  18a 3

37 Solutions  196y 2 = ± 18y  36x 8 = ± 6x 4  m 2 -6m + 9 = ±(m -3)  18a 3 = ± 3a  2a

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

39 Examples 9x 2 = 64 45r 2 – 500 = 0 81y 2 – 16= 0

40 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. a 2 + b 2 = c 2

42 Example a c b

43 8 c 15

44 Solution a 2 + b 2 = c = c =c =c 2 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 a 2 + b 2 = c 2 a = 53 2 a =2809 a 2 =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 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 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 3   13 7    48

56 Solution 9  7 8    3 -4  6

57 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   7) 2 3/(5 - 2  7)

61 Solution  – 20   7

62 Radical Expressions 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 (  11x 2 – 63) -2x = 0

66 End…End…End…End…End... End


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