CHAPTER 11 Rational and Irrational Numbers Rational Numbers

CHAPTER 11 Rational and Irrational Numbers

Rational Numbers 11 -1 Properties of Rational Numbers

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

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

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

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

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

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

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

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

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

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

Rational Numbers 11 -2 Decimal Forms of Rational Numbers

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

Decimal Forms • Terminating • Nonterminating

Examples Express each fraction as a terminating or repeating decimal 5/6 7/11 3 2/7

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.

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.

Express as a fraction • . 38 • . 425

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

Express a Repeating Decimal as a fraction • . 542 • let N = 0. 542 • Multiply both sides of the equation by a power of 10

Continued • Subtract the original equation from the new equation • Solve

Rational Numbers 11 -3 Rational Square Roots

Rule 2 a If = b, then a is a square root of b.

Terminology • Radical sign is • Radicand is the number beneath the radical sign

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

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

Examples • 36 • 100 • - 81/1600 • 0. 04

Irrational Numbers 11 -4 Irrational Square Roots

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

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

Rational or Irrational • 17 • 49 • 1. 21 • 5 + 2 2

Simplify • 63 • 128 • 50 • 6 108

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

Rational Numbers 11 -5 Square Roots of Variable Expressions

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

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

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

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

Irrational Numbers 11 -6 The Pythagorean Theorem

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

Example c a b

Example c 8 15

Solution 2 a 2 b 2 c + = 2 2 2 8 + 15 = c 64 + 225 =c 2 289 =c 2 17 = c

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.

Solution 2 a 2 b 2 c + = 2 2 2 a + 28 = 53 a 2 + 784 =2809 a 2 =2025 a = 45

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.

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

Rationalization The process of eliminating a radical from the denominator.

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

Simplify • 3/ 5 • 7/ 8 • 3 3/7 • 9 3/ 24

Solution • 3/ 5 = 3 5 /5 • 7/ 8= 14/4 • 3 3/7= 2 2 • 9 3/ 24 = 9 2/4

Radical Expressions 11 -8 Adding and Subtracting Radicals

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

Examples • 4 7 + 5 7 • 3 6 - 2 13 • 7 3 - 4 6 + 2 48

Solution • 9 7 • 8 6 - 2 13 • 15 3 -4 6

Radical Expressions 11 -9 Multiplication of Binomials Containing Radicals

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

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

Simplify • (6 + 11)(6 - 11) 2 • (3 + 5) • (2 3 - 5 7) 2 • 3/(5 - 2 7)

Solution • 25 • 14 + 6 5 • 187 – 20 21 • -5 - 2 7

Radical Expressions 11 -10 Simple Radical Equations

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

Examples • d = 1000 • x = 3 • x = ± 3

Solutions • 140 = 2(9. 8)d • ( 5 x +1) + 2 = 6 • ( 11 x 2 – 63) -2 x = 0

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