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

If a2 = 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

36100

• - 81/16000.04

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

17491.21

• 5 + 2 2

Simplify6312850

• 6108

Simplify63 = 9 7 = 37 128 = 64 2 = 82 50 = 25 5 = 55

• 6108= 636 3=36 3

Rational Numbers

11-5 Square Roots of Variable

Expressions

Simplify196y2

36x8

m2-6m + 9 18a3

Solutions196y2 = ± 18y36x8 = ± 6x4

m2-6m + 9 = ±(m -3)18a3 = ± 3a 2a

Solve by factoring

• Get the equation equal to zero

• Factor

• Set each factor equal to zero and solve

Examples

• 9x2 = 64

• 45r2 – 500 = 0

• 81y2 – 16= 0

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. a2 + b2 = c2

Example

ac

b

Example

8c

15

Solution

a2 + b2 = c2

82 + 152 = c2

64 + 225 =c2

289 =c2

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

a2 + b2 = c2

a2 + 282 = 532

a2 + 784 =2809

a2 =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

• 3/57/ 83 3/7

• 9 3/ 24

Solution

• 3/5 = 3 5 /57/ 8= 14/43 3/7= 22

• 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

• 47 + 57

• 36 - 213

• 73 - 46 + 248

Solution

• 97

• 86 - 213

• 153 -46

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)

• (3 + 5)2

• (23 - 57) 2

• 3/(5 - 27)

Solution

• 25

• 14 + 65

• 187 – 2021

• -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

• (5x +1) + 2 = 6

• (11x2 – 63) -2x = 0

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