Rational and Irrational Numbers CHAPTER 11 Rational and Irrational 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.

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.

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

11-3 Rational Square Roots 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 36 100 - 81/1600 0.04

11-4 Irrational Square Roots 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

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

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

Solutions 196y2 = ± 18y 36x8 = ± 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

11-6 The Pythagorean Theorem 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 c a b

Example c 8 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.

11-7 Multiplying, Dividing, and Simplifying Radicals 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

11-8 Adding and Subtracting Radicals 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

11-9 Multiplication of Binomials Containing Radicals 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 (23 - 57) 2 3/(5 - 27)

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

11-10 Simple Radical Equations 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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