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CHAPTER 11 Rational and Irrational Numbers

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Rational Numbers 11-1 Properties of Rational Numbers

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

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Examples 7 = 7/1 5 2/3 = 17/3.43 = 43/ /5 = -9/5

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Write as a quotient of integers 3 48% /5

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

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

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

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

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

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

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

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Rational Numbers 11-2 Decimal Forms of Rational Numbers

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

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Decimal Forms Terminating Nonterminating

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

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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.

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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.

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Express as a fraction

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

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

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Continued Subtract the original equation from the new equation Solve

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Rational Numbers 11-3 Rational Square Roots

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

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

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

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

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Examples 36 81/1600 0.04

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Irrational Numbers 11-4 Irrational Square Roots

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

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

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Rational or Irrational 17 49 2

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Simplify 63 128 50 6 108

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

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Rational Numbers 11-5 Square Roots of Variable Expressions

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

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

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

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

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Irrational Numbers 11-6 The Pythagorean Theorem

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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

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Example a c b

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8 c 15

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Solution a 2 + b 2 = c = c =c =c 2 17 = c

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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.

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

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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.

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

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Rationalization The process of eliminating a radical from the denominator.

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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

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

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

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Radical Expressions 11-8 Adding and Subtracting Radicals

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

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Examples 4 7 3 13 7 48

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Solution 9 7 8 3 -4 6

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Radical Expressions 11-9 Multiplication of Binomials Containing Radicals

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

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

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

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Solution – 20 7

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Radical Expressions Simple Radical Equations

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

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Examples d = 1000 x = 3 x = ± 3

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

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End…End…End…End…End... End

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