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**Rational and Irrational Numbers**

CHAPTER 11 Rational and Irrational Numbers

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**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 = 43/100 -1 4/5 = -9/5

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**Write as a quotient of integers**

3 48% .60 - 2 3/5

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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.**

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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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**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/ / /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 .38 .425

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

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**Express a Repeating Decimal as a fraction**

.542 let N = 0.542 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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**11-3 Rational Square Roots**

Rational Numbers 11-3 Rational Square Roots

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Rule If a2 = 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 36 100 - 81/1600 0.04

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**11-4 Irrational Square Roots**

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 1.21 5 + 2 2

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Simplify 63 128 50 6108

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**Simplify 63 = 9 7 = 37 128 = 64 2 = 82 50 = 25 5 = 55**

6108= 636 3=36 3

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**11-5 Square Roots of Variable Expressions**

Rational Numbers 11-5 Square Roots of Variable Expressions

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

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**Solutions 196y2 = ± 18y 36x8 = ± 6x4 m2-6m + 9 = ±(m -3)**

18a3 = ± 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 9x2 = 64 45r2 – 500 = 0 81y2 – 16= 0

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**11-6 The Pythagorean Theorem**

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. a2 + b2 = c2

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Example c a b

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Example c 8 15

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Solution a2 + b2 = c2 = c2 =c2 289 =c2 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 a2 + b2 = c2 a = 532 a =2809 a2 =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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**11-7 Multiplying, Dividing, and Simplifying Radicals**

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= 22 9 3/ 24 = 9 2/4

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**11-8 Adding and Subtracting Radicals**

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 47 + 57 36 - 213 73 - 46 + 248

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Solution 97 86 - 213 153 -46

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**11-9 Multiplication of Binomials Containing Radicals**

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 (23 - 57) 2 3/(5 - 27)

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Solution 25 14 + 65 187 – 2021 7

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**11-10 Simple Radical Equations**

Radical Expressions 11-10 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 (11x2 – 63) -2x = 0

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End…End…End…End…End... End

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