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**Rational and Irrational Numbers**

4-8 The Real Numbers Evaluating Algebraic Expressions Rational and Irrational Numbers

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**Rational and Irrational Numbers Essential Question**

4-8 The Real Numbers Rational and Irrational Numbers Essential Question Evaluating Algebraic Expressions How do I distinguish between rational and irrational numbers?

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Vocabulary Evaluating Algebraic Expressions real number irrational number

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers. Evaluating Algebraic Expressions Irrational numbers Rational numbers Real Numbers Integers Whole numbers

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The Real Number System Natural Numbers - The set of natural numbers is often referred to as the set of counting numbers, because they are those numbers that we use to count. Also notice that 0 is not included in the natural numbers. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …∞} 5

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The Real Number System Whole Numbers - the natural numbers together with 0. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …∞} 6

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The Real Number System Integers - If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers. {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} 7

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Evaluating Algebraic Expressions Our next group of numbers is called the rational numbers. Rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 4 5 23 3 = 3.8 = 0.6 1.44 = 1.2 So...rational numbers include fractions, terminating decimals, and repeating decimals.

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers So, the rational numbers include the natural numbers, whole numbers, and integers. Evaluating Algebraic Expressions Reals Rationals 1/2 -2.65 Integers -3 -19 Wholes 1, 2, 3... Naturals

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**Rational Numbers {p/q : p and q are integers, q is not zero} Number**

As a fraction Rational? 5 5/1 .75 .001 1/1000 … 2/9 … ? 10

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers (a simple fraction). If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational. Evaluating Algebraic Expressions A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution!

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**What other examples of irrational numbers are there?**

4-8 The Real Numbers What other examples of irrational numbers are there? Evaluating Algebraic Expressions 12

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Let’s Review all of our sets of numbers in the real number system - Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Evaluating Algebraic Expressions Reals Rationals -2.65 Integers -3 -19 Wholes Irrationals Naturals 1, 2, 3...

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Additional Example 1: Classifying Real Numbers Write all classifications that apply to each number. Evaluating Algebraic Expressions A. 5 5 is a whole number that is not a perfect square. irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 = = 2 4 2 16 2 C. whole, integer, rational, real

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Check It Out! Example 1 Write all classifications that apply to each number. Evaluating Algebraic Expressions A. 9 9 = 3 whole, integer, rational, real B. –35.9 –35.9 is a terminating decimal. rational, real 81 3 = = 3 9 3 81 3 C. whole, integer, rational, real

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Evaluating Algebraic Expressions A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Additional Example 2: Determining the Classification of All Numbers Evaluating Algebraic Expressions State if each number is rational, irrational, or not a real number. A. 21 irrational 0 3 0 3 = 0 B. rational

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Additional Example 2: Determining the Classification of All Numbers Evaluating Algebraic Expressions State if each number is rational, irrational, or not a real number. 4 0 C. not a real number

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Check It Out! Example 2 Evaluating Algebraic Expressions State if each number is rational, irrational, or not a real number. A. 23 23 is a whole number that is not a perfect square. irrational 9 0 B. undefined, so not a real number

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**Evaluating Algebraic Expressions**

4-8 The Real Numbers Check It Out! Example 2 Evaluating Algebraic Expressions State if each number is rational, irrational, or not a real number. 8 9 = C. rational

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