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**Rational and Irrational Numbers**

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**Rational and Irrational Numbers Essential Question**

How do I distinguish between rational and irrational numbers?

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Vocabulary real number irrational number

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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. Irrational numbers Rational numbers Real Numbers Integers Whole numbers

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Recall that 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

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Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. 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. A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution!

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**Reals Make a Venn Diagram that displays the following sets of numbers:**

Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Reals Rationals -2.65 Integers -3 -19 Wholes Irrationals Naturals 1, 2, 3...

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**Additional Example 1: Classifying Real Numbers**

Write all classifications that apply to each number. 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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**whole, integer, rational, real**

Check It Out! Example 1 Write all classifications that apply to each number. 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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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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**Additional Example 2: Determining the Classification of All Numbers**

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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**Additional Example 2: Determining the Classification of All Numbers**

State if each number is rational, irrational, or not a real number. 4 0 C. not a real number

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**State if each number is rational, irrational, or not a real number.**

Check It Out! Example 2 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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**State if each number is rational, irrational, or not a real number.**

Check It Out! Example 2 State if each number is rational, irrational, or not a real number. 64 81 8 9 = 64 81 C. rational

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