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**Rational Vs. Irrational**

Making sense of rational and Irrational numbers

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**Biologists classify animals based on shared characteristics**

Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number. Animal Reptile Lizard Gecko

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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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**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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The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.

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**Additional Example 3: Applying the Density Property of Real Numbers**

Find a real number between and 3 5 2 5 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 2 5 ÷ 2 3 5 5 5 = ÷ 2 1 2 = 7 ÷ 2 = 3 A real number between and is 3 . 3 5 2 5 1 2 Check: Use a graph. 3 1 5 2 5 4 3 5 4 5 3 1 2

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**Find a real number between 4 and 4 . 4 7 3 7**

Check It Out! Example 3 Find a real number between and 4 7 3 7 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 3 7 ÷ 2 4 7 7 7 = ÷ 2 1 2 = 9 ÷ 2 = 4 A real number between and is 4 7 3 7 1 2 Check: Use a graph. 4 2 7 3 7 4 7 5 7 1 7 6 7 4 1 2

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**1. 2. – 4. 3. 5. Lesson Quiz 2 16 2 real, irrational**

Write all classifications that apply to each number. 1. 2 2. – 16 2 real, irrational real, integer, rational State if each number is rational, irrational, or not a real number. 25 0 4. 3. 4 • 9 not a real number rational 5. Find a real number between –2 and –2 . 3 8 3 4 Possible answer: –2 5 8

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