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Rational Vs. Irrational Making sense of rational and Irrational numbers

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Animal Reptile 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. 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 numbersRational 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. 3 = = = 1.2

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A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution! 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 2 is irrational.

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

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Check It Out! Example 1 Write all classifications that apply to each number. 9 whole, integer, rational, real –35.9 is a terminating decimal. –35.9 rational, real 81 3 whole, integer, rational, real = = A. B. C. 9 = 3

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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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State if each number is rational, irrational, or not a real number. 21 irrational 0303 rational 0303 = 0 Additional Example 2: Determining the Classification of All Numbers A. B.

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not a real number Additional Example 2: Determining the Classification of All Numbers 4040 C. State if each number is rational, irrational, or not a real number.

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23 is a whole number that is not a perfect square. 23 irrational 9090 undefined, so not a real number Check It Out! Example 2 A. B. State if each number is rational, irrational, or not a real number.

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64 81 rational 8989 = C. Check It Out! Example 2 State if each number is rational, irrational, or not a real number.

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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 ÷ There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.5 = 6 ÷ = 7 ÷ 2 = Find a real number between 3 and A real number between 3 and 3 is Check: Use a graph.

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Check It Out! Example ÷ There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.7 = 8 ÷ = 9 ÷ 2 = Find a real number between 4 and A real number between 4 and 4 is Check: Use a graph.

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Lesson Quiz Write all classifications that apply to each number – State if each number is rational, irrational, or not a real number Find a real number between –2 and – not a real number rational real, irrational real, integer, rational Possible answer: –2 5858

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