Rational and Irrational Numbers

Rational and Irrational Numbers Essential Question How do I distinguish between rational and irrational numbers?

Vocabulary real number irrational number

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

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

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

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...

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

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

A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

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

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

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

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