How Do I Distinguish Between Rational And Irrational Numbers? N1.h How Do I Distinguish Between Rational And Irrational Numbers? Course 3 Lesson Presentation

Learn to determine if a number is rational or irrational.

Vocabulary real number rational number irrational number

Recall that rational numbers can be written as fractions Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat. 4 5 23 3 = 3.8 = 0.6 1.44 = 1.2

Irrational numbers cannot be written as a fraction & can only be written as decimals that do not terminate or repeat (NON-TERMINATING/NON-REPEATING DECIMALS. The square root of a non-perfect square is an irrational number. For example, 2 ≈1.4142135623730950488016… A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution!

The set of real numbers consists of the set of rational numbers and the set of irrational numbers. Integers Whole numbers

Example 1: Classifying Real Numbers Write all names 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

Write all names that apply to each number. Check It Out: Example 1 Write all names 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

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

Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. C. –4 not a real number 4 9 2 3 = 4 9 D. rational

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. not a number, 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. C. –7 not a real number 64 81 8 9 = 64 81 D. rational

The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.

Example 3: Applying the Density Property of Real Numbers Find a real number between 3 and 3 . 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 3 + 3 ÷ 2 3 5 5 5 = 6 ÷ 2 1 2 = 7 ÷ 2 = 3 3 1 5 2 5 4 3 5 4 5 3 1 2 A real number between 3 and 3 is 3 . 3 5 2 5 1 2

Find a real number between 4 and 4 . 4 7 3 7 Check It Out: Example 3 Find a real number between 4 and 4 . 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 4 + 4 ÷ 2 4 7 7 7 = 8 ÷ 2 1 2 = 9 ÷ 2 = 4 4 2 7 3 7 4 7 5 7 1 7 6 7 4 1 2 A real number between 4 and 4 is 4 . 4 7 3 7 1 2

Lesson Quiz Write all names 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