Rational and Irrational Numbers

LESSON. Rational and Irrational Numbers.NS. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion;... lso.ns.2,.ee.2? ESSENTIL QUESTION How do you rewrite rational numbers and decimals, take square roots and cube roots, and approximate irrational numbers? Expressing Rational Numbers as Decimals rational number is any number that can be written as a ratio in the form a _ b, where a and b are integers and b is not 0. Examples of rational numbers are 6 and 0.5. 6 can be written as 6 _. 0.5 can be written as 2. Every rational number can be written as a terminating decimal or a repeating decimal. terminating decimal, such as 0.5, has a finite number of digits. repeating decimal has a block of one or more digits that repeat indefinitely. EXMPLE.NS. Write each fraction as a decimal. 4 0.25 4.00-20 -20 0 4 = 0.25 0..000 = 0. _ Remember that the fraction bar means divided by. Divide the numerator by the denominator. Divide until the remainder is zero, adding zeros after the decimal point in the dividend as needed. = 0... Divide until the remainder is zero or until the digits in the quotient begin to repeat. dd zeros after the decimal point in the dividend as needed. When a decimal has one or more digits that repeat indefinitely, write the decimal with a bar over the repeating digit(s). My Notes 7

YOUR TURN Online ssessment Write each fraction as a decimal.. 5 2.. 2 My Notes Expressing Decimals as Rational Numbers You can express terminating and repeating decimals as rational numbers. EXMPLE 2 Write each decimal as a fraction in simplest form. 0.25.NS. The decimal 0.25 means 25 thousandths. Write this as a fraction. 25 00 To write 25 thousandths, put 25 over 00. Then simplify the fraction. 25 25 00 25 = 40 Divide both the numerator and the denominator by 25. 0.25 = 40 0. _ 7 Let x = 0. _ 7. The number 0. _ 7 has 2 repeating digits, so multiply each side of the equation x = 0. _ 7 by 2, or 0. x = 0. _ 7 (0)x = 0(0. _ 7 ) 0x = 7. _ 7 ecause x = 0. _ 7, you can subtract x from one side and 0. _ 7 from the other. 0x = 7. _ 7 x 0. _ 7 99x = 7 0 times 0. _ 7 is 7. _ 7. 7. _ 7 minus 0. _ 7 is 7. Now solve the equation for x. Simplify if necessary. 99x 99 = 7 99 x = 7 99 Divide both sides of the equation by 99. Unit

YOUR TURN Write each decimal as a fraction in simplest form. 4. 0.2 5. 0. _ 57 6..4 Finding Square Roots and Cube Roots The square root of a positive number p is x if x 2 = p. There are two square roots for every positive number. For example, the square roots of 6 are 6 and 6 because 6 2 = 6 and ( 6) 2 = 6. The square roots of 25 are and 5 5. You can write the square roots of as ± 25 5. The symbol _ 5 indicates the positive, or principal square root. Online ssessment number that is a perfect square has square roots that are integers. The number is a perfect square because its square roots are 9 and. The cube root of a positive number p is x if x = p. There is one cube root for every positive number. For example, the cube root of is 2 because 2 =. The cube root of 27 is because ( ) = 27. The symbol _ indicates the cube root. number that is a perfect cube has a cube root that is an integer. The number is a perfect cube because its cube root is 5. EXMPLE Solve each equation for x..ee.2 x 2 = 2 x 2 = 2 x = ± _ 2 x = ± The solutions are and. x 2 = 6 x 2 = 6 x = ± 6 x = ± 4 The solutions are 4 and 4 Solve for x by taking the square root of both sides. pply the definition of square root. Think: What numbers squared equal 2? Solve for x by taking the square root of both sides. pply the definition of square root. Think: What numbers squared equal 6?. Math Talk Mathematical Practices Can you square an integer and get a negative number? What does this indicate about whether negative numbers have square roots? 9

C 729 = x _ 729 = _ x _ 729 = x 9 = x Solve for x by taking the cube root of both sides. pply the definition of cube root. Think: What number cubed equals 729? D The solution is 9. x = _ x = _ x = x = 2_ 5 The solution is 2_ 5. Solve for x by taking the cube root of both sides. pply the definition of cube root. Think: What number cubed equals? YOUR TURN Online ssessment Solve each equation for x. 7. x2 = 96. x2 = 9 256 9. x = 52. x = 64 4 EXPLORE CTIVITY Estimating Irrational Numbers Irrational numbers are numbers that are not rational. In other words, they cannot be written in the form _ a, where a and b are integers and b is not 0. b Square roots of perfect squares are rational numbers. Square roots of numbers that are not perfect squares are irrational. The number _ is irrational because is not a perfect square of any rational number. Estimate the value of _ 2. C D Since 2 is not a perfect square, _ 2 is irrational. To estimate _ 2, first find two consecutive perfect squares that 2 is between. Complete the inequality by writing these perfect squares in the boxes. Now take the square root of each number. Simplify the square roots of perfect squares. _ 2 is between and..ns.2,.ee.2 < 2 < _ < _ 2 < _ < _ 2 < Unit

E F 2.5 Estimate that _ 2.5. 0 2 4 To find a better estimate, first choose some numbers between and 2 and square them. For example, choose.,.4, and.5.. 2 =. 4 2 =. 5 2 = Is _ 2 between. and.4? How do you know? Is _ 2 between.4 and.5? How do you know? _ 2 is between and, so _ 2. G Locate and label this value on the number line...2..4.5 Reflect. How could you find an even better estimate of _ 2? 2. Find a better estimate of _ 2. Draw a number line and locate and label your estimate. _ 2 is between and, so _ 2.. Estimate the value of _ 7 to two decimal places. Draw a number line and locate and label your estimate. _ 7 is between and, so _ 7.