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Transcript
2-1 Rational Numbers California Evaluating Algebraic Expressions Standards NS1.5 Know that every rational number is either a terminating or a repeating decimal and be able to convert terminating decimals into reduced fractions. NS1.3 Convert fractions to decimals and percents and use representations in estimations, computations, and applications. 2-1 Rational Numbers Evaluating Algebraic Expressions Vocabulary rational number terminating decimal repeating decimal 2-1 Rational Numbers Evaluating Algebraic Expressions A rational number is any number that can n be written as a fraction , where n d and d are integers and d 0. Any fraction can be written as a decimal by dividing the numerator by the denominator. If the division ends or terminates, because the remainder is zero, then the decimal is a terminating decimal. 2-1 Rational Numbers Evaluating Algebraic Expressions If the division leads to a repeating block of one or more digits (where all digits are not zeros) after the decimal point, then the decimal is a repeating decimal. A repeating decimal can be written with a bar over the digits that repeat. So 0.13333… = 0.13. 2-1 Rational Numbers Additional Example 1A: Writing Fractions as Decimals Evaluating Algebraic Expressions Write the fraction as a decimal. 11 9 1 .2 9 11 .0 –9 20 –1 8 2 The fraction The pattern repeats. 11 is equivalent to the decimal 1.2. 9 2-1 Rational Numbers Additional Example 1B: Writing Fractions as Decimals Evaluating Algebraic Expressions Write the fraction as a decimal. 7 20 0.3 5 This is a terminating decimal. 20 7.0 0 –0 70 –6 0 1 00 –1 0 0 0 The remainder is 0. 7 The fraction is equivalent to the decimal 0.35. 20 2-1 Rational Numbers Check It Out! Example 1A Write the fraction as a decimal. Evaluating Algebraic Expressions 15 9 1 .6 9 15 .0 –9 60 –5 4 6 The pattern repeats, so draw a bar over the 6 to indicate that this is a repeating decimal. 15 The fraction is equivalent to the decimal 1.6. 9 2-1 Rational Numbers Check It Out! Example 1B Write the fraction as a decimal. Evaluating Algebraic Expressions 9 40 0.2 2 5 This is a terminating 40 9.0 0 0 decimal. –0 90 –8 0 1 00 – 80 200 – 2 00 0 The remainder is 0. 9 The fraction is equivalent to the decimal 0.225. 40 2-1 Rational Numbers Evaluating Algebraic Expressions To write a terminating decimal as a fraction, identify the place value of the digit farthest to the right. Then write all of the digits after the decimal point as the numerator with the place value as the denominator. 2-1 Rational Numbers Additional Example 2: Writing Terminating Decimals as Fractions WriteEvaluating each decimal Algebraic as a fraction Expressions in simplest form. A. 5.37 37 5.37 = 5 100 B. 0.622 622 0.622 = 1000 311 = 500 7 is in the hundredths place, so write hundredths as the denominator. 2 is in the thousandths place, so write thousandths as the denominator. Simplify by dividing by the greatest common divisor. 2-1 Rational Numbers Evaluating Algebraic Expressions Remember! A fraction is in reduced, or simplest, form when the numerator and the denominator have no common divisor other than 1. 2-1 Rational Numbers Check It Out! Example 2 Write each decimal as a fraction in simplest form. Evaluating Algebraic Expressions A. 8.75 5 is in the hundredths place, 75 so write hundredths as the 8.75 = 8 100 denominator. 3 Simplify by dividing by the = 8 4 greatest common divisor. B. 0.2625 5 is in the 2625 0.2625 = 10,000 ten-thousandths place. Simplify by dividing by the 21 = greatest common divisor. 80 2-1 Rational Numbers Additional Example 3: Writing Repeating Decimals as Fractions _ Algebraic Expressions WriteEvaluating 0.4 as a fraction in simplest form. x = 0.44444… 10x = 10(0.44444…) 10x = 4.444444… -x = -0.44444… 9x = 4 9x = 4 9 9 4 x= 9 Let x represent the number. Multiply both sides by 10 because 1 digit repeats. Subtract x from both sides to eliminate the repeating part. Since x = 0.44444…, use 0.44444… for x on the right side of the equation. Since x is multiplied by 9, divide both sides by 9. 2-1 Rational Numbers __ Check It Out! Example 3 Write 0.36 as a fraction in simplest form. Evaluating Algebraic Expressions x = 0.363636… 100x = 100(0.363636…) 100x = 36.363636… -x = -0.363636… 99x = 36 99x = 36 99 99 x = 36 = 4 99 11 Let x represent the number. Multiply both sides by 100 because 2 digits repeat. Subtract x from both sides to eliminate the repeating part. Since x = 0.363636…, use 0.363636… for x on the right side of the equation. Since x is multiplied by 99, divide both sides by 99. Write in simplest form.