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Rational Numbers converting fractions to decimals Every number has a decimal expansion 1 12 1 4 1 2 2 5 6 3 16 1 10 1 6 14 27 Rational Numbers A rational number is the ratio of 2 integers Rational Numbers a b rational number Rational numbers can terminate in 0 or repeat. Which of these numbers terminate in 0? Which of these numbers repeat? 3 4 5 6 12 7 10 11 20 4 fraction 8 decimal 1 7 terminating recurring ends in 0 repeats converting decimals to rational converting decimals to rational Recurring decimals Terminating decimals 0.7 0.7 let x equal the reccuring decimal let x = 0.777777..... (1) 10x = 7.777777..... 0.045 multiply (1) by 10 {tenths} subtract (2) - (1) 2.34 0.625 (2) 10x = 7.777777..... x = 0.777777..... 9x = 7 x= 7 9 converting decimals to fractions 0.564 Recurring decimals let x = EASY (1) 0.564564564..... 1000x = 564.564564..... (2) multiply (1) by 1000 {thousandths} Change 0.3 into fraction form MEDIUM Change 0.45 into fraction form TRICKY Change 0.4714285 into fraction form subtract (2) - (1) 1000x = 564.564564..... x = 0.564564564..... 999x = 564 x = 564 999 Change 0.317 into fraction form CHALLENGE Change 2.16 into fraction form = 188 333 Change 0.333 into fraction form x = 0.333 10x = 3.333 9x = 3 x = 3= 1 9 3 Change 0.45 into fraction form x = 0.4545... 100x = 45.4545... 99x = 45 x = 45 = 15 99 33 Change 0.31717... into fraction form x = 0.31717... 100x = 31.717... 99x = 31.4 x = 31.4 = 314 99 990 Change 2.166... into fraction form Change 0.4714285 into fraction form x= 0.4714285... 10000000x = 4714285.4714285... 9999999x = 4714285 x = 4714285 9999999 0.4 0.45 0.432 0.409 0.428571 0.461538 EXTENSION 4 /9 5 /11 16 /37 9 /22 3 /7 6 /13 x = 2.1666... 10x = 21.6666... 9x = 19.5 x = 19.5 = 195 = 9 90 21/6 Pick any terminating decimal. Label it D. Count how many places there are after the decimal point and label this number P. Now multiply D by 10P 10P Reduce the fraction. What have you demonstrated? Decide whether each of the following numbers is rational or irrational. If it is rational, explain how you know. If 12 = 1 Then what does √1 equal? √36? What about √4? If the square root of a number is an integer then it is a perfect square. What about √2? It's not an integer. It's not a terminating decimal. It's not a repeating decimal. Because it does neither of the above it is an irrational number. What about -√2? A B C D E F G 0.333 √4 √2 π = 3.141592.. 11 1/7 = 0.142857 12.3456565656 What other irrational numbers can you think of? Graph √½ and ½ on a number line. | 0 | | | | | | Complete this statement. √½ ½ | | | | Graph √ and | 0 < = > | | | on a number line. | | | | | Complete this statement. √ < | | = > Ordering real numbers Complete this statement. We are going to order these decimals from least to greatest. 4 √8 < => 0.47, 0.474, 0.47, √0.23 First, decide if these numbers are rational or irrational 4 5 √ 45 < = > 1 4 √41 < => 3 < = > √7 Ordering real numbers We are going to order these decimals from least to greatest. 0.52, 0.525, 0.525, √0.276 First, decide if these numbers are rational or irrational Write them to 6 decimal places Write them to 6 decimal places 0.477777... 0.474474... 0.474747... 0.479583... Textbook Page 477 Exercise 3-15 Page 478 Ex 16, 19, 32, 34, 36, 38, 40 0.525252... 0.525525... 0.525555... 0.525357... For each pair of numbers, decide which is larger without using a calculator. Explain your choices. Without using your calculator, label approximate locations for the following numbers on the number line.