* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Rational numbers and their decimal representation
Survey
Document related concepts
Infinitesimal wikipedia , lookup
Large numbers wikipedia , lookup
History of logarithms wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Location arithmetic wikipedia , lookup
Mechanical calculator wikipedia , lookup
Approximations of π wikipedia , lookup
Transcript
1 Surds and indices Rational numbers and their decimal representation All rational numbers can be expressed as either & decimals which terminate, such as 0.72, or & recurring decimals, such as 0:102 432 432 43 . . . 0:102 432 43 . . . can be written _ The dots indicate as 0:102_ 43. the set of digits which recur. Conversely, all decimals that terminate or recur represent rational numbers. A method for converting a terminating decimal into a fraction is shown in the Example below. Irrational numbers cannot be expressed as terminating or recurring decimals. Conversely an infinite decimal that never recurs cannot be expressed as a rational number. For example, in 0:101 001 0001 . . . the number of zeros increases by one before each digit 1, so the decimal never recurs and it cannot be expressed as a ratio of two integers. To sum up: Rational numbers can be represented by terminating or recurring decimals and vice versa. Irrational numbers are non-terminating, non-recurring decimals and vice versa. Example Express decimals as fractions in their lowest terms. a b 43:72 0:7_ Let Then 4372 100 43.72 is 4372 hundredths. 0:2_ 4_ Let Then 7 1 100 2 The fraction cannot be cancelled further. It is in its lowest terms. x 0:777_ 1 10x 7:777_ 2 Divide both sides by 9. 1 100x 24:2_ 4_ 2 99x 24 24 x 99 8 33 Multiply both sides by 10. Since one digit recurs, this lines up the recurring digits. Subtract line 1 from line 2 . x 0:2_ 4_ x 1 10 Divide numerator and denominator by 4. 1093 25 9x 7 7 x 9 c 10 1 4 3 Multiply both sides by 100. Since two digits recur, this lines up the recurring digits. Subtract line 1 from line 2 . Divide both sides by 99. Cancel to lowest terms. AS Core for OCR # Pearson Education Ltd. 2004 1 1 Surds and indices For an alternative method for converting recurring decimals to fractions, see Example 27 on page 324. Example Express fractions as decimals. a 3 12 0:12 25 100 b 6 7 By multiplying numerator and denominator by 4, the denominator can be made into a power of 10 (i.e. 10, 100, 1000, etc. . . .). The conversion to decimal form is then simple. When the denominator cannot be made into a power of 10, the numerator must be divided by the denominator. Divide 6 by 7 0:8 5 7 1 4 2 8 . . . 7j6:04 05 01 03 02 06 0 . . . The group of six digits will recur. 6 _ 142_ 0:857 7 Note Division of the numerator by the denominator can always be used to convert a fraction to a decimal. Use of calculators Many calculators have a fraction button and, with practice, all fractions and their equivalent decimal form can be found using a calculator. The serious student must aim, however, to be fluent with numerical work and this means acquiring the ability to calculate speedily and accurately with and without a calculator. Being skilful with numbers is satisfying as well as an important basis for further mathematical development. The Exercise below should be completed without a calculator. A calculator can be used to check that an answer is correct or, at least, to provide supporting evidence. For example, p p 2 6 2 p 17 3 2 1 Evaluating both these expressions on the calculator gives 0.4361 correct to 4 d.p. This does not prove that the expressions are equivalent but it does provide some support and reassurance! Exercise 2 1 Express each of these decimals as a fraction in its lowest terms. a 0:6_ b 0:4_ c 0:1_ 2_ d _ _ _ _ _ e 0:12 f 3:407 g 0:48 h AS Core for OCR # Pearson Education Ltd. 2004 0:305_ 4:2_ 1 2 Express each of these fractions in decimal form. 2 b 125 c 59 a 507 e 44 7 f 1 6 g 3 8 Surds and indices d 4 11 h 5 7 3 Explore the decimal representation of all proper fractions (< 1) with these denominators. a 7 b 9 c 11 d 13 e 19 4 What must be true of the integers a, and/or b, if ab terminates when expressed in decimal form (assume that ab is in its lowest terms)? AS Core for OCR # Pearson Education Ltd. 2004 3