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Transcript
07. Decimals 07-1 Decimals (1) Definition: Decimals are numbers expressed with a decimal point. The decimal system is based on powers of ten (base ten system). whole number part Place Value: The value of a digit depends on its place in the number. Each place has a value of 10 times the place to its right decimal part ex.) 5.26 5 0.2 0.06 (5 1) (2 0.1) (6 0.01) (5 100) (2 101) (6 102) Every number to the right of the decimal point is a decimal fraction. ex.) 0.75 “seventy-five hundredths” ex.) 45.321 “forty-five and three hundred twenty-one thousandths” (2) Rounding off Decimals Rounding off is a kind of estimating. Look at the digit to the right of the “rounding digit”. If the digit is 5 or more, round up by adding one to the rounding digit and drop all digits to the right of it; otherwise, do not change the rounding digit but drop all digits to the right of it. [Example] Round off 3815.296. Round to the nearest: thousand: 4000 hundred: 3800 tenth: 3815.3 ten: 3820 hundredth: 3815.30 one: 3815 07-2 Classification of Decimals Decimals Finite decimals (or terminating) Rational Numbers Repeating decimals Infinite decimals Non-repeating decimals Irrational Numbers Terminating decimals and repeating decimals are rational numbers and can be written in the form of a fraction. Irrational numbers cannot be expressed as fractions. [Note] When a fraction is converted to a decimal, it is as either a terminating decimal or a repeating decimal. (a) Fractions representing terminating decimals (i) Terminating decimals can be represented by the fraction of which the denominator has a power of 10. 216 216 4 1485 1485 ex.) 0.4 , 2.16 , 1.485 10 1000 100 103 102 (ii) When terminating decimals can be represented by fractions in simplest form, there are only prime factors of 2 and 5 in the denominator. 3 3 ex.) There are only prime factors of 2 and 5 in the denominator. Terminating decimal 50 2 52 3 2 52 32 2 52 2 6 22 5 2 6 0.06 The denominator has a power of 10. 100 (b) Fractions representing repeating decimals: If there is any prime factor other than 2 or 5 in the denominator, the fraction cannot be a terminating decimal but a repeating decimal. 7 7 ex.) There is a prime factor of 3 other than 2 or 5 in the denominator. Repeating decimal 15 35 Repeating Decimals (a) Repeating decimals: An infinite repeating decimal is one that has a specified sequence of digits that repeat. ex.) 0.263737373737… (b) Repeating block (c) The repeating decimal can be represented by putting a dot over the digit or the sequence of digits that repeat. ex.) 0.166666666 can also be written as 0.16 I. Number and Operations (d) Fractions that can be represented by repeating decimals: Fractions of which the denominator has any prime factor other than 2 or 5. ex.) 7 7 0.2333··· 0.23 30 2 3 5 Repeating decimals Repeating block 3.1232323··· 23 0.555··· Representation 0.5 5 1.234234234··· 3.123 1.2 34 234 Converting Repeating Decimals to Fractions (a) Let N be the given repeating decimal (b) Make two equations by multiply both sides of the equation by a power of 10 so that the repeating block cancels out (c) Subtract N from both sides ex.) Converting a repeating decimal of a .bcd to a fraction (i) N a .bcd (ii) (i) 1000 1000 N abcd.cdcdcd ··· (i) 10 10 N ab.cdcdcd ··· [Formula] (iii) subtract both sides: whole number without decimal point abcd ab a .bcd 990 2 repeating decimals 1 non-repeating decimal after decimal point number without a repeating decimal 990 N abcd ab N abcd ab 990 The number of digits in the repeating block 2 Denominator: write 9 for every repeating decimal and 0 for every number that does not have a repeating decimal Numerator: (whole number) (number without a repeating decimal) a ab (i) 0. a (ii) 0. a b 9 99 ab a (iii) 0. a b 90 ex.) 3.792 3792 3 999 591 59 a b c a b (iv) 0. a b c 900 3789 421 999 111 532 266 90 90 45 472 4 468 26 0.472 990 990 55 5.91 I. Number and Operations