12-Representations-of-Numbers-4UP

12-Representations-of-Numbers-4UP - Decimal representations...

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Decimal representations Alternative bases Conversions Representations of Numbers Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) Representations of Numbers MATH 2070U 1 / 26 Decimal representations Alternative bases Conversions Representations of Numbers 1 Decimal representations of numbers 2 Alternative representations of number with other bases 3 Conversion between numeric representations c ± D. Aruliah (UOIT) Representations of Numbers MATH 2070U 2 / 26 Decimal representations Alternative bases Conversions Decimal representations of numbers Recall: 71.32 actually means 7 1 . 3 2 = 7 × 10 + 1 × 1 + 3 10 + 2 100 = 7 × 10 1 + 1 × 10 0 + 3 × 10 - 1 + 2 × 10 - 2 7 tens + 1 unit + 3 tenths + 2 one hundredths Decimal point separates integer part from fractional part Decimal system favoured by most modern humans c ± D. Aruliah (UOIT) Representations of Numbers MATH 2070U 4 / 26 Decimal representations Alternative bases Conversions Relevance of representations XLIII × XXXVII MDXCI 43 × 37 301 1290 1591 c ± D. Aruliah (UOIT) Representations of Numbers MATH 2070U 5 / 26

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Decimal representations Alternative bases Conversions Rational numbers Rational numbers : ratio of two integers (fractions) e.g., 1 3 , 2 5 , 1 4 , - 21 137 , . . . Q R denotes set of all rational numbers Decimal representations: ﬁnite or repeating 1 4 = 0.25 = 2 × 10 - 1 + 5 × 10 - 2 2 5 = 0.4 = 4 × 10 - 1 1 3 = 0. 3 = 0. 333 ··· = 3 × 10 - 1 + 3 × 10 - 2 + 3 × 10 - 3 + ··· = k = 1 3 × 10 - k = 3 10 k = 0 ± 1 10 ² k = 3 10 ± 1 1 - 1/10 ² c ± D. Aruliah (UOIT) Representations of Numbers MATH 2070U 6 / 26 Decimal representations Alternative bases Conversions Irrational numbers Irrational numbers : real numbers not expressible as ratios 3 2 = 4.24264068711929 ··· , π = 3.14159265358979 ··· , e = 2.71828182845905 ··· , etc. Irrational numbers not ﬁnitely representable in decimal form Computer decimal representations of numbers have to be approximate c ± D. Aruliah (UOIT) Representations of Numbers MATH 2070U 7 / 26 Decimal representations Alternative bases Conversions Alternative number bases Not restricted to decimal (base 10) representations Different civilisations have used different number bases
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This note was uploaded on 02/23/2010 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Spring '10 term at UOIT.

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12-Representations-of-Numbers-4UP - Decimal representations...

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