05 - Lecture 5 Today’s class • Representations of...

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Unformatted text preview: Lecture 5 Today’s class: • Representations of numbers in different bases • Floating-point numbers & machine representations • Arithmetic with floating-point numbers Material on floating-point numbers, machine arithmetic, and rounding errors introduces many extremely technical subtleties. Unfortunately, it is often required (even for non- computer scientists) to understand these technicalities as consumers of numerical software! • Taylor’s theorem and truncation errors (next time) Representations of numbers XLIII × XXXVII MDXCI 43 × 37 301 1290 1591 Decimal representations • Recall: 71 . 32 actually means 7 1 . 3 2 = 7 · 10 + 1 · 1 + 3 10 + 2 100 = 7 · 10 1 + 1 · 10 + 3 · 10- 1 + 2 · 10- 2 7 tens + 1 unit + 3 tenths + 2 one hundredths integer part = 71 , fractional part = 0 . 32 • Positional notation needs digit zero & decimal point • Decimal system favoured by most modern humans Floating-point format • Floating-point format : scientific notation • Normalised: shift decimal point ( × powers of 10)- 1 . 3701542 · 10 3 =- 1370 . 1542 9 . 376 · 10- 7 = 0 . 0000009376 • Terminology: if x = s · 10 e = ( a .a 1 a 2 ...a t ) 10 · 10 e – s = ( a .a 1 a 2 ...a t ) 10 = significand – f = (0 .a 1 a 2 ...a t ) 10 = mantissa (fraction) – t = precision = # digits in mantissa – e = exponent of base b = 10 Different number systems • Different civilisations used different numeral bases • Number systems distinguished by...
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This note was uploaded on 06/20/2011 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Winter '10 term at UOIT.

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05 - Lecture 5 Today’s class • Representations of...

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