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l14 - Real Numbers How do we represent real numbers Several...

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Unformatted text preview: Real Numbers How do we represent real numbers? Several issues: • How many digits can we represent? • What is the range? • How accurate are mathematical operations? • Consistency... Is a + b = b + a ? Is ( a + b ) + c = a + ( b + c ) ? Is ( a + b )- b = a ? Real Numbers How do we represent real numbers? Several issues: • How many digits can we represent? • What is the range? • How accurate are mathematical operations? • Consistency... Is a + b = b + a ? Is ( a + b ) + c = a + ( b + c ) ? Is ( a + b )- b = a ? Fixed Point Basic idea: 0 1 0 0 1 0 1 0 radix point is here Choose a xed place in the binary number where the radix point is located. For the example above, the number is (010 . 01010) 2 = 2 + 2- 2 + 2- 4 = (2 . 3125) 10 How would you do mathematical operations? Floating-Point Some problematic numbers.... 6 . 023 × 10 23 6 . 673 × 10- 11 6 . 62607 × 10- 34 Scientic computations require a number of digits of precision... But they also need range ⇒ permit the radix point to move ⇒ oating-point numbers Floating-Point: Scientic Notation sign 6.023 x 10 23 exponent mantissa radix (base) + • Number represented as: – mantissa, exponent • Arithmetic – multiplication, division: perform operation on mantissa, add/subtract exponent – addition, subtraction: convert operands to have the same exponent value, add/subtract mantissas Fixed Point Basic idea: 0 1 0 0 1 0 1 0 radix point is here Choose a xed place in the binary number where the radix point is located.radix point is located....
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