Computation using irrational numbers mostly produce irrational results. Double precision floating point uses 52 bits for mantissa with and implied 1 (53 bits of precision for fraction). Some fractional numbers cannot be exactly represented using a finite number of bits (some whole cannot either). A good example of this is the binary value of the decimal 0.1, this is 0.1100110011001100... This is an infinite repeating fraction. In DP FP this fraction will be left shifted by 1 bit and binary exponent diminished by 1 (to account for the shift). the 1. is removed as excepting for 0 and denormalized numbers, it will always be 1. this leave a binary fraction of:
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1001100110011001100110011001100110011001100110011001100110011001100... (infinite irrational)
0000000001111111111222222222233333333334444444444555 (10's bit counter)
1234567890123456789012345678901234567890123456789012 (1's bit counter)
1001100110011001100110011001100110011001100110011010 (rounded to 52 bits)
00000000000000000000000000000000000000000000000000000110011001100... (+error)
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Depending on how and how often you manipulate these numbers, the error grows. Using longer floating point formats postpone the error from creaping into the results beyond acceptible levels, but will not eliminate it from happening. Programmers can work around these errors if need be but in many cases the error is within an acceptible range and can be ignored.
Even before the days of binary computers, numerical computations had to take into consideration error in values. SIN, COS, LN tables were published to finite number of places.
Jim Dempsey
