Numeral system

:Occasionally the term "number system" is used for this concept, but that is also the name of an altogether different concept; see number system.

Change of radix

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7:

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1020304 / 7 = 145757 r 5

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145757 / 7 = 20822 r 3

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20822 / 7 = 2974 r 4

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2974 / 7 = 424 r 6

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424 / 7 = 60 r 4

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60 / 7 = 8 r 4

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8 / 7 = 1 r 1 => 11446435

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E.g., 10110111 base 2 into base 5:

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10110111 / 101 = 100100 r 11 (3)

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100100 / 101 = 111 r 1 (1)

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111 / 101 = 1 r 10 (2) => 1213

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To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately even a terminating fraction in one base may not terminate in another. E.g., 0.1A4C base 16 into base 9:

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0.1A4C × 9 = 0.ECAC

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0.ECAC × 9 = 8.520C

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0.520C × 9 = 2.E26C

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0.E26C × 9 = 7.F5CC

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0.F5CC × 9 = 8.A42C

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0.A42C × 9 = 5.C58C => 0.082785...

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