radix
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000:
The hardware and software on this system was updated September 4th.  Please let me know of any problem you encounter. <caldwell@utm.edu>

We usually write numbers using a place-value system. For example, in decimal 1101 means

1.103 + 1.102 + 0.10 + 1

and in binary 1101 would mean
1.23 + 1.22 + 0.2 + 1 (= 13 in decimal).
In general, a position numbering system encodes the numbers as
anbn + an-1bn-1 + . . . + a2b2+ a1b+ a0 (0 < ai < b, i=0,1,2,...,n)
where the integer b > 1 is the radix or base. So decimal is "radix 10" and binary is "radix 2." Other common systems include octal (radix 8) and hexadecimal (radix 16, here we need to add some more digits: 'a'=10, 'b'=11, 'c'=12, 'd'=13, 'e'=14 and 'f'=15). If it is not clear for the context which radix we are using, we usually add it as a subscript. For example, (23)4 = (21)5 = (11)10 = (b)16.

We can extend this to cover all reals if we allow the inclusion of a negative sign and a radix-point (decimal point):

(anan-1. . . a2a1a0. a-1a-2a-3 . . . )b = anbn + an-1bn-1 + . . . + a2b2 + a1b + a0 + a-1b- 1 + a-2b-2 + a-3b-3 + . . .




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