# radix

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

1^{.}10^{3}+ 1^{.}10^{2}+ 0^{.}10 + 1

and in binary 1101 would mean

1^{.}2^{3}+ 1^{.}2^{2}+ 0^{.}2 + 1 (= 13 in decimal).

In general, a **position numbering system** encodes
the numbers as

a+_{n}b^{n}a-1_{n}b-1 + . . . +^{n}a_{2}b^{2}+a_{1}b+a_{0}(0 ≤a_{i}<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):

(a-1_{n}a_{n}^{. . .}a_{2}a_{1}a_{0}.a_{-1}a_{-2}a_{-3}^{ . . . })=_{b}a+_{n}b^{n}a-1_{n}b-1 + . . . +^{n}a_{2}b^{2}+a_{1}b+a_{0}+a_{-1}b^{- 1 }+a_{-2}b+^{-2}a_{-3}b^{-3}+ . . .

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