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base:fixed_point_arithmethic

# Fixed point arithmethic

A fixed-point number representation is a number that has a fixed number of digits before and after the radix point (e.g. “.” in English decimal notation).

In terms of binary numbers, each magnitude bit represents a power of two, while each fractional bit represents an inverse power of two. Thus the first fractional bit is ½, the second is ¼, the third is ⅛ and so on.

8:8 Fixed Point representation is the most straightforward approach (in fact the only sane approach when coding on the c64).

for example:

integer.fractional

`00001101.01010000`

represents the number:

integer part:

`1*2^3+1*2^2+0*2^1+1*2^0`

fractional part:

`0*(2^-1)+1*(2^-2)+0*(2^-3)+1*(2^-4)`

giving us:

`1*2^3+1*2^2+0*2^1+1*2^0  +  0*(2^-1)+1*(2^-2)+0*(2^-3)+1*(2^-4) = 13.3125`

It's easyer to think of a 8.8 fixed number in a way that you have a 1 byte integer part, and a 1 byte fractional part where the fractional part represents a number which is: fractional part* 1/256.

Repeating the example above:

integer.fractional

`00001101.01010000`
`%01010000 = 80 decimal => 80*1/256 = 0.3125`

You may totally forget about fractional parts and just threat the two 8 bit numbers as a straight representation of numbers from 0-65536: a 16 bit number when working with numbers like this. In reality a fixed point number will be always just a bunch of bits, and what makes it fixed point is only how you think about it. :) 