Representing $-2.5$ as a floating point number

Question

I am trying to understand the following:

For example the number $-2.5 = -1*1.25*2^1$ is strored as:
$S = 1$, Exponent $= 1+127 = 128$, Mantissa = $0.25 $

This got me. How do you connect all these numbers to yield $-2.5$ ?

You understand that the computer processor numbers are in binary? The above corresponds to the standard conventions for 32 bit float numbers. See stack-overflow for daily discussions of properties and limitations of finite length floating point numbers. — Lutz Lehmann, Feb 23 '14 at 17:23
The Wikipedia page on single-precision numbers goes into great detail about this and gives several examples. — AnonSubmitter85, Feb 23 '14 at 17:33

score 3 · Accepted Answer · answered Feb 23 '14 at 18:10

3

If you are given a sign bit $S$, 127-offset exponent $e$, and mantissa $m$, you calculate the value as

$$(-1)^S2^{e - 127}(1 + m)$$

So for your values $S=1$, $e=128$, and $m=0.25$,

$$(-1)^1 2^{128-127}(1 + 0.25) = -2.5$$

answered Feb 23 '14 at 18:10

NovaDenizen

score 1 · Answer 2 · answered Feb 23 '14 at 17:31

1

Actually, the number $-2.5$ is stored according to the division 32=1+8+23 as

1 | 1000 0000 | 0100 0000 0000 0000 0000 000

according to the bit pattern that you found, i.e., $-2.5=(-1)^1\cdot 2^{128-127}\cdot (1.01)_2$.

answered Feb 23 '14 at 17:31

Lutz Lehmann

2 Answers2