How to convert ( 11011.0101 ) from binary to decimal?

How do we convert this binary number to decimal? I am confused about the decimal point, the one before the decimal I got (27), how about the part after the decimal point?

• Context is required. What is the number supposed to represent? Jan 18 '17 at 7:15
• <pedantry>It's not a 'decimal point' if the number is binary- you can call it a radix point.</pedantry> Jan 18 '17 at 14:01

4 Answers

Often the expansion is performed like this:

$$1\times2^4+1\times2^3+0\times2^2+1\times2^1+1\times2^0+0\times2^{-1}+1\times2^{-2}+0\times2^{-3}+1\times2^{-4}$$

This is analogous to the usual base-10 positional representation except the 10s are replaced with 2s.

The positions to the right of the point are treated the same way as the ones before: each step to the right halves the weight. Hence

binary   | decimal
------------------
100.     |    4
10.     |    2
1.     |    1
0.1    |  1/2
0.01   |  1/4
0.001  |  1/8


With these weights you should be able to calculate the decimal value of any binary number, including ones like you show with a .

• +1, all the answers say essentially the same thing (they have to) but this one says it most clearly Jan 18 '17 at 9:47
• @Neil_UK Wouter is a teacher, after all. +1. Jan 18 '17 at 14:05

The place values go with $2^n$, where $n$ is positive on the left side of the decimal point and negative on the right side of the decimal point.

In this case,

         1   1   0   1   1  .  0      1      0      1
power    4   3   2   1   0    -1     -2     -3     -4
value   16   8   4   2   1     0.5    0.25   0.125  0.0625


Result: 27.3125

Split it in 2 parts: a) 11011 which is 1 + 2 + 8 + 16 = 27 b) 0101 which would be converted this way:

 0 * 1/2 + 1 * 1/4 + 0 * 1/8 + 1 * 1/16 = 5/16 = 0.3125


Number in decimal is 27.3125

you can check the following link for more examples Floating To Decimal

• How did you get 27.9375? Jan 18 '17 at 7:29
• @alex.forencich typo! sorry Jan 18 '17 at 7:38