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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?

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    \$\begingroup\$ Context is required. What is the number supposed to represent? \$\endgroup\$
    – Passerby
    Commented Jan 18, 2017 at 7:15
  • \$\begingroup\$ <pedantry>It's not a 'decimal point' if the number is binary- you can call it a radix point.</pedantry> \$\endgroup\$ Commented Jan 18, 2017 at 14:01

4 Answers 4

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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.

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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 .

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    \$\begingroup\$ +1, all the answers say essentially the same thing (they have to) but this one says it most clearly \$\endgroup\$
    – Neil_UK
    Commented Jan 18, 2017 at 9:47
  • \$\begingroup\$ @Neil_UK Wouter is a teacher, after all. +1. \$\endgroup\$ Commented Jan 18, 2017 at 14:05
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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

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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

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    \$\begingroup\$ How did you get 27.9375? \$\endgroup\$ Commented Jan 18, 2017 at 7:29
  • \$\begingroup\$ @alex.forencich typo! sorry \$\endgroup\$
    – W1ldworm
    Commented Jan 18, 2017 at 7:38

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