How convert 1/2 to binary?

Question

How convert 1/2 to binary? isn't it 0.1111 $1/2$ result $0$ $(1/2)*2=1$ result $0.1$ $2*2>1$ result $0.11 $ etc.

Answer 1

To expand upon Gerry Myerson's excellent comment,

If you're working in base 10, and want to get 1/10, you do two steps, as shown in the diagram below

• Write the numerator, 1, as 01. (add in the decimal point and a leading zero)

• Move the decimal point one space to the left, to get 0.1

So if you're working in base 2, and want to get 1/2, you need to do similar: write 1 in base 2, add the binary point and leading zero, and then move the binary point.

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Each digit to the right of the binary point is some negative power of 2. So

$$ \text{binary } 0.1111 = 0\times2^0 + 1\times2^{-1} + 1\times2^{-2} + 1\times2^{-3} + 1\times2^{-4}$$

In decimal this is 0.9375. The more 1s you add after the binary point the closer you get to 1.

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In case you're wondering how 1/2 is stored in a computer it is held in "scientific notation" style:

$$ 1/2 = 1 \times 2^{-1}$$

The computer stores the mantissa (1) and the exponent($-1$). (In the case of signed numbers things are a little more complicated. Easy to find info on the internet, e.g. here.)