Subtracting binary numbers

The most common way of subtracting binary numbers is done by first taking the second value (the number to be subtracted) and apply what is known as two's complement, this is done in two steps:

1. complement each digit in turn (change 1 for 0 and 0 for 1).
2. add 1 (one) to the result.

note: the first step by itself is known as one's complement.

By applying these steps you are effectively turning the value into a negative number, and as when dealing with decimal numbers, if you add a negative number to a positive number then you are effectively subtracting to the same value.

In other words 25 + (-8) = 17, which is the same as writing 25 - 8 = 17.

An example, let's do the following subtraction 11101011 - 01100110 (235₁₀ - 102₁₀)

note: When subtracting binary values it is important to maintain the same amount of digits for each number, even if it means placing zeroes to the left of the value to make up the digits, for instance, in our example we have added a zero to the left of the value 1100110 to make the amount of numerals up to 8 (one byte) 01100110.

First we apply two's complement to 01100110

which gives us 10011010.

Now we need to add 11101011 + 10011010, however when you do the addition you always disregard the last carry, so our example would be:

which gives us 10000101, now we can convert this value into decimal, which gives 133₁₀

So the full calculation in decimal is 235₁₀ - 102₁₀ = 133₁₀ (correct !!) 

reference: helpwithpcs.com