How to subtract binary numbers

Subtracting binary is another fundamental operation that is used to subtract one binary number from another.

Just like in decimal subtraction, the corresponding bits of the numbers are subtracted from right to left, taking into account the borrows.

Binary subtraction can result in a negative result if the number to be subtracted is greater than the number it is being subtracted from.

Borrow

“Borrow” occurs when a number cannot be subtracted from another due to its value. This happens when the number being subtracted is greater than the number it is being subtracted from.

In binary, this only happens when we try to subtract a 1 from a 0. In this case, the result is ‘1’, and a value is “borrowed” from the next more significant bit.

Example

For example, to subtract the binary number 1101 (13 in decimal) from 1011 (11 in decimal):

  1011
- 1101
______
  1110

The result is 1110, which is equal to -2 in decimal.

Binary subtraction with two’s complement

Binary subtraction is not difficult, but it can be a bit of a pain. It is generally faster using the two’s complement method. This method allows us to convert subtraction into an addition, and we know how to quickly add two binary numbers “by hand”.

To subtract binary numbers using the two’s complement, we follow these steps:

• Convert the number to be subtracted into its two’s complement
• Add the number from which it is being subtracted to the two’s complement
• Discard any overflow bit at the end

Example

Let’s subtract the binary number 1101 (13 in decimal) from the binary number 0100 (11 in decimal) using the two’s complement:

  1101
- 0100
_____
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First, we convert 1011 into its two’s complement. To do this, we invert all the bits and then add 1:

1011 (Original number) => Two's complement

0100 (Inverted bits)
+  1 (Add 1)
_____
0101 (Two's complement of 1011)

2. We add 1101 to the two’s complement of 1011:

       1101 (Original number)
     + 0101 (Two's complement of 1011)
     ______
     1 0010 (Result)

By discarding the overflow bit on the left, we get the final result: 0010, which is equal to 2 in decimal.

The advantages of using the two’s complement for subtractions are that it simplifies the process and eliminates the need to handle borrows.

Bonus: How to quickly subtract 1 in binary

Special bonus, how to quickly subtract 1 from a binary number. A very simple operation, but since it is very common, it is convenient to know how to do it.

Fortunately, subtracting 1 in binary is “a piece of cake”. Simply

• Go through the binary number from the right
• Look for the first 1 you find, and change it to 0
• From that position, all 1s to the right.

For example, if we had to subtract 1 from 1000100000

We look for the first 1

We change it to 0 and, from there to the right, all 1s.

So the result is 1000011111. As you can see, adding one is very quick.