In many computer science related fields, you are going to encounter or require the knowledge of Two’s complement.

Two’s complement is a way of representing negative numbers in binary.

Two’s complement is the way every computer chooses to represent integers.

**Implementation**

For example, let’s use an 8 bit long binary number:

1 2 3 |
0 0 0 0 0 0 0 0 | 8 bit long 128 64 32 16 8 4 2 1 | Value Represented 0 0 0 1 0 1 0 0 | Number chosen is 20 |

So the 8 bit number we chose is equivalent to `20`

in decimal format. But what if we are looking for the negative counterpart; `-20`

?

To find `-20`

in binary form, the most convenient way is as follows:

First you find the least significant bit- the smallest non-zero bit, which is `100`

of binary (equivalent to 4 in decimal). You keep the `100`

of binary as is, but the values of the rest of the bits to the left is flipped. Like so:

1 |
1 1 1 0 1 1 0 0 | This is Two's Complement for -20 |

**Binary Addition with Two’s Complement**

The great thing about Two’s Complement is that math works the same way when you are adding and subtracting numbers in binary; whether they are in Two’s Complement or not. So to become familiar with how to do binary addition with Two’s Complement, let’s evaluate `24 + (-20)`

.

1 2 3 4 5 6 |
0 0 0 1 1 0 0 0 | Number chosen is 24 + 1 1 1 0 1 1 0 0 | Two's Complement -20 ---------------------------------- 1 0 0 0 0 0 1 0 0 | We are working with 8 bits, so drop the first 1 0 0 0 0 0 1 0 0 | equivalent to 4 |

But there is a caveat! The number `1`

at the far left of the resulting number is dropped or doesn’t count! That’s because we are working with Two’s Complement or 8 bit numbers, not 9 bits. The result you get is `00000100`

, which is equivalent to 4.