Subtraction as Addition
To make the discussion simple I'm going to use 4 bit integer, int4
throughout the post.
NOTE: I expect you to have a basic understanding of binary number system. If you're not familiar with it, take a moment and have a look at this article first.
What's two's complement?
It's one way of representing signed integers^{1}. int4
can represent 2^4 = 16
different numbers, and two's complement assigns 16 integers ranging from 8 to 7 to each bit pattern. The mapping rules go like this:
 Map from 0 to 7 to binary numbers starting from
0000
to0111
. This is same as how we represent positive integers in binary.  For negative numbers, assign the smallest negative number 8 to
1000
, then assign 7 to the one bigger binary1001
, 6 to1010
…and so on up to1111
.1111
is going to be 1.
If they aren't very clear, have a look at the table showing full mapping below and examine the rules again.
Decimal  Binary 

7  0111 
6  0110 
5  0101 
4  0100 
3  0011 
2  0010 
1  0001 
0  0000 
1  1111 
2  1110 
3  1101 
4  1100 
5  1011 
6  1010 
7  1001 
8  1000 
Just by looking at it, it doesn't make much sense why it does such a tricky mapping for negative numbers. There's a reason powerful enough to justify this trickiness.
Why is it so widely used?
The reason why two's complement is widely used is it makes hardware simple in a way that it enables hardware to use the addition unit to also calculate subtraction.
You may be surprised to know that computers don't do subtraction. What it actually does instead is addition of the negative number. For example, 5  3
is calculated as 5 + (3)
inside a computer. By converting subtraction to addition, it's now free from minding how to go about subtraction, which is a big deal from hardware engineering point of view.
However, we need to carefully pick a way to represent negative numbers in order to get correct answers. Two's complement is one of such representations^{2} that makes it happen. Let's look at an example.
Let's do 5 + (3)
. In two's complement 5 and 3 are represented as 0101
and 1101
respectively. The addition of these two is 0101 + 1101 = 0010
. If you look back at the table in the previous section, you can find it represents 2, which corresponds to the answer of 5 + (3)
.
You may still wonder "so what? what's so special about this?" because my first impression was like that. Let's see what happens if it did NOT use two's complement.
Another way of representing minus numbers
I'm going to use the first bit solely as a flag that indicates sign: 0 to be positive, and 1 to be negative. The rest of three bits are used to represent number part, for example, 0001
is 1 and 1001
is 1.
This is the mapping table of this system.
Decimal  Binary 

7  0111 
6  0110 
5  0101 
4  0100 
3  0011 
2  0010 
1  0001 
0  0000 
0  1000 
1  1001 
2  1010 
3  1011 
4  1100 
5  1101 
6  1110 
7  1111 
Probably this is more intuitive way of representing negative numbers. This system is called signed magnitude representation, and was actually used in early computers. Be aware it has two zeros.
Now let's see how 5 + (3)
looks like in this system.
In this system, 5 and 3 are represented 0101
and 1011
respectively. If we add these two binary numbers, the result is 0101 + 1011 = 0000
. 0000
is simply 0, which is clearly wrong answer for 5 + (3)
.
This means it can't use the addition hardware to perform this calculation, and to get the right answer it needs another hardware dedicated for subtraction.
Being able to calculate subtraction as addition is a very unique characteristic two's complement has, and it's the reason why it's so popular in computing systems.
Footnotes

Signed numbers: Numbers that consist of both positive and negative numbers ↩

Ones' complement is another way to realise subtraction as addition. See Stack Overflow — Advantage of 2's complement over 1's complement? if you're interested in why two's complement is more preferred. ↩