# Two's Complement Explained

**Binary makes it really easy to add two numbers together, but adding a negative number is a little trickier**. We run into a number of problems when trying to represent negative numbers in binary. To solve this we use Two's Complement, or **signed notation**. Two's complement is how we convert numbers to negative. Let's look at an 8-bit example. Using 8-bit means we can use numbers up to 255.

```
// Below is how we represent 9
0000 1001
// This means 2^0 + 0^1 + 0^2 + 2^3 + 0^4 + 0^5 + 0^6 = 9
```

Two's Complement is essentially a way to take these 8 bits, and also represent negative numbers. It will also work for other lengths of bits (i.e. 16-bit, or 32-bit) **In Two's Complement, the first bit is used to represent if the number is positive or negative**, so we can't use it to store data. As such, for our 8-bits, we can store values between 127 and -128 using 8 bits and Two's Complement.

**To do Two's Complement**, the first thing we do is invert the number except the first bit.

```
// Below is how we represent 9
0000 1001
// This means 2^0 + 0^1 + 0^2 + 2^3 + 0^4 + 0^5 + 0^6 = 9
// Let's invert our binary, excluding the first bit..
0111 0110
// And since we want to indicate it's negative, we set the first bit to 1.
1111 0110
// Finally, we add 1 to this number.
// So negative 9 (-9) can be represented by the following:
1111 0111
```

**In short, to calculate Two's Complement:**

- Invert the binary, and change the first digit to a 1, so
**0000 1001**becomes**1111 0110** - Add 1 to the number, so
**1111 0110**becomes**1111 0111**

So our representation of **-9** in Binary using Two's Complement is **1111 0111**. To calculate this mathematically, **we need to subtract the number in the first position from the total**. That means:

**2 ^{0} + 2^{1} + 2^{2} + 0^{3} + 2^{4} + 2^{5} + 2^{6} - 2^{7} which is equal to -9**

Let's look at how we might use this in calculation:

```
// Let's try doing some arithmetic with Two's Complement.
// The below number represents 8
0000 1000
// And then we have our -9
1111 0111
// Let's try and add these, i.e. 8 + -9. Adding binary is the same as with decimals
// That means we just add the numbers above and below each other, and carry over anything else.
1111 0111
0000 1000 +
-----------
1111 1111
```

We know that **8 + -9 = -1**. If we convert **1111 1111** back to decimal, using the method we previously used:

**2 ^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{4} + 2^{5} + 2^{6} - 2^{7} = -1**

This is a simple example, so here is another version with carries:

```
// The below number represents 14
0000 1110
// And then we have our -9
1111 0111
// Let's try adding 14 + -9, or 14 - 9
// Since we have to carry numbers here, that working out is shown as well
1111 11
0000 1110
1111 0111 +
-----------
0000 0101
// The final carried 1 is ignored since it is outside our 8 bit space, leaving us with only 8 bits rather than 9.
```

So we know that **14 - 9 = 5**, is that what our binary shows too? Let's see:

**2 ^{0} + 0^{1} + 2^{2} + 0^{3} + 0^{4} + 0^{5} + 0^{6} - 0^{7} = 5**

In summary, **Two's Complement** makes it super easy to add and subtract numbers.