Two’s complement

This is a way to represent signed integers. Assuming the left most digit is the most significant bit then:

sign digit | numerical representation.

So assume you store your signed integer as a byte then left most bit would be for the sign and the other 7 bits would be for the number.

Positive numbers

For example when representing 10 in 7 bits you would use

therefore as 10 is positive it’s full representation would be

0|000 1010

with the 0 representing positive numbers.

Negative numbers

This is where stuff gets freaky - lets first start with the formula and go backwards to explain why this works.

Lets say we want to represent -10:

1. Start with the representation of the absolute value of number in binary e.g. 10 = 0000 1010
2. Flip all the digits e.g. 1111 0101
3. Add 1 to the representation and ignore overflow (this is pretty key as is a legit option) e.g. 1111 0110

Ok great, so it has the left most digit being 1 (so not positive) and this seems to be unique. This might take a bit more maths but we actually use all the ‘space’ we have in the byte. 0111 1111 is , 0000 0000 is , 1000 0000 is - and 1111 1111 is .

Lets just cover how to go decimalise a binary representation of a negative number as well suppose we wanted to convert 1100 1100:

1. First subtract 1 from the expression (or add -1 but we will come back to that) e.g. 1100 1011.
2. Flip all the digits e.g. 0011 0100.
3. Convert the binary number into decimal representation and put a negative sign on it e.g. 32 + 16 + 4 = 52, so the number is -52.

That seems easy enough but there has to be simpler way to do this right?

Addition

You add two numbers just as you would unsigned integers by starting from the least significant bit (right most in our example) and carry the overflows forward. This works even when adding two negatives or a negative with a positive!

Two positive

Lets add 10 = 0000 1010 with 12 = 0000 1100

Negative with a positive

Lets add -5 = 1111 1011 with 10 = 0000 1010

Two negative

Lets add -5 = 1111 1011 with -16 = 1111 0000

Over flow detection

You can use the sign digit to detect overflow i.e. if you add two positive numbers the outcome should be positive!

Comparison

This essentially works the same as unsigned integers starting from the most significant bit work your way to the least until you see a difference. Which ever is 1 in the different place is larger.

The only exclusion to this is if the signed bits of the two numbers are different. Then the positive number is larger!

Conclusion

This does seem like a pretty good method to encode negative numbers … you don’t need to add in much additional logic on top of unsigned integers.