Negative Numbers

Two's Complement

Two's Complement uses a similar number system to binary except the msb or left hand bit is a negative value, meaning for 8 bit two's complement it would be -128 instead of 128 like it is in regular 8bit binary.

-128  	64	32	16	8	4	2	1

We can see that a 1 in the msb position, or the position of -128 would result in the binary number being negative as the other bits 64-1 only total 127. This means that even if there was a 1 in every position a two's complement number of 11111111 in binary would equal -1.

this means that in two's complement if the msb is a 0 the number is positive and if it is a 1 the number is negative.

Therefore we know that the smallest possible value in 8bit two's complement binary is 10000000 = -128 and the largest value is 01111111 = +127.

Method 1

Two's Complement can be used to convert binary numbers from positive to negative, to do this we need to:

1) Write the number is its equivalent positive binary form
2) Add 0's to the number to make it 8 bit
3) Invert each bit, changing 0's to 1's and 1's to 0's
4) Add 1 to the number to make it a two's complement number

For example, represent -41 in two's complement form:

First calculate +41 in binary using your preferred method

41= 32+0+8+0+0+1 = 101001

Then add 0's to make it 8 bit. 00101001

Then Invert the bits. 11010110.

Then Add 1 to the number. 11010110+1= 11010111.

To check our answer we can convert the number to denary, remembering that the msb represent -128.

11010111 = -128+64+16+4+2+1 = -41

Method 2

There is one other method for representing numbers in two's complement form, without using calculations. To do this we need to:

1) Write the number is its equivalent positive binary form
2) Add 0's to the number to make it 8 bit
3) Starting from the right and going left find the first 1 and keep it
4) Invert each bit after, changing 0's to 1's and 1's to 0's, but don't invert the 1 you kept or any 0's to the right of it

For example, represent -46 in twos complement:

First calculate +46 in binary using your preferred method

46= 32+0+8+4+2+0 = 101110

Then add 0's to make it up to 8bit

Then find the first one and keep it 00101110

Then invert the bits excluding the 1 you kept and all 0's to the right of it. 11010010

To check our answer we can convert the number to denary, remembering that the msb represent -128.

-128+64+16+2= -46