Two's complement an negative numbers

Negative numbers: Two's complement

Another method of representing?signed?numbers is?two's complement. Most computers use this method to represent negative numbers. This method can be more effective when performing mathematical operations like adding and subtracting.

With two's complement, the?bit?at the far left of the bit pattern - the?most significant bit?or?MSB?- is used to indicate positive or negative and the remaining bits are used to store the actual size of the number. Positive numbers always start with a 0.

Four-bit, positive, two's complement numbers would be 0000 = 0, 0001 = 1, up to 0111 = 7. The smallest positive number is the smallest?binary?value.

Negative numbers always start with a 1. The smallest negative number is the largest binary value. 1111 is -1, 1110 is -2, 1101 is -3, etc down to 1000 which represents -8.

Computers sometimes need to work with negative numbers, and as binary is only made up of 0s and 1s, showing negative numbers could be a problem. Most computers use ‘two’s complement’ to solve this. With this method, the most significant bit – the bit on the far left of the sequence - is used to indicate positive or negative – 0 is positive, 1 is negative.?To work out a negative number with two’s complement, you first have to find its equivalent positive binary number. Let’s say we want to show -1.?The number 1 is represented as 001 in binary.??You then add another 0 to the front of that number to show it is positive, and then you invert, or ‘find the complement of’, each bit in that number. Then add 1.??As the most significant bit is a 1, we know that this is a negative number. That’s two’s complement!?

Using two's complement for negative numbers

1. Find the positive binary value for the negative number you want to represent.
2. Add a 0 to the front of the number, to indicate that it is positive.
3. Invert or find the complement of each bit in the number.
4. Add 1 to this number.

Examples

Find -1 using two's complement numbers

1. 1 = 001
2. Adding 0 to the front becomes?0001
3. 'Inverted' becomes?1110
4. Add 1 =?1111?(-8 + 4 + 2 + 1 = -1)

Find -4 using two's complement numbers

1. 4 = 100
2. Adding 0 to the front becomes?0100
3. 'Inverted' becomes?1011
4. Add 1 =?1100?(-8 + 4 = -4)