Representing Negative Numbers

The issue with the binary system is that there is no way to represent negative numbers, here we will introduce 3 systems designed to solve that problem:

1. Sign-and Magnitude
2. 1s Complement
3. 2s Complement

Signed-and-Magnitude

Use the first bit to represent the sign.

• 0 for + (positive)
• 1 for - (negative) For example,
• 00110100 represents 52
• 10110100 represents -52

Given n-bits, the range of value it can represent is: $[-(2^{n-1}-1),2^{n-1}-1]$

1s Complement

• Represent the negation of a positive binary value as $-x=2^n-x-1$
• Or, can think of it as “flipping” the binary numbers around
• e.g. negation of 00001100 is 11110011, representing 1210 and -1210 respectively

Given n-bits, the range of value it can represent is: $[-(2^{n-1}-1),2^{n-1}-1]$

2s Complement

• Represent the negation of a positive binary value as $-x=2^n-x$
• Or, can think of it as “flipping” the binary numbers around, then add 1
  • Alternatively, (from LSB to MSB), copy the numbers to the first 1, and invert the rest.
• e.g. negation of 00001100 is 11110100, representing 1210 and -1210 respectively

Given n-bits, the range of value it can represent is: $[-(2^{n-1}),2^{n-1}-1]$

Note

You can also extend the idea of complement on fractions!

Excess Representation

• Allows range of values to be distributed evenly between the positive and negative values.
• E.g. Excess-4 representation on 3-bits number

Actual value = Bit pattern (unsigned) - 4

3-bit pattern	Unsigned value	Actual value (Excess-4)
000	0	–4
001	1	–3
010	2	–2
011	3	–1
100	4	0
101	5	+1
110	6	+2
111	7	+3