Sign and Magnitude

In sign and magnitude representation, the most significant bit (MSB) is used as a sign bit:

• 0 in the MSB = positive number
• 1 in the MSB = negative number

The remaining 7 bits store the magnitude (absolute value) of the number.

Sign and Magnitude Examples (8-bit)

Range using 8-bit sign and magnitude: −127 to +127. Note: there are two representations of zero (+0 = 00000000 and −0 = 10000000).

Two's Complement

Two's complement is the most widely used method for representing negative integers in real computer systems. It has the advantage of having only one representation of zero, and addition works naturally without special rules for negative numbers.

Converting Positive to Negative (Two's Complement)

To negate a number in two's complement:

1. Write the positive number in binary
2. Invert all bits (flip 0s and 1s — this is the one's complement)
3. Add 1 to the result

Example: Represent −45 in 8-bit two's complement:

• +45 in binary: 00101101
• Invert all bits: 11010010
• Add 1: 11010011
• Result: 11010011 = −45 in two's complement

Reading a Two's Complement Number

If the MSB is 1 (negative number): invert all bits, add 1, then add a minus sign.

Example: What is 11100110 in denary (two's complement)?

• MSB = 1 → negative
• Invert: 00011001
• Add 1: 00011010
• 00011010 = 16+8+2 = 26
• Result: −26

Two's Complement Range

In 8-bit two's complement: range is −128 to +127. The MSB has a weight of −128 (rather than +128).