Signed Numbers

Signed Numbers

Overview

Signed numbers in digital systems allow representation of both positive and negative values. Various methods exist for representing signed numbers, with two’s complement being the most widely used in modern computers. Understanding signed number representations is crucial for arithmetic operations and data processing in digital systems.

Detailed Explanation

Representation Methods

1. Sign-Magnitude
   +5: 0101  (0 = positive)
   -5: 1101  (1 = negative)

2. One's Complement
   +5: 0101
   -5: 1010  (invert all bits)

3. Two's Complement
   +5: 0101
   -5: 1011  (invert + add 1)

Sign Bit Conventions

MSB (Most Significant Bit):
0 → Positive number
1 → Negative number

8-bit Example:
0xxx xxxx → Positive range
1xxx xxxx → Negative range

Comparison Table

Method         | Range (8-bit)  | Zero Forms
---------------|---------------|------------
Sign-Magnitude | -127 to +127  | +0, -0
One's Comp     | -127 to +127  | +0, -0
Two's Comp     | -128 to +127  | One zero

Sign Extension

4-bit to 8-bit:
Positive: 0101 → 0000 0101
Negative: 1011 → 1111 1011

Rule: Copy the sign bit to all new positions

Practice Problems

  1. Convert between representations:

    • +42 in all three formats
    • -15 in all three formats
    • Compare ranges and ease of use

  2. Perform sign extension:

    • 0110 to 8 bits
    • 1101 to 8 bits
    • 0011 to 12 bits

References

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