One's Complement

One’s Complement

Overview

One’s complement is a method of representing negative numbers in binary by inverting all bits of the positive number. This system provides a way to perform subtraction using addition circuits, though it has the peculiarity of having two representations for zero (+0 and -0). Understanding one’s complement is crucial for digital arithmetic and forms the basis for the more commonly used two’s complement system.

Detailed Explanation

Formation Rules

1. Positive numbers: Same as regular binary
2. Negative numbers: Invert all bits (1→0, 0→1)

Example (4-bit):
+5: 0101
-5: 1010 (invert all bits)

Range (n bits)

Positive: 0 to +(2ⁿ⁻¹-1)
Negative: -(2ⁿ⁻¹-1) to -0

4-bit example:
+7 (0111) to +0 (0000)
-7 (1000) to -0 (1111)

Addition Examples

1. Regular Addition:
   0101 (+5)
 + 0011 (+3)
 -------
   1000 (+8)

2. With End-Around Carry:
   0101 (+5)
 + 1010 (-5)
 -------
   1111 (-0)
   0000 (+0)

Truth Table (4-bit)

Number | Binary | One's Complement
-------|--------|----------------
+7     | 0111   | 0111
+6     | 0110   | 0110
+5     | 0101   | 0101
+4     | 0100   | 0100
+3     | 0011   | 0011
+2     | 0010   | 0010
+1     | 0001   | 0001
+0     | 0000   | 0000
-0     | 1111   | 1111
-1     | 1110   | 1110
-2     | 1101   | 1101
-3     | 1100   | 1100
-4     | 1011   | 1011
-5     | 1010   | 1010
-6     | 1001   | 1001
-7     | 1000   | 1000

Practice Problems

  1. Find one’s complement:

    • 1100 1010
    • 0011 0110
    • 1111 0000

  2. Perform using one’s complement:

    • (+6) + (-3)
    • (-4) + (+4)
    • (-7) + (-1)

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