Digital systems use two's complement to represent negative numbers. This method allows for simple addition and subtraction operations without needing special handling for negative numbers. The leftmost bit (most significant bit) serves as the sign bit, where 1 indicates negative and 0 indicates positive.
Converting 42 to 8-bit Two's Complement
Step 1: Original Number
0010 1010
Start with absolute value: 42
Step 2: One's Complement
1101 0101
Flip all bits
Step 3: Two's Complement
1101 0110
Add 1 to one's complement
Final Result:
Binary:
0010 1010
Decimal:
42
Two's Complement Properties
Most significant bit (leftmost) is the sign bit (0 = positive, 1 = negative)
Positive numbers are represented normally
Negative numbers use two's complement representation
Range for 8-bit: -128 to 127
Adding a number and its two's complement equals 0
No special handling needed for addition/subtraction
Common Gotchas
Overflow can occur when result exceeds the bit width
Right-shifting negative numbers may need special handling
Converting between different bit widths needs sign extension
The smallest negative number has no positive counterpart