Two's complement

What you need to know

• ✓Two's complement represents both positive and negative integers using standard binary adder circuits.
• ✓In an 8-bit number, the Most Significant Bit (MSB, bit 7) has a place value of -128 (the sign bit).
• ✓If the MSB is 0, the number is positive (from 0 to +127). If the MSB is 1, the number is negative (from -128 to -1).
• ✓The total 8-bit range is -128 to +127 (256 distinct values).
• ✓To negate a number: invert all bits (0 becomes 1, 1 becomes 0) and add 1.

What you don’t need

• Sign and magnitude representation
• One's complement arithmetic
• Numbers larger than 8 bits in two's complement

Worked examples

1. Convert the 8-bit two's complement binary number 1101 0110 into denary.

   1. Write the place values: -128, 64, 32, 16, 8, 4, 2, 1.
   2. Add the active values: (-128 × 1) + (64 × 1) + (16 × 1) + (4 × 1) + (2 × 1).
   3. -128 + 64 + 16 + 4 + 2 = -42.

   Answer: -42

2. Represent the denary number -19 as an 8-bit two's complement binary number.

   1. Step 1: Write +19 in 8-bit binary: 0001 0011 (16 + 2 + 1).
   2. Step 2: Invert all bits: 1110 1100.
   3. Step 3: Add 1: 1110 1100 + 1 = 1110 1101.

   Answer: 1110 1101

Common mistakes

• ✕Thinking the range is -128 to +128 — zero takes up one positive pattern, so the maximum positive is +127.
• ✕Confusing two's complement with sign and magnitude (which is out of scope on AQA 8525).
• ✕Forgetting that bit 7 has a negative value (-128), not +128.