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Binary Arithmetic and Complements: Addition to 2's

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Computers do all their arithmetic, from adding a bill to rendering games, with only the digits 0 and 1. Binary arithmetic follows the same ideas as school decimal arithmetic but with simpler rules, since 1 + 1 is the biggest sum you ever handle. These rules show exactly how a CPU's arithmetic unit works.

Negative numbers are the tricky part. Instead of a minus sign, computers use complements. The 2's complement is brilliant: it lets the CPU subtract using the same addition circuit, with no separate subtractor. Every negative integer in your computer is stored this way.

Key Concepts

  • Binary addition: 0+0=0, 0+1=1, 1+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1).
  • Binary multiplication: shift-and-add, like decimal long multiplication with simpler digits.
  • 1's complement: flip every bit; 1001 becomes 0110.
  • 2's complement: 1's complement plus 1; 1001 becomes 0111. This represents the negative of the number.
  • Subtraction by addition: A − B = A + (2's complement of B); discard the final carry out.
  • Signed range: n bits in 2's complement cover −2n−1 to +2n−1 − 1; eight bits give −128 to +127.
  • Overflow: a result needing more bits than available loses the extra bit and comes out wrong.

Formulas

  • 2's complement: negative(B) = (bitwise NOT of B) + 1, within a fixed bit width.
  • Subtraction: A − B = A + 2's complement(B), discarding carry beyond n bits.
  • Sign bit: the leftmost bit: 0 means zero or positive, 1 means negative.

Worked Example

Compute 13 − 9 using 4-bit 2's complement.

  1. 13 = 1101, 9 = 1001. 2's complement of 1001: flip to 0110, add 1 to get 0111.
  2. Add: 1101 + 0111 = 10100. Discard the final carry: 0100.
  3. 0100 is 4, and 13 − 9 = 4. The CPU never needed a subtraction circuit.

Common Misconceptions

  • The discarded carry means an error. It is a normal part of the method; in 2's complement subtraction the carry out is always thrown away.
  • 1's and 2's complement are interchangeable. They are not: 1's complement has two zeros and needs correction, while 2's complement has one zero and plain addition just works.
  • CPUs borrow like school subtraction. They convert subtraction into complement addition because one adder circuit is cheaper and faster than two circuits.

Real-World Applications

  • Every CPU ALU adds, subtracts, and compares using 2's complement.
  • Audio and video DSP chips do millions of complement operations per second.
  • Network checksums rely on binary addition with discarded carries.
  • Microcontroller firmware uses these methods for multi-byte arithmetic.

To recap: binary arithmetic uses tiny rule sets, and the 2's complement turns subtraction into addition so one circuit does both. Flip the bits, add one, add, discard the carry: that recipe is literally how your processor subtracts.

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