Binary arithmetic and signed integers

Doing maths in binary

• A computer adds, subtracts and stores negative numbers — all in binary.
• The clever trick is two's complement, which turns subtraction into addition.
• Let's build it up from simple binary addition.

Binary addition

• Add column by column from the right, carrying a 1 just like in denary.
• The rules: $0+0=0$, $0+1=1$, $1+1=10$ (write 0, carry 1), $1+1+1=11$ (write 1, carry 1).
• Worked: 0101 $+$ 0011 $=$ 1000 (i.e. $5 + 3 = 8$).
• Overflow happens when the result needs more bits than the register holds — the carry out of the leftmost column is lost.

Subtraction by two's complement

• To do $A - B$: form the two's complement of $B$ (invert every bit, then add 1), then add it to $A$, and discard any final carry-out.
• Worked, 4-bit: $7 - 3$. Two's complement of 0011 → invert 1100 → add 1 → 1101.
• 0111 $+$ 1101 $=$ 1 0100 → discard the leading 1 → 0100 $= 4$. ✓

Two's complement signed integers

• The most significant bit (MSB) is the sign bit: 0 = positive, 1 = negative.
• To read a negative number: invert all bits, add 1, then put a minus sign on the result.
• e.g. 11111101 → invert 00000010 → add 1 → 00000011 $= 3$, so the value is $-3$.
• For $n$ bits the range is $-2^{n-1}$ to $+2^{n-1}-1$; for 8 bits, $-128$ to $+127$.

Binary Coded Decimal (BCD)

• In BCD, each denary digit is stored as its own 4-bit pattern (0000–1001 only).
• $93$ in BCD is 1001 0011 — not the same as binary 93 (0101 1101).
• Used in calculators, digital clocks and 7-segment displays, and for currency (it avoids the rounding errors of storing $0.1$ in pure binary).

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