Signed number representation (2's complement)

1. Definition

The 2's complement of an $n$ -bit binary number is its 1's complement plus 1:

2's complement = (2^{n} - 1) - X + 1 = 2^{n} - X

The MSB is still the sign bit (0 = positive, 1 = negative), but unlike the other schemes there is exactly one zero, and no special carry handling is needed. Every modern CPU — Intel, AMD, ARM, RISC-V — stores negative integers this way.

2. Two Ways to Find the 2's Complement

Method A — invert every bit, then add 1:

Decimal	8-bit magnitude	1's complement	+1 → 2's complement
+45	0010 1101	1101 0010	1101 0011 = −45
+9	0000 1001	1111 0110	1111 0111 = −9
+1	0000 0001	1111 1110	1111 1111 = −1
−128	1000 0000 (the only negative pattern with no positive twin)	0111 1111	1000 0000 = −128

Method B — "copy up to the first 1 from the LSB, invert the rest": scan the bits from right to left; copy each bit until (and including) the first 1, then invert all the remaining higher-order bits. No addition is required.

For 0010 1101 (+45), reading from the LSB: 1, 0, 1, 1, 0, 1, 0, 0

Bit position	b0	b1	b2	b3	b4	b5	b6	b7
Original	1	0	1	1	0	1	0	0
Action	first 1 → copy	invert	invert	invert	invert	invert	invert	invert
Result	1	1	0	0	1	0	1	1

Reading the result MSB-first: 1101 0011 = −45 ✓ (identical to Method A)

Negation is self-inverse: the 2's complement of −45 (1101 0011) is +45 again, because $2^{n} - (2^{n} - X) = X$ .

3. Range and the Single Zero

For $n$ bits:

-2^{n-1} to +2^{n-1} - 1

Bits	Range	Number of values
4	−8 to +7	16
8	−128 to +127	256
16	−32768 to +32767	65536
32	−2147483648 to +2147483647	~4.3 × 10⁹

The asymmetric range exists because the pattern 1000…0 is used for the most negative value (−128 in 8 bits), while zero has only one representation (0000 0000). This gives 2's complement one extra negative value compared with 1's complement and sign-magnitude.

All 4-bit values:

Pattern	Value	Pattern	Value
0000	0	1000	−8
0001	1	1001	−7
0010	2	1010	−6
0011	3	1011	−5
0100	4	1100	−4
0101	5	1101	−3
0110	6	1110	−2
0111	7	1111	−1

4. Addition — Plain Binary Addition

Add the patterns as ordinary binary and discard the carry beyond the MSB (there is no end-around carry).

Example 1: 9 + (−4)

   0000 1001     (+9)
 + 1111 1100     (−4)
 ───────────
 1 0000 0101  → discard the carry out → 0000 0101 = +5   ✓

Example 2: (−12) + 5

   1111 0100     (−12)
 + 0000 0101     (+5)
 ───────────
   1111 1001     no carry → negative → 2's comp = 0000 0111 = 7 → −7   ✓

Example 3: (−5) + (−2)

   1111 1011 + 1111 1110 = 1 1111 1001 → discard carry → 1111 1001
   2's comp of 1111 1001 = 0000 0111 = 7 → answer −7   ✓

5. Subtraction

Use the identity A − B = A + (2's complement of B) — the same adder hardware, no borrow logic.

Example: 12 − 5

   0000 1100     (+12)
 + 1111 1011     (2's comp of +5)
 ───────────
 1 0000 0111 → discard carry → +7   ✓

Example: 5 − 12

   0000 0101 + 1111 0100 = 1111 1001 → negative
   2's comp of 1111 1001 = 0000 0111 = 7 → answer −7   ✓

6. Overflow Detection

Overflow happens only when two numbers of the same sign produce a result of the opposite sign. Two equivalent hardware tests:

Carry test — the carry into the MSB differs from the carry out of the MSB.
Sign test — the sign bit of the result is wrong for the operands' signs.

Example (4-bit range −8…+7): $5 + 5$ → 0101 + 0101 = 1010 → both operands positive but the result is negative (−6 in 2's complement) ⇒ overflow, because the true answer +10 lies outside the range. Adding a positive and a negative number can never overflow.

7. Sign Extension

When a smaller 2's complement number is moved into a wider register, the sign bit must be copied into all new high-order bits (sign extension), not zero-filled:

-5 (4-bit 1011) → 8-bit 1111 1011 (still −5, not 251)

8. Why 2's Complement is Used Everywhere

Criterion	Sign-magnitude	1's complement	2's complement
Number of zeros	2	2	1
Subtraction hardware	Separate subtractor	Adder + end-around carry	Adder only
Range (8 bits)	−127…+127	−127…+127	−128…+127
Overflow detection	Compare signs	Compare signs	Carry-in vs carry-out of MSB
Extra hardware step	Magnitude comparison	End-around carry	None
Used in modern CPUs	No	No	Yes

9. Exam Points

Define 2's complement and find the 2's complement of given numbers by both methods — 5 marks, almost certain.
Add/subtract signed numbers using 2's complement and show the discarded carry — 5-7 marks, very frequent.
State the range for n bits and explain why there is one extra negative value — 3-5 marks.
What is overflow? How is it detected using carries? Show with a 4-bit example — 5 marks.
Compare sign-magnitude, 1's complement and 2's complement (tabular) — 5 marks, frequent.
What is sign extension and why is it needed? — 3 marks.

Signed number representation (2's complement)