3.4 · Two's Complement

Goal: convert any integer to its n-bit two's-complement form, and convert it back.

The problem of negative numbers

Binary has only 0 and 1 — there is no − sign. How can a computer store negative integers?

Three solutions existed historically:

Scheme	Idea	Drawback
Sign-magnitude	Reserve the leftmost bit as the sign (0 = +, 1 = −); rest is magnitude	Two zeros (+0, −0); subtraction is awkward
One's complement	Flip every bit to negate	Still two zeros; carry handling messy
Two's complement	Flip every bit, then add 1	Single zero; addition and subtraction work uniformly

Modern computers all use two's complement for signed integers. The HKEAA explicitly requires you to know it.

How to encode a negative integer

The 3-step recipe

To represent −x in n-bit two's complement:

1. Write +x in n-bit binary.
2. Invert every bit (one's complement).
3. Add 1.

Example · −5 in 8 bits

Step	Working
1. Write +5	0000 0101
2. Invert	1111 1010
3. Add 1	1111 1011

So −5 = 1111 1011₂ in 8-bit two's complement.

Example · −1 in 8 bits

Step	Working
1. Write +1	0000 0001
2. Invert	1111 1110
3. Add 1	1111 1111

−1 is all ones. (This is true for any number of bits.)

Example · −128 in 8 bits

Step	Working
1. Write +128	1000 0000 (already negative interpretation, but bit pattern is the same)
2. Invert	0111 1111
3. Add 1	1000 0000

The bit pattern 1000 0000 represents −128 in two's complement — the smallest 8-bit signed value.

How to decode a two's complement number

If the MSB is 0 → it's a positive number; read as normal binary.

If the MSB is 1 → it's negative. Two ways to convert:

Method 1 · Apply the same recipe (it's symmetric!)

1111 1011₂ → flip → 0000 0100 → +1 → 0000 0101 = 5 → so original is −5.

Method 2 · Use the negative MSB place value

For an n-bit two's complement number, the MSB has the place value −2ⁿ⁻¹ instead of +2ⁿ⁻¹.

For 8-bit:

Place	−128	64	32	16	8	4	2	1
Bit	1	1	1	1	1	0	1	1

Value = −128 + 64 + 32 + 16 + 8 + 0 + 2 + 1 = −5. ✓

Range of n-bit two's complement

Bits	Range
4	−8 to +7
8	−128 to +127
16	−32,768 to +32,767
32	−2,147,483,648 to +2,147,483,647

Formula: for n bits, range is −2ⁿ⁻¹ to +2ⁿ⁻¹ − 1.

Notice the asymmetry: there is one more negative value than positive because 0 lives in the positive half (0000 0000).

Why two's complement is brilliant

One representation for zero

Sign-magnitude has +0 and −0. Two's complement has only 0000 0000 = 0.

Subtraction is just addition

A − B = A + (two's-complement of B).

The same circuitry adds positive and negative numbers — no separate subtractor needed.

Sign extension works naturally

If you copy an 8-bit signed number into a 16-bit register, you just copy the MSB into the new upper bits:

8-bit  −5 = 1111 1011
16-bit −5 = 1111 1111 1111 1011

Worked examples

Example A · Compute 7 − 3 in 4-bit two's complement

+7 = 0111+3 = 0011, so −3 = 0011 → 1100 → 1101.

Add: 0111 + 1101 = 1 0100. Discard the carry out → 0100 = +4. ✓

Example B · Compute 5 − 9 in 8-bit two's complement

+5 = 0000 0101−9 = 0000 1001 → 1111 0110 → 1111 0111

Add: 0000 0101 + 1111 0111 = 1111 1100 = −4. ✓

Common student mistakes

• Forgetting step 3 (add 1) — that's the difference between one's and two's complement.
• Confusing MSB = 1 as always negative — only in two's complement; in unsigned binary the MSB is just another bit.
• Asymmetric range bug: trying to represent +128 in 8-bit signed (it's out of range).
• Mixing widths — comparing a 4-bit signed 1011 with an 8-bit signed 1011 gives different values.

Practice activity

In 8-bit two's complement, find:

1. The bit pattern for −1.
2. The bit pattern for −42.
3. The decimal value of 1110 0000.
4. The decimal value of 1000 0000.
5. The range of values representable.

Answers

1. 1111 1111
2. 1101 0110 (+42 = 0010 1010 → invert 1101 0101 → +1 → 1101 0110)
3. −32
4. −128
5. −128 to +127

Exam-style question

Q (4 marks): Using 8-bit two's complement, express −45₁₀ in binary. Show your working.

Sample answer:

Step	Working
1. Write +45 in 8-bit binary	0010 1101
2. Invert every bit	1101 0010
3. Add 1	1101 0011

−45₁₀ = 1101 0011₂ in 8-bit two's complement.

Key takeaways

• Two's complement = invert all bits, then add 1.
• The MSB of a two's-complement number has place value −2ⁿ⁻¹.
• One single representation of zero.
• Addition and subtraction use the same hardware.
• Range: −2ⁿ⁻¹ to +2ⁿ⁻¹ − 1.

➡️ Next: 3.5 Character Encoding