Represent signed integers using two's complement, convert to and from denary, and perform subtraction by addition, explaining range and overflow

What this dot point is asking

SEAB wants you to represent signed whole numbers in two's complement, convert both ways with denary, perform subtraction by adding a negation, and explain the representable range and how overflow is detected. The key insight is that two's complement lets one ordinary binary adder handle both positive and negative numbers, which is why every modern processor uses it.

The answer

What two's complement means

In an $n$-bit two's complement number the most significant bit has a negative weight of $-2^{n-1}$, while all other bits keep their usual positive weights. For 8 bits:

So $11011011_2 = -128 + 64 + 16 + 8 + 2 + 1 = -37$. A leading $1$ always means the number is negative; a leading $0$ means it is non-negative.

Negating a number

To form the negative of any two's complement number: invert every bit, then add one. The same procedure undoes itself, so it converts $+x$ to $-x$ and $-x$ back to $+x$.

+5  = 0000 0101
invert: 1111 1010
add 1:  1111 1011  = -5

Subtraction is addition

Because the negative is built into the encoding, $a - b$ is computed as $a + (\text{two's complement of } b)$. The processor has no separate subtractor: it negates and adds, discarding any carry out of the top bit.

The representable range

For $n$ bits the range is asymmetric:

For 8 bits that is $-128$ to $+127$. There is exactly one representation of zero ($00000000$), which is a key advantage over sign-and-magnitude.

Detecting overflow

Signed overflow happens when adding two numbers of the same sign produces a result of the opposite sign. Equivalently, overflow occurred if the carry into the sign bit differs from the carry out of it. Adding numbers of opposite signs can never overflow.

Examples in context

Example 1. Temperature sensors. A microcontroller reading that can go below zero stores its value in two's complement, so a reading of $-3$ degrees is the byte $11111101$. The same arithmetic unit that adds positive readings handles the negative one without extra circuitry.

Example 2. The year-2038 problem. Many systems counted seconds since 1970 in a 32-bit signed integer. Two's complement caps that count at $2^{31} - 1$ seconds, which is reached in 2038; past it the value overflows to a large negative number, flipping the date to 1901. It is a direct consequence of the asymmetric signed range.

Try this

Q1. Represent $-1$ in 8-bit two's complement. [2 marks]

• Cue. $+1 = 00000001$; invert to $11111110$, add 1 to get $11111111$, so $-1$ is all ones.

Q2. State the range of a 16-bit two's complement integer. [1 mark]

• Cue. $-2^{15}$ to $2^{15} - 1$, that is $-32768$ to $+32767$.

Q3. Does adding $01111111$ and $00000001$ (8-bit signed) overflow? Explain. [2 marks]

• Cue. $+127 + 1$ gives $10000000 = -128$; two positives produced a negative, so yes, signed overflow occurred.