Number Systems

So far every binary number has been unsigned, meaning all the bits represent positive values. To store negative numbers, Cambridge IGCSE uses two's complement, an 8-bit format where the leftmost bit's place value is negative.

The key idea

In an 8-bit two's complement value, the column headings are:

Notice the leftmost column is −128, not +128. The other columns keep their normal positive values. The leftmost bit is therefore the sign bit: if it is 0, the number is positive (or zero); if it is 1, the number is negative.

The range of 8-bit two's complement

So the range is −128 to +127 inclusive.

Reading a two's complement number

To find the denary value of a two's complement number, sum every column heading whose bit is set to 1, treating the leftmost column as −128.

Example — What denary value does 11111111 represent?

−128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = −1

Example — What denary value does 10110100 represent?

−128 + 32 + 16 + 4 = −76

Writing a negative number in two's complement

The fastest method on paper is the copy-and-invert trick:

1. Write the positive equivalent of the number in 8-bit binary.
2. Working from the right-hand end, copy each bit unchanged until you have written down the first 1 you meet (the 1 itself is also copied).
3. Invert every bit to the left of that 1 (turn 0s into 1s and 1s into 0s).

Example — Represent −76 in 8-bit two's complement.

Positive 76 in 8-bit binary:

Apply the copy-and-invert trick: working right to left, copy 100 (the two trailing zeros and the first 1). Then invert all the bits to the left of that first 1:

Sanity check: −128 + 32 + 16 + 4 = −76. ✓

Alternative method: invert all bits and add 1

A second standard method gives the same answer:

1. Write the positive equivalent in 8-bit binary.
2. Invert every bit (flip 0 ↔ 1). This gives the one's complement.
3. Add 1 to the result. The final 8-bit value is the two's complement.

Both methods are accepted in the exam; pick whichever you find easier and stick with it.

Why two's complement is used

Two's complement is the dominant way of storing signed integers in modern computers because:

• One representation for zero. Some other schemes (like sign-and-magnitude) have both a +0 and a −0, wasting a bit pattern. Two's complement has only one 0.
• Addition works the same for positive and negative numbers. The CPU can use one binary adder for both signed and unsigned arithmetic; no special "subtract" hardware is needed.
• Negative numbers behave naturally. Adding a number and its negative gives 0 (with the carry-out discarded), exactly as expected.