Understand how computers represent and manipulate unsigned integers and two's complement signed integers, and convert between denary and 8-bit binary numbers (0 to 255, and -128 to +127).

What this dot point is asking

Edexcel wants you to represent whole numbers in binary both as unsigned integers (0 and up) and as two's complement signed integers (which can be negative), and to convert fluently between denary and 8-bit binary, covering the unsigned range 0 to 255 and the signed range -128 to +127.

Unsigned integers and conversion

Binary to denary is direct addition: $01100100 = 64 + 32 + 4 = 100$. Denary to binary is repeated subtraction: for 100, 64 fits (leaving 36), 32 fits (leaving 4), 4 fits (leaving 0), giving $01100100$. Always pad the answer to the requested number of bits (8 bits here), and cross-check by adding the place values back.

Two's complement signed integers

The clever part is that ordinary binary addition then works for negative numbers with no special cases, which is why processors use two's complement. The most significant bit doubles as a sign bit: 0 means positive (or zero), 1 means negative. To read a negative two's complement value, add the place values as usual but treat the leftmost as -128: $11011000 = -128 + 64 + 16 + 8 = -40$.

Finding a two's complement representation

Ranges to remember

These ranges are worth memorising because questions often ask whether a value "fits". For example, +200 cannot be stored as an 8-bit two's complement number, because the maximum is +127; it would need more bits or an unsigned representation.

Try this

Q1. Convert the 8-bit two's complement number $11111111$ to denary. [1 mark]

• Cue. -1, because $-128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = -1$.

Q2. State the range of values that 8-bit two's complement can represent. [1 mark]

• Cue. -128 to +127.