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01 · At a glance

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Cambridge · Computer Science

The whole syllabus, mapped

Open any topic, pick a note, and read it right here. Topic one is free to preview.

Syllabus tree

6 topics
01Data representation3 notes · ~7 minFree
Binary & hexadecimal
Two’s complement
Text, sound & images
02Data transmission3 notes · ~5 min
03Hardware3 notes · ~6 min
04Software3 notes · ~5 min
05Algorithm design3 notes · ~6 min
06Programming3 notes · ~8 min

TOPIC 01 · DATA REPRESENTATION · ~7 MIN

Two’s complement

The elegant trick that lets a fixed-width binary pattern hold negative numbers — and the single step students forget under exam pressure.

The sign bit carries weight

A plain 8-bit number runs 0 to 255. Two’s complement reclaims the top half of that range for negatives by giving the most significant bit a negative place value of −128. Every other column stays positive, so reading a value is still just addition.

-128
1
64
1
32
1
16
1
8
0
4
0
2
0
1
0
−128 + 64 + 32 + 16 = −16

Worked example

To encode −45, start from the positive value and apply the two-step rule:

  1. Write +4500101101
  2. Invert bits11010010
  3. Add one →11010011

Where marks are lost

Inverting the bits alone gives one’s complement. You must add one afterwards. Examiners reserve a full mark for this final step — skip it and the whole answer is wrong.

Check your understanding

What is the denary value of the 8-bit two’s complement number 11110000?

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Data representationRevision note

Binary prefixes

Storage counts in powers of two. A kibibyte is 2¹⁰ = 1024 bytes — not 1000. Each prefix climbs by another factor of 1024.

kibi
2¹⁰
mebi
2²⁰
gibi
2³⁰
tebi
2⁴⁰

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