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Number Bases: Binary, Decimal and Hexadecimal

FoundationHigherAQA

Understand Number Bases: Binary, Decimal and Hexadecimal for GCSE Computer Science with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA GCSE Computer Science (8525). Computers use base-2 (binary), people use base-10 (decimal), and base-16 (hexadecimal) is a shorthand for binary.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA specification. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Computers store all data in binary, using only the digits 0 and 1, because each digit maps directly onto a circuit being off or on.

Denary, also called decimal, is base 10 and uses digits 0 to 9. Binary is base 2 and uses 0 and 1. Hexadecimal is base 16 and uses 0 to 9 followed by A to F, where A represents 10 and F represents 15.

Hexadecimal exists for human convenience rather than machine necessity. One hex digit represents exactly four binary digits, so a byte becomes two hex characters instead of eight binary ones. That makes long binary values far quicker to read, write and check — and mistakes far easier to spot.

Revision notes

The three bases

Denary (base 10): digits 0 to 9. Binary (base 2): digits 0 and 1.

Hexadecimal (base 16): digits 0 to 9 then A to F, where A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.

Why binary

Each binary digit maps directly onto a circuit being off or on, or a low or high voltage.

Only two states must be distinguished, which makes the hardware reliable and cheap. A base 10 system would need ten distinguishable voltage levels.

Why hexadecimal

One hex digit represents exactly four binary digits, because 2⁴ = 16.

So a byte becomes two hex characters rather than eight binary ones. Hex is shorter to write, easier to read, and errors are easier to spot — but it is for humans, not machines.

Key points

  • Computers store all data in binary.
  • Binary is base 2 using 0 and 1.
  • Denary is base 10 using 0 to 9.
  • Hexadecimal is base 16 using 0 to 9 and A to F.
  • One hex digit represents four binary digits.
  • Hexadecimal is for human convenience.

Worked examples

Example 1

State the denary value represented by the hexadecimal digit C. [1 mark]

Working

12A is 10, B is 11, so C is 12

Example 2

Explain why computers use binary rather than denary. [2 marks]

Working

Each binary digit corresponds to a circuit being off or on, so only two states must be distinguishedstate the reason
which makes the hardware simpler and more reliable than distinguishing ten different voltage levelsexplain the benefit

Example 3

Explain why hexadecimal is used by programmers. [2 marks]

Working

One hexadecimal digit represents exactly four binary digits, so a byte needs only two hex charactersstate the relationship
making long binary values much shorter to write and easier to read without errorexplain the benefit

Common mistakes

  • Thinking computers work in hexadecimal.

    They work in binary; hex is a convenience for humans.

  • Getting the letter values wrong.

    A is 10 through to F is 15.

  • Saying hex is base 10.

    It is base 16.

  • Forgetting one hex digit is four bits.

    This relationship is what makes hex useful.

Exam tips

  • Learn A to F as 10 to 15.
  • Remember one hex digit equals four bits.
  • Say hex is for humans, not machines.
  • State the base when naming a number system.

Key terms

Binary
Base 2, using digits 0 and 1.
Denary
Base 10, using digits 0 to 9.
Hexadecimal
Base 16, using 0 to 9 and A to F.
Bit
A single binary digit.

Related topics

Written and reviewed against the current AQA specification. Spotted an error? Let us know.