Number bases: Understand the following number bases:

The idea

This heading introduces the number bases you must recognise and work with in GCSE Computer Science: decimal (base 10), binary (base 2) and hexadecimal (base 16). Understanding a base means knowing how many distinct digits it uses, how each position carries a place value that is a power of the base, and how the same quantity can be written differently in each system. For example, the decimal value 13 is 1101 in binary and D in hexadecimal. You should be able to state the base of a given representation, identify which digits are legal in it, and convert between bases. Recognising the base lets you interpret a string of digits correctly rather than assuming it is decimal.

What earns the marks

• Decimal is base 10 and uses the ten digits 0–9.
• Binary is base 2 and uses only the digits 0 and 1.
• Hexadecimal is base 16 and uses 0–9 plus A–F, where A represents 10 and F represents 15.
• Each position in a number has a place value that is a power of the base: in base 10 the columns are 10⁰, 10¹, 10² …; in base 2 they are 2⁰, 2¹, 2² …; in base 16 they are 16⁰, 16¹, 16² ….
• The same quantity can be represented in different bases, for example decimal 13 equals binary 1101 and hexadecimal D.
• The base tells you how many distinct digit symbols the system uses before a new column is needed.

Common mistakes and how to put them right

• Assuming every number written without a label is decimal.

  Put it right: Check the context, because 10 in binary means two and 10 in hexadecimal means sixteen.

• Thinking hexadecimal uses only the letters A–F with no digits.

  Put it right: Hexadecimal uses 0–9 as well as A–F, sixteen symbols in total.

• Believing the digit 2 is valid in binary.

  Put it right: Binary uses only 0 and 1, so any digit of 2 or above signals a different base.

Exam tips

• Learn the digit set for each base so you can spot an invalid representation quickly.
• Remember the place-value columns are powers of the base, which explains why 10 in any base equals that base in decimal.
• Practise converting a small decimal number such as 13 into binary and hexadecimal to confirm you understand the relationship.