Converting between number bases: decimal and hexadecimal.

The idea

This row covers conversion between decimal (base 10) and hexadecimal (base 16). Hexadecimal uses sixteen symbols: 0 to 9 then A to F for ten to fifteen, with each place worth a power of 16 (1, 16, 256 and so on). To convert decimal to hexadecimal, repeatedly divide by 16, recording each remainder; read the remainders from last to first. For example, 200 divided by 16 gives 12 remainder 8, and 12 is C, so 200 is C8. To convert hexadecimal to decimal, multiply each digit by its place value and add: C8 is 12 × 16 + 8 = 200. Practise both directions with values that cross the 16 and 256 boundaries so place value is secure.

What earns the marks

• States that hexadecimal is base 16 and uses digits 0 to 9 plus letters A to F for ten to fifteen.
• Converts decimal to hexadecimal by repeated division by 16, reading remainders from last to first.
• Converts hexadecimal to decimal by multiplying each digit by its place value (powers of 16) and summing.
• Applies place values 16⁰ = 1, 16¹ = 16, 16² = 256 correctly.
• Handles letters A to F by substituting their decimal values ten to fifteen during calculation.

Common mistakes and how to put them right

• Reading remainders in the order produced rather than reversing them.

  Put it right: The final quotient's remainder is the most significant digit.

• Treating a letter such as C as an unknown rather than the value twelve.

  Put it right: Substitute the decimal value before multiplying by the place value.

• Using powers of 10 for place value in a hexadecimal number.

  Put it right: Each place is a power of 16.

Exam tips

• Show the division steps with remainders so a single slip is easy to spot and correct.
• Convert your answer back to the original base as a check.
• Write the place values 1, 16, 256 above the hex digits before multiplying.