Converting between number bases: binary and decimal

The idea

This row names binary and decimal as a conversion pair you must handle. Binary uses only the digits 0 and 1, with each place value a power of two (1, 2, 4, 8, 16, ...). Decimal uses digits 0 to 9 with place values that are powers of ten. To convert binary to decimal, add the place values where a 1 appears; for example 1011₂ = 8 + 0 + 2 + 1 = 11. To convert decimal to binary, repeatedly subtract the largest power of two that fits, or divide by 2 and read the remainders upwards; for example 11 gives 1011. A multiple-choice question will test whether you can carry out these conversions accurately and select the correct value.

What earns the marks

• Convert binary to decimal by summing the powers of two (1, 2, 4, 8, 16, ...) corresponding to each 1 bit, for example 1011₂ = 11.
• Convert decimal to binary by repeatedly subtracting the largest fitting power of two, or by dividing by 2 and reading remainders from bottom to top.
• Recognise that binary place values double from right to left and that only the digits 0 and 1 are valid in binary.
• Check the answer against the stated equivalent maximum values so a result that is too large is rejected.

Common mistakes and how to put them right

• Reading binary place values left to right as increasing.

  Put it right: The place values increase from right to left, so correct the direction before adding.

• Forgetting that a 0 bit contributes nothing.

  Put it right: Only add place values where the bit is 1.

• Reading division remainders in the wrong order when converting decimal to binary.

  Put it right: Read them from the last remainder upwards.

Exam tips

• Write the place values (1, 2, 4, 8, 16, ...) above the binary digits before adding.
• For decimal to binary, subtract the largest power of two that fits, then continue with the remainder.
• Check your chosen option by converting it back to the original base.