SPM Mathematics: How to Master Number Bases

In base 5 the columns are 1, 5, 25, 125 instead of 1, 10, 100. To turn a number into base 10, multiply each digit by its column value and add; to go the other way, divide repeatedly by the base and read the remainders from bottom to top.

Once you write the column headings first, most questions solve themselves.

Two errors show up again and again: writing a digit that is not allowed in the base (there is no digit 5 in base 5), and carrying the wrong amount when adding within a base. Work one column at a time and resist rushing.

Your calculator has no base button that does this for you, so clear written working is the only safe method.

When a question asks you to convert straight from, say, base 5 to base 2 without mentioning base 10, the safest route is still to go through base 10 as a middle step, even though it's an extra line of working. Convert the base 5 number to base 10 first, then convert that base 10 value into base 2.

Trying to convert directly between two non-decimal bases without this middle step is where most errors creep in, because it's easy to lose track of which place-value system you're multiplying against. The extra line costs a little time but removes a common source of wrong answers.

Addition and subtraction in a non-decimal base follow the same column method you already use in base 10, except that a column carries or borrows at the base's own value instead of ten. In base 5, for instance, a column total of 5 or more means you carry 1 to the next column and keep the remainder, just as a column total of 10 or more triggers a carry in base 10.

Writing the base clearly next to every number in the working, not just the final answer, prevents the easy mistake of finishing a calculation in the wrong base by accident.

Number bases usually appears as a short, self-contained Paper 1 item or a small part of a Paper 2 question, rather than a long standalone question. Its real value is as a fast, reliable few marks: the method is short and mechanical once place value is solid, so it is one of the more efficient topics to revise close to the exam, since a short refresher tends to bring the skill back quickly.