Number Bases · Form 4
Number Bases: Revision Notes
A tight revision summary of Number Bases for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
Everyday numbers are written in base ten. This chapter shows that the same quantity can be written in other bases, two, five and eight, and teaches you to convert between them, add and subtract within a base, and solve problems built on the idea of place value.
Key ideas to revise
- Place value is the whole idea. In base five each column is worth five times the one to its right, just as base ten uses tens. Get this and conversions follow.
- Converting to base ten. Multiply each digit by its place value and add, the safest first step for almost any base problem.
- Converting from base ten. Repeatedly divide and record the remainders, then read them upward. Practice makes this quick.
One worked mini-example for each key idea
- Place value is the whole idea: in 132 (base five) the columns are worth 25, 5 and 1, so 132 (base five) = 1x25 + 3x5 + 2x1 = 25 + 15 + 2 = 42 (base ten).
- Converting to base ten: for 1101 (base two) the columns are worth 8, 4, 2, 1, so 1101 (base two) = 1x8 + 1x4 + 0x2 + 1x1 = 8 + 4 + 0 + 1 = 13 (base ten).
- Converting from base ten: to write 47 (base ten) in base five, divide repeatedly: 47 / 5 = 9 remainder 2, 9 / 5 = 1 remainder 4, 1 / 5 = 0 remainder 1. Read the remainders upward: 142 (base five). Check: 1x25 + 4x5 + 2 = 47.
A pre-paper checklist for this chapter
- Re-derive the column values from scratch: in any base b the columns from the right are 1, b, b squared, b cubed. Do not memorise them, rebuild them, so base five is 1, 5, 25, 125.
- Fix the largest allowed digit in your head: in base b the digits run 0 to b minus 1, so base five uses 0-4 and base eight uses 0-7. Any digit equal to or above the base is an automatic error.
- What the exam gives you: nothing but the numbers and their bases, shown as a small subscript. There is no formula sheet entry for this chapter, so the method must come from you.
- When unsure, route every conversion through base ten: to go from base five to base eight, first make it base ten, then divide out to base eight. Slower, but far harder to get wrong.
- The one habit that saves marks: after every conversion, convert the answer straight back and check it equals the original. This single check catches almost every slip in this topic.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How many worked conversions should I practise before the paper?
Aim for fluency, not a number. Practise until you can convert a three-digit number both ways without pausing to think about the column values.
Ten mixed conversions a day for a week usually gets most students there, covering base two, five and eight in both directions so no single case feels unfamiliar.
Should I show the division-remainder working or can I do it in my head?
Always write it. The remainders are the answer, and a marker can only award method marks for working they can see.
Set the divisions out vertically with the remainder beside each line, then draw the arrow showing you read upward. Mental arithmetic here saves seconds but risks the whole answer.
Is converting through base ten ever penalised for being the long way?
No. The exam marks a correct answer with valid working, not the shortest route.
Going base five to base ten to base eight earns full marks exactly like a direct conversion would, provided each step is shown and correct. Choose the method you trust under pressure; a right answer by a longer path beats a wrong one by a clever shortcut.