Form 4 · Number and Operations
Number Bases
Number bases ask you to count in something other than tens, base two, five and eight. It feels strange at first, then clicks.
What is Number Bases?
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.
Content standards (DSKP)
The DSKP KSSM sets these content standards for this chapter:
- 2.1 Number Bases
The key ideas
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.
How this chapter is examined
This is usually a short, self-contained topic in Paper 1 and a structured question in Paper 2. Marks are lost less to difficulty than to carelessness, a digit five written in base five, or an arithmetic slip during conversion.
Neat, checked working scores well here.
How to study this chapter
Common mistakes to avoid
- Using a digit that is too large for the base (like 5 in base five)
- Reading division remainders in the wrong order
- Mixing up which base you are converting to and from
Adding and subtracting without leaving the base
You do not always have to detour through base ten. In base five, a column carries when it reaches five, not ten: if a column adds to seven, you write 2 and carry 1, because seven is one five and two ones.
Subtraction borrows a whole base, borrowing in base eight brings across eight, not ten. The mechanics feel exactly like ordinary column arithmetic; only the size of a full column changes.
Line the place values up neatly and name the base you are working in before the first carry, and the process stays reliable.
Use base ten as a bridge between any two bases
There is no need to memorise a direct route from base five to base eight. Convert the base-five number to base ten first by multiplying each digit by its place value, then convert that base-ten value into base eight by repeated division.
Base ten is the language you already think in, so mistakes are easier to spot there. The bridge is slower than a direct conversion, but in an exam a slower method you can check beats a quicker one you cannot.
Whenever a question mixes two unfamiliar bases, route both through base ten.
How to open a base question under exam pressure
Before writing anything, do three quick things. First, name the base of every number in the question, because a stray digit, a 6 sitting inside a base-five number, is usually a misread rather than a hard step.
Second, decide the direction: are you going to base ten, from base ten, or between two bases through it. Third, sanity-check the size of your final answer: a base-two number equal to a small base-ten value should be noticeably longer, since base two packs less into each column.
These habits turn a topic that punishes carelessness into one that rewards a calm, ordered method.
A worked exam-style example
Here is a Paper 2-style question that walks through three conversions built on the same value.
- (a) Multiply each digit by its place value in base five: 2 × 5² + 3 × 5¹ + 4 × 5⁰.
- That is 2 × 25 + 3 × 5 + 4 × 1 = 50 + 15 + 4 = 69. So 234₅ = 69 in base ten.
- (b) Convert 69 to base two by repeated division by 2, recording remainders: 69÷2 = 34 r 1, 34÷2 = 17 r 0, 17÷2 = 8 r 1, 8÷2 = 4 r 0, 4÷2 = 2 r 0, 2÷2 = 1 r 0, 1÷2 = 0 r 1.
- Read the remainders upward: 1000101. So 69 = 1000101₂.
- (c) Convert 69 to base eight by repeated division by 8: 69÷8 = 8 r 5, 8÷8 = 1 r 0, 1÷8 = 0 r 1.
- Read the remainders upward: 105. So 69 = 105₈.
Study Number Bases
Frequently asked questions
How this chapter is examined
SPM Mathematics assesses this chapter across Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective), drawing on the DSKP content standards above. Paper 2 gives marks for working, so showing every step matters.
Common mistakes to avoid
Using a digit that is too large for the base (like 5 in base five); Reading division remainders in the wrong order; Mixing up which base you are converting to and from.
Are any Number Bases formulae given in the exam?
This chapter has no formula on the exam formula sheet, the working is expected from memory and method.
Why do we even learn base two and base five when everyday life uses base ten?
Base two is the language of computers, and working in unfamiliar bases forces you to truly understand place value rather than rely on habit. In the SPM syllabus it is a self-contained topic where careful, tidy working earns full marks, so it is a reliable place to build points.
In base five, am I ever allowed to write the digit 5?
No. A base tells you how many digit symbols exist, so base five uses only 0, 1, 2, 3 and 4.
The moment a column reaches five you carry one to the next column and write 0. Seeing a 5 or larger inside a base-five number is a sure sign of a slip.
What is the quickest safe way to convert straight from base five to base eight?
Go through base ten. Convert the base-five number to base ten by multiplying each digit by its place value, then divide repeatedly by eight to reach base eight.
A direct route exists but is easy to bungle; the base-ten bridge is a step longer and far easier to check under exam pressure.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)· SPM: Format Pentaksiran mulai 2021, Matematik (1449)
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