Number Bases · Form 4
How does converting between bases work?
Converting between bases is really about re-grouping the same quantity into different-sized bundles, with base ten acting as the convenient meeting point in between.
It's re-bundling, not new numbers
The quantity itself never changes when you convert, only the way it is packaged. In base ten a number is really each digit times its place value added up; a different base just packs the same total into different-sized bundles.
It is like swapping an RM10 note for coins: the money is unchanged, only the denominations differ.
Base ten as the bridge
To move between two non-decimal bases, say base five and base eight, base ten often serves as the middle step, because base ten is where we add and multiply most comfortably. This is why writing a number in expanded form, each digit times a power of the base, matters so much: it expresses the number in base ten, a familiar footing you can then re-group into the target base.
The bridge is a convenience, not a rule of the number itself.
The usual misconception
Students often think converting changes the number's value, or that you can swap it digit-by-digit like a secret code (1 stays 1, 2 stays 2). You cannot, because the place values are different, so the whole number must be re-grouped as one quantity.
A useful check: if your converted answer contains a digit too big for its base, a 5 in a base-five answer, say, the re-grouping went wrong somewhere.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What's the fastest way to convert a number from another base into base ten?
Multiply each digit by its place value, a power of the base, starting from the rightmost digit as the power of zero, then add the results. For example, 1011₂ = (1×2³)+(0×2²)+(1×2¹)+(1×2⁰) = 11.
Missing out a power, especially a zero digit, is the most common slip in Paper 1.
How do you convert a base ten number into another base, such as base two?
Divide repeatedly by the target base and write down each remainder, dividing the quotient again until it reaches zero. Reading the remainders from the last division back to the first gives the converted number.
Forgetting to reverse that remainder order is the exam mistake most students make.
Can I convert directly between two bases that aren't ten, like base eight and base two?
Yes, but only directly when one base is a power of the other, base eight is 2³, so each base-eight digit becomes exactly three binary digits. For any other pair of bases, convert through base ten first, since it's the only reliable meeting point the SPM syllabus expects you to use.