Number Bases · Form 4
Number Bases: Common Mistakes
The mistakes that quietly cost marks in Number Bases, and how to avoid each one in the SPM exam.
In our experience teaching Number Bases, most lost marks come from a small set of repeated slips, not from a lack of understanding. Here they are, with the fix for each.
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
Six more slips that quietly cost marks
- What students write: 23 (base five) + 14 (base five) = 37 (base five). Why it loses marks: they carried in tens, not fives, so the answer is impossible in base five. Correct working: 3 + 4 = 7 = 12 (base five), write 2 carry 1; 2 + 1 + 1 = 4; answer 42 (base five). Check: 13 + 9 = 22, and 4x5 + 2 = 22.
- What students write: 32 (base five) minus 14 (base five) = 22 (base five). Why it loses marks: when borrowing they added ten to the units instead of five. Correct working: 2 minus 4 needs a borrow, so 2 + 5 = 7, then 7 minus 4 = 3; the tens become 2 minus 1 = 1; answer 13 (base five). Check: 17 minus 9 = 8, and 1x5 + 3 = 8.
- What students write: the value 59 in base eight as 18 (base eight). Why it loses marks: the digit 8 does not exist in base eight, whose digits are 0-7. Correct working: 59 / 8 = 7 remainder 3, 7 / 8 = 0 remainder 7, so the answer is 73 (base eight). Check: 7x8 + 3 = 59.
- What students write: 47 (base ten) to base five as 241 (base five). Why it loses marks: they read the division remainders top-down instead of bottom-up. Correct working: remainders came out 2, then 4, then 1, so reading upward gives 142 (base five), not 241. Check: 1x25 + 4x5 + 2 = 47.
- What students write: 1101 (base two) grouped into base eight as 118 (base eight). Why it loses marks: they grouped bits loosely instead of in threes from the right. Correct working: 1101 becomes 001 101, which is 1 then 5, giving 15 (base eight). Check: 1101 (base two) = 13, and 1x8 + 5 = 13.
- What students write: 234 (base five) = 2x100 + 3x10 + 4 = 234. Why it loses marks: they used base-ten place values (100, 10) inside a base-five number. Correct working: the columns are 25, 5, 1, so 2x25 + 3x5 + 4 = 50 + 15 + 4 = 69 (base ten).
- What students write: 142 with no base label. Why it loses marks: without the subscript the marker cannot tell which base is meant, and identical digits mean different values in different bases. Correct working: always tag the base, for example 142 (base five), so the value is unambiguous.
The costliest slip to guard against
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
When I add in base five, why did my answer come out larger than in base ten?
Because the same value needs more digits in a smaller base. Twenty-two is a two-digit number in base ten but 42 in base five, which looks bigger yet is not.
Do not compare the appearance of the digits; convert both to base ten and you will see they match. The digit pattern grows as the base shrinks.
How do I check an addition or subtraction I did inside a base?
Convert every number to base ten, do the same operation there, then convert the result back to the original base and compare. If the two answers agree, your in-base working is sound.
It takes an extra minute but turns a nervous guess into a certainty, and it catches carrying and borrowing errors that are otherwise invisible.
Does forgetting the subscript really cost a mark if the digits are right?
It can, because 142 alone is not one number; it is different values in base five, base eight and base ten. When the question asks for a specific base, the base label is part of the answer, not decoration.
Get into the habit of writing the subscript the moment you write the digits, so it is never left off in the rush of the final line.