Number Bases · Form 4
What is a number base and place value?
A number base is how many different digits you count with before carrying to the next column; place value means each column is worth a power of that base.
Digits and the base
In base ten we count with ten digits, 0 to 9; when we run out we carry into the next column. The base is simply how big each bundle is before a carry happens: base five allows only 0 to 4, and base two allows only 0 and 1.
So a smaller base means you carry sooner and use fewer digit symbols.
Place value is powers of the base
In base ten the columns are ones, tens and hundreds, that is 10⁰, 10¹, 10². In base five the same columns become 5⁰, 5¹, 5², so 23₅ means 2 fives plus 3 ones, which is 13 in base ten.
The small subscript tells you the base, and the value of each column is always a power of that base.
The usual misconception
Many students read 23₅ as "twenty-three", but that is base-ten thinking. A key rule is that no digit may be equal to or larger than the base, so you never see a 5 in base five or a 2 in base two.
If a number contains a digit that is too big for its base, it is written incorrectly, a fast way to catch mistakes.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why can't base five use the digit 5?
In base five only digits 0–4 exist, because reaching a count of five means you carry over to the next column, the same way base ten carries at ten. Writing a digit equal to or larger than the base is a common exam mistake, e.g.
"352" is not a valid base-five number.
How does place value change between bases?
In any base, each column is worth a power of that base, units, then base¹, then base², and so on, but the value jumps faster in a small base like two than in base ten. For example the third column from the right is worth 100 in base ten but only 4 in base two, which is why base-two numbers look much longer for the same quantity.
What's the biggest pitfall when reading a number "in base n"?
The biggest pitfall is treating a number like 23 in base five as if it were an ordinary base-ten number, it actually means (2 × 5) + 3 = 13 in base ten, not twenty-three. Always convert to base ten mentally, or work directly with place values, before comparing or calculating with numbers written in different bases.