Position carries meaning
Why Does 25 Mean Twenty-Five?
A digit's value depends on where it sits. That idea makes big numbers possible.
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Step 1 of 5
We only have ten digit symbols.
Roman numerals needed a new letter as numbers grew. This never does. 0 through 9, and that is all. Yet we can write any number, however enormous — because we reuse the same ten symbols in different places.
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Step 2 of 5
Each place is worth ten times the one to its right.
Ones, then tens, then hundreds, then thousands. Move a digit one place left and it becomes worth ten times as much.
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Step 3 of 5
So the same digit means different amounts.
Shift the left-hand 2 one place further left and it gains 1,800 by itself. In 222 the three 2s are worth 200, 20 and 2. They look identical; their positions do all the work.
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Step 4 of 5
Zero holds an empty place open.
Take the zero away and 205 quietly collapses into 25 In 205 there are no tens. Without a symbol for that gap, 205 would collapse into 25. Zero is what keeps the columns lined up.
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Step 5 of 5
Not every system worked this way.
Roman numerals gave each symbol a fixed value, so big numbers got long and multiplication was miserable. MMXXIV is 2024 — try multiplying that.
Place value reached Europe around 1200 and made arithmetic accessible to ordinary merchants for the first time.
The short version
A digit's worth comes from its position, and that is why ten symbols can write any number.
Try it yourself
Write 5, 50, 500 in a column. Same digit every time — only the seat it is sitting in changed.