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Position carries meaning

Why Does 25 Mean Twenty-Five?

A digit's value depends on where it sits. That idea makes big numbers possible.

The number 25322 hundreds, 5 tens, 3ones — position sets thevalue
Ten symbols, infinite numbers.

  1. Step 1 of 5

    We only have ten digit symbols.

    Ten symbols do everything

    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.

  2. Step 2 of 5

    Each place is worth ten times the one to its right.

    Hundreds — worth 100eachthird from therightTens — worth 10 eachsecondOnes — worth 1 eachfirst

    Ones, then tens, then hundreds, then thousands. Move a digit one place left and it becomes worth ten times as much.

  3. Step 3 of 5

    So the same digit means different amounts.

    The 2 on the left Worth 200The 2 on the right2Worth 2

    In 222 the three 2s are worth 200, 20 and 2. They look identical; their positions do all the work.

  4. Step 4 of 5

    Zero holds an empty place open.

    Without zero?2 5 — is that 25 or205?With zero0205, unambiguously

    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.

  5. Step 5 of 5

    Not every system worked this way.

    Roman numerals Symbol value neverchanges — clumsyPlace value Position does the work— compact

    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.

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