Place Value

Tip out a big jar of buttons and try to say how many there are. Counting them one at a time takes forever — so we bundle them into groups of ten, and a huge pile suddenly becomes easy to read. That bundling trick is how we write every big number, from a house number to the price on a bike.

There are only ten digits in the whole world: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. So how can we write enormous numbers like a thousand, or a million? The secret is one brilliant trick: where a digit sits tells you how much it is worth. That trick is called place value.

When you can count past nine, you run out of single digits. So we do something clever: we bundle ten ones together and call the bundle one ten. From then on, a two-digit number is just a pile of tens and a pile of ones.

Take 23. The 2 is not really "two" — it is sitting in the tens place, so it means two tens, which is twenty. The 3 is in the ones place, so it means three ones:

23 = 2 \text{ tens} + 3 \text{ ones} = 20 + 3

Look at your hands. Ten fingers! Long ago people counted on their fingers, ran out at ten, and started a new bundle — so we built our whole number system in groups of ten. If we had eight fingers we might count in eights instead. There is nothing magic about ten; it is just the number of fingers we happen to have. Computers, which have no fingers, count in twos.

See it: tens and ones as blocks

Mathematicians draw a ten as a tall rod made of ten little cubes stuck together, and a one as a single cube. Count the rods, count the loose cubes, and you can read the number. Press Refresh to build a new one.

Here is the same idea as an animation. Press play: first ten loose dots bundle up into a single ten, then we build a number from a few tens and a few ones — and say it aloud. Replay it for a different number each time.

Reading a few numbers

Once you see the two columns, every two-digit number splits the same way:

47 is 4 tens and 7 ones (forty-seven).
80 is 8 tens and 0 ones — the 0 is important: it says "no ones", which keeps the 8 in the tens place.
60 is 6 tens, which we say as "sixty".

The two biggest place-value traps:

The 2 in 23 is not "two" — it is twenty. The place changes the value.
The 0 is a real worker, not "nothing". In 30 it holds the ones place empty so the 3 stays in the tens place — without it you'd just have 3.

It is the same amount — thirty single cubes — but we never write it as 30 ones. We always trade ten ones for one rod, as many times as we can. Thirty ones become three neat rods: 3 tens and 0 ones, written 30. Tidying loose ones into tens is the whole job of place value.

The most tens you can have before you'd bundle again is nine (ten tens would become a hundred), and the most ones is nine. So the biggest is 9 tens and 9 ones — 99. Add one more and it rolls over to 100: a brand-new hundreds column begins.

Khan Academy breaks a two-digit number into its tens and ones here: