The same digit means something different depending on where it stands.
Here is a small puzzle. The number 55 is written with two of the very same digit — two fives. Yet the two fives are not worth the same. The five on the right is worth five; the five on the left is worth fifty. A digit, all on its own, does not tell you its value: you also have to look at the place it stands in. This is the idea of place value, and it is the engine that lets ten little symbols — 0 through 9 — name every number there is, from 7 to seven billion. In this lesson we will see exactly how much each place is worth, why we need the digit 0 to hold an empty place, and how to read and write any number by saying its places out loud from the largest down to the ones.
Back in 1.2 we bundled ten loose ones into one new unit — a ten. Place value is what we do with those bundles: we give each kind of bundle its own column. The rightmost column counts loose ones; the next column to the left counts tens (each one a bundle of ten). So a digit's worth is its face value times the worth of its column.
Take the digit 5. Put it in the ones place and it means five loose ones — just 5. Slide that same 5 one column to the left into the tens place and it now counts five bundles of ten, which is 50. The digit did not change; its place changed, and that multiplied its worth by ten.
A digit's value = its face value × the value of its place. The ones place is worth 1; the tens place is worth 10. So the 5 in 53 is worth 50, and the 3 is worth 3.
The columns do not stop at tens. Bundle ten tens together and you get a hundred; bundle ten hundreds and you get a thousand. Reading from the right, the places run ones, tens, hundreds, thousands, and on. The rule never changes: each place is worth ten times the place just to its right.
That is why a chart is so handy. Write 3,452 in its columns and each digit lights up its own worth: 3 × 1,000 = 3,000, 4 × 100 = 400, 5 × 10 = 50, 2 × 1 = 2. Add those four pieces and you are back to 3,452. Writing a number as a sum of its place pieces like this is called its expanded form.
3,452 = 3 × 1,000 + 4 × 100 + 5 × 10 + 2 × 1 = 3,000 + 400 + 50 + 2. Each step left is worth ten of the step before: 1 → 10 → 100 → 1,000.
Now we meet the most important job of the digit 0. Suppose a number has three hundreds, no tens, and five ones. The hundreds column gets a 3 and the ones column gets a 5 — but the tens column is empty. We cannot just leave a gap and write "3 5," because then nobody can tell which column the 3 and the 5 belong to. Instead we write a 0 in the tens place to say "no tens here." The number is 305.
That little 0 is doing real work. Drop it and the 3 slides over into the tens column, the 5 stays in the ones, and you have written a completely different number: 35 — three tens and five ones, which is far smaller than three hundred five. The 0 is not "nothing": it is a placeholder that keeps every other digit in its proper place.
A digit is not worth its face value. The 3 in 305 is worth three hundred, not three. And erasing the placeholder 0 changes the number: 305 becomes a different, smaller number, 35.
To read a number, start at the highest place and work down: say how many thousands, then how many hundreds, then tens, then ones. For 3,452 that is "three thousand, four hundred fifty-two." You only say a place out loud when its digit is not zero — a placeholder 0 is silent. So 305 is read "three hundred five" (no "tens" spoken), not "three hundred zero five."
Reading from the top down is what makes the size of a number obvious the moment you start: the leftmost place is the biggest, so it tells you roughly how large the whole number is before you have even finished saying it.
2,008 is read "two thousand eight." There are 2 thousands and 8 ones; the hundreds and tens are empty, so each holds a silent placeholder 0 — but we do not say the word "zero."
Writing is reading run backward. Hear the places named, drop each digit into its own column, and fill any place that was not mentioned with a placeholder 0. Hear "two thousand eight": put 2 in thousands, 8 in ones, and since no hundreds and no tens were named, write 0 in each of those columns. The result is 2,008 — four digits, because the thousands place is the highest one mentioned.
The dialer above is exactly this skill: you place each digit in its column, and the empty columns you leave at 0 hold their spots so the digits you set stay where they belong.
To write a number from its words, give every named place its digit and fill every unnamed place with a 0. The placeholders are what keep "two thousand eight" from collapsing into 28.
The ladder keeps climbing. Ten thousands make one ten-thousand; ten of those make a hundred-thousand; and ten hundred-thousands make a million. As numbers get long, a wall of digits like 1342500 is hard to read. So we group the digits in threes from the right and separate the groups with commas: 1,342,500. The first comma marks the thousands, the second marks the millions — then come billions, and on.
The commas do not change the number at all; they are just rest stops for the eye. (Some traditions group large numbers by ten-thousands instead, but the English-speaking convention — and the one we use throughout — is groups of three.)
Every place is worth ten times the place to its right; a digit's value is its face value times its place; the digit 0 holds an empty place so the others stay put; read from the highest place down; and group big numbers in threes with commas.
| Idea | What it says | Example |
|---|---|---|
| Each place ×10 | a place is worth ten of the one to its right | ones 1 → tens 10 → hundreds 100 |
| Digit × place | a digit's worth depends on its column | the 5 in 50 is worth 50 |
| 0 holds a place | an empty place needs a placeholder | 305 ≠ 35 |
| Read top-down | highest place first; skip silent 0s | 2,008 → "two thousand eight" |
| Group in threes | commas every three digits from the right | 1,342,500 |
Six questions to lock it in. Tap the answer you think is right.
This lesson develops the base-ten place-value system. It addresses CCSS 2.NBT.A.1 (understanding hundreds, tens, and ones in three-digit numbers) and 2.NBT.A.3 (reading and writing numbers to 1000 using base-ten numerals, number names, and expanded form), and extends toward 4.NBT.A.2 (reading and writing multi-digit whole numbers, grouped in threes with commas). The recurring emphases — that a digit's worth is its face value times its place, and that a 0 is a genuine placeholder — guard against the two most common errors, treating digits at face value and dropping placeholder zeros.