Place value
Place value is why 24 and 42 are different numbers made of the same digits: the position of a digit decides how much it is worth. Everything we do with bigger numbers rests on it.
The place value tool
Bundle them into tens
Tap a square to add or remove a counter at that point
0 tens 0 ones
0 10 0 0
Tap a square to put a counter down.
How place value works
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Count in ones, then bundle
Counters are counted one at a time until there are ten, and then those ten are bundled into a single ten. Nothing is added or thrown away: ten loose ones and one bundle are the same amount in two shapes. Doing this twice gives two tens.
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Read the digits as bundles and leftovers
In 24, the left digit counts the tens and the right digit counts the ones: two tens and four left over, which is twenty-four. The digits do not mean 2 and 4 added together. The 2 is worth twenty because of where it stands, and that is the whole idea of place value.
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Compare by tens first
To compare 62 and 26, look at the tens: six tens beats two tens, so 62 is the bigger number, whatever the ones digits say. Two-digit numbers are compared from the left because the left digit is worth ten times more than the one beside it.
Worked examples
24
Two tens and four ones. Bundle twenty loose counters into two groups of ten and four are left over, so 24 is 2 × 10 + 4. The 2 is worth twenty, not two.
13
One ten and three ones. A single bundle of ten plus three loose counters is thirteen, and that is why the number is written with a 1 in front of a 3 rather than as "three-teen". Every number from ten to nineteen is one ten and a few ones.
30
Three tens and no ones. The zero is not nothing to write — it holds the ones place open, so the 3 stays in the tens column and keeps its value of thirty. Without the zero, 30 would read as 3.
Where it goes wrong
- Adding the digits instead of reading the places Reading 24 as 2 + 4 = 6 turns a tidy method into nonsense. The left digit tells you how many tens there are, so 24 is two tens, which is twenty, and four more.
- Swapping the digits without noticing Writing 26 when the answer is 62 is a place-value slip, not a careless one, and it always deserves a second look: the same two digits in the other order are a different amount, thirty-six apart. Check by bundling: two tens or six tens?
- Dropping the placeholder zero Thirty counters bundled into three tens is written 30, not 3. The zero keeps the 3 in the tens column, and a missing zero changes the amount by a factor of ten.
Questions
What does place value mean in plain terms?
It means a digit is worth different amounts depending on where it sits in the number. In 24 the 2 stands for twenty because it is in the tens place, while in 42 the same 2 stands for two. Our number system is built in tens, so each place to the left is worth ten times the place to its right.
Why do ten ones make one ten?
Because we count in a base of ten. There is no law of nature that says so; it is a choice our number system makes, and it is a good one because we have ten fingers to count on. Once ten ones become one ten, a single symbol can stand for a whole bundle, which is what makes writing big numbers possible.
Why is thirteen not written as "three-teen"?
The words are irregular — eleven and twelve come from older systems, and the teens put the ones before the tens when we say them — but the digits never are. 13 is always one ten and three ones, written left to right, tens first. Learning to match the words to the digits is a real step, and it is worth practising out loud.
What is the zero doing in 20?
It holds the ones place. Twenty is two tens and no ones, and the zero marks that empty place so the 2 stays in the tens column and keeps its value of twenty. A placeholder zero is doing work: it is the difference between two and twenty.
How does place value help with adding?
It makes adding columns possible. To add 24 and 13 you add the ones first (4 + 3 = 7), then the tens (2 + 1 = 3), and the answer is 37. When the ones make ten or more, those ten bundle into another ten and move left — the carry — which is the same bundling you do on this page with counters.