Whole Numbers and Place Value
Ten digits write every whole number there has ever been. That is not a coincidence and it is not a convention we could easily have chosen differently — it is a single idea, applied over and over: a digit's worth depends on where it sits, and each column is worth ten of the one to its right.
This page follows that idea from the first count to a million, then out to the two things people find genuinely awkward — rounding to the right level of detail, and numbers below zero. Each rule comes with a worked example and the reason it holds.
What is place value, and why does it matter?
Place value is the rule that position carries size. In 3,507 the 3 is three thousand, the 5 is five hundred, the 0 holds the tens column open, and the 7 is seven ones. The digits count; the columns weigh.
Children meet it in stages, and each stage is a genuine step rather than more of the same. Counting comes first: Counting to Ten settles that the last word you say is how many there are, and One and One makes the first addition a picture rather than a fact to memorise. Writing Numbers to Twenty is where the written symbol and the spoken word are matched up, which is a surprising amount of work for a five-year-old — fourteen says the four first and writes it second.
Comparison arrives alongside counting rather than after it. More and Fewer teaches the comparison by matching things one to one, so the answer does not depend on counting accurately. And the number bonds — Number Bonds to Five, then Number Bonds to Ten — are the small set of facts everything later leans on, because ten is the number the whole system is built around.
How does a number get bigger than ten?
By bundling. Ten ones become one ten, and the ten goes in its own column. Tens and Ones is the lesson where that bundle becomes visible, and Counting to a Hundred is where the pattern repeats often enough to be believed.
Then it happens again, unchanged. Ten tens make a hundred (Hundreds), ten hundreds make a thousand (The Thousand Cube, which draws the thousand as a solid block so the jump from a flat hundred is something you can see), and the same move carried out three more times reaches a million — the subject of Numbers to a Million.
| Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 4 | 2 | 0 | 7 | 0 | 0 | 0 |
4,207,000 reads as four million, two hundred and seven thousand. The commas are not decoration — they group the digits in threes so the word thousand and the word million land in the right places when you say it aloud.
How do you round a number, and to what?
Look at the single digit just after the place you are keeping: five or more rounds up, four or less leaves it. Everything to the right of the kept place becomes zero.
3,470 to the nearest hundred → the tens digit is ,500. 3,428 to the nearest hundred → the tens digit is ,400.
Rounding to the Nearest Ten starts this on a number line, where "round up or down" is visibly a question about which marker you are closer to, and Rounding to the Nearest Hundred and Thousand shows the same move sliding one column left without changing.
When do you use significant figures instead?
When the size of the number is not known in advance. Decimal places count from the point; significant figures count from the first non-zero digit, so they keep the same proportion of detail whether the number is a million or a thousandth. Significant Figures makes the counting rule precise, including the leading zeros that never count and the trailing ones that sometimes do.
One significant figure is enough to check almost any calculation before you trust it — Estimating with One Significant Figure turns 38 × 512 into 40 × 500 = 20,000 in a second, which is all you need to catch an answer that came out as 1,945 or 194,560. The estimation guide covers that habit at length.
Rounding also has a cost, and the cost can be stated exactly. Bounds and Error Intervals shows that a length recorded as 24 cm to the nearest centimetre is really somewhere in 23.5 ≤ x < 24.5, and Percentage Error turns that gap into a proportion, which is the form that lets you compare an error on a bridge with an error on a bolt.
What are negative numbers?
Negative numbers are what you get when the number line does not stop at zero. They are as real as the positive ones and behave under exactly the same rules — the awkwardness is entirely about direction, and it clears up once direction is the thing you are tracking.
Below Zero introduces them where they already live for most people: temperature, floors below ground, money owed. Ordering Negative Numbers then fixes the one comparison that catches everyone — −7 is less than −3, because it is further down, even though 7 is more than 3.
Adding and subtracting across zero
Addition moves right and subtraction moves left, whichever side of zero you start on. Adding and Subtracting Across Zero works the crossings: −4 + 7 = 3 is four steps up to zero and three more beyond it.
Subtracting a negative is the move that needs a reason rather than a rule, and Subtracting Negative Numbers gives two. The first is a pattern: 5 − 1 = 4, 5 − 0 = 5, 5 − (−1) = 6, climbing by one each time with no gap where the sign changes. The second is meaning: taking away a debt of three leaves you three better off, so 5 − (−3) = 8.
Absolute Value is the tool for the times when only the distance matters. |−7| = 7 — how far from zero, not which way — and it is what lets a tolerance be written once instead of twice.
Why two negatives multiply to a positive
Because multiplying by a negative reverses direction, and reversing twice leaves you facing forwards. Multiplying and Dividing Negative Numbers shows the same argument as a pattern that refuses to break:
3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Each answer is 2 higher than the last. Keeping that steady climb going gives (−1) × (−2) = 2.
The four mistakes worth naming
- Reading zeros as nothing. A zero holds a column open. Dropping it from 4,207,000 does not shorten the number, it shrinks it by a factor of ten thousand.
- Rounding in two steps. 3,449 to the nearest hundred is 3,400. Rounding to 3,450 first and then to 3,500 lets a digit vote twice.
- Treating −7 as bigger than −3. Further from zero is not further up. Compare positions on the line, not sizes with a sign stuck on.
- Counting leading zeros as significant. 0.00427 is three significant figures, not five. Significance starts at the first digit that is not a zero.
Where this leads next
Place value is the foundation the written methods stand on: carrying and borrowing are place value in motion, and they are worked through in the guide to mental and written arithmetic methods. Extending the columns to the right of the ones, instead of the left, gives tenths and hundredths — the subject of the decimals guide and of the fuller decimals, percentages and interest ladder. And asking which whole numbers divide which leads into factors, multiples and primes.
Practise it in the game
Math Challenge is a mental-math game with an illustrated lesson for every step above — pictures, a worked derivation, and try-it problems that re-teach the exact question you missed rather than a nearby one. The lessons named on this page are part of a catalog of 800+ across the whole ladder, from first counting to university-entrance topics.
For drilling rather than reading, the Negative Numbers topic runs the sign rules until they stop needing thought, and Rounding does the same for the place-value decisions.
Your turn
Three to try — tap what you get.
What is the 7 worth in 3,742?
Round 4,861 to the nearest hundred.
Which is largest?