Whole Numbers and Place Value

Updated 4 October 2026

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.

What is place value, and why does it matter?

Place value is the rule that a digit's worth depends on the column it sits in, and each column is worth ten times the column to its right. In 3,507 the 3 sits in the thousands column, so it means 3,000; the 5 means 500; the 0 means no tens; and the 7 means 7 ones. Add the parts and the number comes back: 3,000 + 500 + 0 + 7 = 3,507. It works because ten ones make a ten, ten tens make a hundred and ten hundreds make a thousand, so ten digits are enough to write any whole number. The zero is the case that catches people. It holds the tens column open, and without it the 3, 5 and 7 each slide one column to the right, turning 3,507 into 357, about ten times smaller. To check a number, say it aloud column by column: three thousand, five hundred, no tens, seven ones.

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 memorize. 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?

A number gets bigger than ten by bundling: ten ones make one ten, and that ten is written in a new column to the left. Count 8, 9, 10 and the ones column runs out of digits, so it goes back to 0 while the tens column shows 1. Tens and Ones draws the bundle as a stick of ten cubes, so 34 is 3 sticks and 4 loose cubes, and 3 × 10 + 4 = 34. It works because each column holds only the digits 0 to 9, and a tenth item always becomes one of the next column. A full column carries into the next one. Add 1 to 99: the ones reach ten and carry, that fills the tens, and they carry too, so 99 + 1 = 100. Counting to a Hundred repeats that pattern until it feels ordinary. To check, regroup the number: 100 is ten tens, and 10 × 10 = 100.

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.

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?

To round a number, look at the one digit just after the place you are keeping: 5 or more rounds up, 4 or less leaves the kept digit as it is, and every digit after the kept place becomes zero. To round 2,650 to the nearest thousand, keep the 2 in the thousands, look at the hundreds digit, 6, and round up to 3,000. It works because 2,650 lies between 2,000 and 3,000, and every number from 2,500 upward is nearer to 3,000. The job decides the place: the nearest ten for a quick sum, a hundred or thousand for a large amount. A 9 that rounds up carries. Rounding 3,960 to the nearest hundred turns the 9 hundreds into ten hundreds, which carry, so the answer is 4,000. To check, the answer should be a multiple of the place you kept and no more than half a step from the start.

3,470 to the nearest hundred $\to$→ the tens digit is $7\to 3$7 → 3,500. 3,428 to the nearest hundred $\to$→ the tens digit is $2\to 3$2 → 3,400.

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 centimeter is really somewhere in $23.5\le x<24.5$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 the numbers below zero on the number line, written with a minus sign: −3 is three steps below zero, just as 3 is three steps above it. A temperature of 2 degrees that falls by 5 degrees ends at −3, because 2 − 5 = −3: two steps down reach zero and three more go below it. They follow the same rules as positive numbers because they sit on the same line; only the direction is new. Adding moves right, subtracting moves left, and the sign tells you which side of zero you are on. Negative numbers turn comparing around. On the line, −8 lies further left than −2, so −8 is the smaller number, even though 8 is bigger than 2. Subtracting a negative is the other case, and it moves right: 4 − (−2) = 6. To check, count the steps on a number line, or work back: −3 + 5 = 2.

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.

How do you solve a whole-number problem without algebra?

Many word problems with whole numbers have a known shape, and each shape has a method that works without letters for the unknowns. Each method is also an equation in disguise, which Forming Equations shows by writing the same kind of problem with a letter. Six shapes cover most of them: gaps and intervals, supposition, excess and shortage, a container weighed twice, working backwards, and elimination. The first is gaps and intervals. A fence 60 m long with a post every 4 m, from one end to the other, has 60 ÷ 4 = 15 gaps, and one more post than gaps, so 16 posts. Count the gaps first, because the gaps are what the measurement divides. A clock that takes 4 seconds to strike 3 o'clock has 2 gaps between its 3 strikes, so each gap lasts 2 seconds, and striking 7 o'clock, with 6 gaps, takes 12 seconds.

Supposition handles two kinds of thing mixed together. A farm has 20 chickens and rabbits with 56 legs between them. Suppose all 20 were chickens: that is 40 legs, which is 16 short. Each rabbit swapped in for a chicken adds 2 legs, so there are 16 ÷ 2 = 8 rabbits and 12 chickens. Check it: 8 × 4 + 12 × 2 = 32 + 24 = 56. Excess and shortage compares two ways of sharing. Giving each child 4 pencils leaves 10 over, and giving each child 6 is 6 short. The second way needs 10 + 6 = 16 more pencils than the first, and each child gets 2 more, so there are 16 ÷ 2 = 8 children and 4 × 8 + 10 = 42 pencils. Check: 6 × 8 = 48, which is 6 more than 42.

A container weighed twice gives the contents by subtraction. A jar full of rice weighs 1500 g, and half full it weighs 900 g. The difference, 600 g, is half the rice, so the full amount of rice is 1200 g and the empty jar weighs 1500 − 1200 = 300 g. Working backwards undoes each step in reverse order. Someone took half of a pile of cards and 3 more, and 9 were left. Before the 3 more were taken there were 9 + 3 = 12, and that 12 was the half that was left, so the pile started with 24. Check forwards: half of 24 is 12, and 12 + 3 = 15 were taken, leaving 24 − 15 = 9. Multiplying and Dividing Undo Each Other is the idea underneath.

Elimination compares two shopping lists that share an item. Two pens and three rulers cost 13 dollars, and two pens and one ruler cost 9 dollars. The pens are the same in both, so the difference of 4 dollars is the price of two rulers, and one ruler costs 2 dollars. Then two pens cost 9 − 2 = 7 dollars, so a pen costs 3 dollars 50 cents. When no item matches, multiply one list first so that it does: Simultaneous by Elimination shows how. A remainder needs reading in its situation, as Remainders teaches: 38 people in taxis that seat 4 need 10 taxis, not 9.5 or 9, because the 2 people left over still need a ride.

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.