Place Value and Rounding

Place value is the idea that where a digit sits changes its value. In 3 472, the 4 is worth 400, but in 3 274 the same digit 4 is worth only 40.

Reading the chart left to right gives the parts: $3000 + 400 + 70 + 2 = 3472$.

Each column is ten times the one to its right. That single fact is what lets you compare numbers: to decide which of two whole numbers is bigger, line them up by their columns and compare from the left. The first column where they differ decides it.

The decimal point does not stop place value — it just continues the pattern. The columns after the point are tenths, hundredths and thousandths, each still ten times smaller than the one before.

So $5.27$ means $5 + 0.2 + 0.07$.

A useful tip: writing extra zeros on the end of a decimal does not change its value, because the new columns are empty. So $0.5 = 0.50 = 0.500$. This makes comparing decimals easier — give them the same number of decimal places, then compare digit by digit. To compare $0.4$ and $0.38$, rewrite $0.4$ as $0.40$; now you can see $0.40$ is larger.

Because each place is ten times the one to its right, multiplying or dividing by powers of 10 just shifts the digits. The decimal point stays put — it is the digits that move.

• $\times 10$ moves every digit one place LEFT — $6.5 \times 10 = 65$.
• $\div 10$ moves every digit one place RIGHT — $6.5 \div 10 = 0.65$.
• $\times 100$ shifts two places, $\times 1000$ shifts three, and so on.

The number of zeros tells you how many places to shift. For $\times 1000$, shift three places: $4.2 \times 1000 = 4200$.

If you run out of digits, fill the gap with zeros. $7 \times 100 = 700$, and $3 \div 100 = 0.03$.

Rounding keeps numbers tidy and useful. To round to a number of decimal places, look at the next digit — the first one you are about to drop.

• If the next digit is 5 or more, round UP.
• If it is 4 or less, round DOWN (leave the digit as it is).

So $3.471$ rounded to 1 decimal place is $3.5$ — the next digit, 7, rounds the 4 up to 5. Rounded to 2 decimal places it is $3.47$ — the next digit, 1, rounds down.

One thing to watch: when rounding up causes a 9 to become 10, you carry. $2.96$ to 1 decimal place is $3.0$, not $2.10$. Keep the trailing zero — it shows the level of accuracy.

Significant figures are the digits that carry the real information in a number. You start counting from the first non-zero digit.

In $3472$, the first significant figure is the 3, the second is the 4, and so on. In $0.00608$, the leading zeros are just placeholders — the first significant figure is the 6.

To round to a number of significant figures, find the last digit you want to keep, then use the next digit and the same up/down rule as before.

• $3472$ to 2 significant figures: keep the 3 and 4, look at the next digit (7), round up — $3500$. The zeros hold the place value.
• $0.00608$ to 1 significant figure: keep the 6, look at the next digit (0), round down — $0.006$.
• $48\,910$ to 2 significant figures: keep 4 and 8, next digit is 9, round up — $49\,000$.

Significant figures are handy because they keep the size of a number while trimming the detail — useful for sensible, easy-to-read estimates.

An estimate is a rough answer you work out quickly by rounding the numbers first — usually to 1 significant figure. It is one of the most powerful checking tools in maths.

Suppose a calculator gives $312 \times 19 = 5928$. Is that sensible? Round each number: $312 \approx 300$ and $19 \approx 20$. Then $300 \times 20 = 6000$. The estimate is close to 5928, so the answer looks right.

Now imagine you had pressed a wrong key and got 59 280. The estimate of 6000 is nowhere near, so you would know to try again.

The same trick works for division. To check $4187 \div 21$, round to $4000 \div 20 = 200$.

Always estimate before trusting a calculator answer, not after. The symbol $\approx$ means "is approximately equal to" — use it whenever you round for an estimate.

Getting confident with place value and rounding now makes lots of later topics easier:

• Measurement uses powers of 10 to convert units — millimetres to metres, grams to kilograms.
• Money problems rely on rounding to the nearest penny or cent, and on sensible estimates.
• Statistics asks you to round averages and to estimate totals from a graph.
• In everyday life, you will use these skills for shopping, reading scales, checking bills and judging whether an answer looks right.

If a topic later feels hard, it is often a place-value step underneath that needs a quick refresh. Come back to this guide whenever you need it — there is no prize for rushing.