Place Value and Rounding

Place value is the idea that where a digit sits decides its worth. The pattern does not stop at the decimal point — it carries straight on, with each column ten times smaller than the one to its left.

So $6.308$ means $6 + 0.3 + 0 + 0.008$. The zero in the hundredths column is doing real work — it holds the place so the 8 stays in thousandths.

To compare and order decimals, give them the same number of decimal places by filling gaps with zeros, then compare column by column from the left. To order $0.7$, $0.68$ and $0.704$, rewrite as $0.700$, $0.680$ and $0.704$ — now the digits line up and the order is clear.

Because each place is ten times the one to its right, multiplying or dividing by a power of 10 simply shifts the digits. The point stays put — the digits move.

• $\times 10$ shifts one place left, $\times 100$ two places, $\times 1000$ three.
• $\div 10$ shifts one place right, $\div 100$ two places, $\div 1000$ three.

The number of zeros in the power of 10 equals the number of places to shift. If you run out of digits, fill the gap with zeros: $5 \times 1000 = 5000$, and $3 \div 1000 = 0.003$.

This is also why a power of 10 can be written as $10^2$ or $10^3$ — the index is the number of places. You will see this shorthand again as standard form.

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 — and use the up/down rule.

So $7.4863$ to 2 decimal places: the digit after the second place is 6, which rounds the 8 up — giving $7.49$. To 1 decimal place, the digit after the first place is 8, so the 4 rounds up to $7.5$.

Two things to watch. First, round in one step — always look at the original next digit, never round twice. Rounding $7.4863$ to $7.49$ and then to $7.5$ is wrong; to 1 decimal place the answer is $7.5$ only because the second digit, 8, rounds up. Second, when rounding up turns a 9 into 10 you carry: $3.96$ to 1 decimal place is $4.0$, and the trailing zero stays to show the accuracy.

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

In $4.0726$, the first significant figure is 4, the second is 0, the third is 7. In $0.00519$, the leading zeros are placeholders — the first significant figure is the 5.

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

• $48\,360$ to 2 significant figures: keep the 4 and 8, the next digit is 3, round down — $48\,000$. The zeros hold the place value.
• $0.0073914$ to 2 significant figures: keep the 7 and 3, the next digit is 9, round up — $0.0074$.
• $5.96$ to 1 significant figure: keep the 5, the next digit is 9, round up — $6$.

Significant figures keep the size of a number while trimming detail, which makes them perfect for quick, sensible estimates.

An estimate is a rough answer found 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 $412 \times 28 = 11\,536$. Is that sensible? Round each number: $412 \approx 400$ and $28 \approx 30$, so $400 \times 30 = 12\,000$. The estimate is close, so the answer looks right.

If you had mis-keyed and got $115\,360$, the estimate of $12\,000$ would be nowhere near, warning you to try again. The same trick works for division: to check $6195 \div 29$, round to $6000 \div 30 = 200$.

Always estimate before trusting an answer. The symbol $\approx$ means "is approximately equal to" — use it whenever you round.

Confident place value and rounding make many later topics easier:

• Measurement uses powers of 10 to convert units — millimetres to metres, grams to kilograms.
• Standard form in later stages relies entirely on powers of 10 to handle very large and very small numbers.
• Statistics asks you to round averages sensibly and to estimate totals from graphs.
• In everyday life, you will use these skills for shopping, reading scales, checking bills and judging whether an answer looks right.

If a later calculation feels off, an estimate is your first line of defence. Come back to this guide whenever you need it — there is no prize for rushing.