Place Value and Rounding

Place value is the idea that where a digit sits decides its worth. The pattern runs straight through the decimal point: each column is ten times smaller than the one to its left.

So $4.207$ means $4 + 0.2 + 0 + 0.007$. The zero in the hundredths column is doing real work — it holds the place so the 7 stays in thousandths.

Because each place is a power of 10, multiplying or dividing by a power of 10 simply shifts the digits: $\times 100$ shifts two places left, $\div 1000$ shifts three places right. This shifting idea is exactly what standard form, later in this guide's family of topics, is built on.

Rounding trims a number to a useful level of detail. To round, look at the next digit — the first one you would drop — and apply the up/down rule: 5 or more rounds up, 4 or less rounds down.

Decimal places count digits after the point. So $5.7382$ to 2 decimal places: the digit after the second place is 8, which rounds the 3 up — giving $5.74$.

Significant figures count the digits that carry the real information, starting at the first non-zero digit. In $0.0064\,17$ the first significant figure is the 6. So $0.0064\,17$ to 2 significant figures is $0.0064$. For $58\,460$ to 2 significant figures, keep the 5 and 8, the next digit is 4, round down — $58\,000$, with zeros holding the place value.

Two rules matter. Round in one step — judge from the original next digit, never round twice. And when rounding up turns a 9 into 10, you carry: $4.96$ to 1 decimal place is $5.0$.

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 $387 \times 52 = 20\,124$. Is that sensible? Round each number: $387 \approx 400$ and $52 \approx 50$, so $400 \times 50 = 20\,000$. The estimate is close, so the answer looks right.

If you had mis-keyed and got $2012$, the estimate of $20\,000$ would be nowhere near, warning you to try again. The same trick works for division and for awkward decimals: to check $\frac{61.4 \times 8.9}{2.1}$, estimate with $\frac{60 \times 9}{2} = 270$.

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

Here is the big new Stage 9 idea. When a measurement is rounded, it is no longer exact — it stands for a range of values it could really be.

A length given as $24$ cm "to the nearest centimetre" could truly be anything that rounds to 24. Anything from $23.5$ cm up to (but not quite) $24.5$ cm rounds to 24.

The two ends of that range are the limits of accuracy: the lower bound ($23.5$ cm) and the upper bound ($24.5$ cm). The rule is simple — the bounds sit half a rounding unit either side of the rounded value.

• To the nearest cm, the unit is 1, so the bounds are $\pm 0.5$.
• To the nearest 10, the unit is 10, so the bounds are $\pm 5$ — a mass of $80$ kg to the nearest 10 kg lies between $75$ kg and $85$ kg.
• To 1 decimal place, the unit is 0.1, so the bounds are $\pm 0.05$.

This idea matters: it tells you how much a rounded measurement can really be trusted.

Knowing the limits of accuracy lets you reason about a rounded measurement instead of treating it as exact.

Suppose a plank is measured as $150$ cm to the nearest 10 cm. Its true length lies between $145$ cm and $155$ cm. So a shelf gap of exactly $148$ cm might fit the plank — but a gap of $160$ cm certainly will not, because the plank cannot be that long.

The same reasoning settles "could this be true?" questions. If two children's heights are both given as $140$ cm to the nearest cm, are they definitely the same height? No — one could be $139.6$ cm and the other $140.4$ cm. They are close, but the rounding hides a possible difference of nearly a whole centimetre.

So whenever a measurement is rounded, write down its bounds before drawing a conclusion. The bounds tell you exactly what the number can, and cannot, be.

Confident place value, rounding and bounds make many later topics easier:

• Standard form depends entirely on powers of 10 to handle very large and very small numbers.
• Measurement uses bounds to judge whether a real object will fit, balance or match within the accuracy given.
• 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, and the limits of accuracy tell you how far a rounded figure can be trusted. Come back to this guide whenever you need it — there is no prize for rushing.