Scientific Notation using Significant Figures
Every question with a fully worked solution.
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Theory
Significant figures are the digits that carry meaning in a number, counted from the first non-zero digit. This Year 9 maths page shows which zeros count, how to round large numbers and small decimals to a given number of significant figures, how to estimate, and how to give calculator answers in scientific notation.
Significant figures (s.f.) are the digits that tell you how accurate a number is. Counting starts at the first non-zero digit.
Every non-zero digit is significant. Zeros depend on where they sit:
Leading zeros at the front, as in \(0.0045\), never count. They only place the decimal point.
Zeros between other digits, as in \(4.07\), always count.
Trailing zeros after a decimal point, as in \(6.30\), count, because they were written on purpose. Trailing zeros in a whole number, as in \(7900\), are usually placeholders and do not count.
Decimal places are different: they count digits after the point. \(0.00412\) has five decimal places but three significant figures.
| Number | Significant figures | Why |
|---|---|---|
| \(0.00560\) | \(3\) | leading zeros do not count; the last \(0\) does |
| \(2.005\) | \(4\) | the zeros sit between significant digits |
| \(7900\) | \(2\) | the zeros are placeholders |
| \(30.40\) | \(4\) | a zero between, and a zero after the point |
How to round to significant figures
- Find the first non-zero digit. That is the first significant figure.
- Count along to the number of significant figures needed and mark a cut.
- Look at the next digit: \(5\) or more, round up; otherwise leave it. A \(9\) that rounds up carries to the left.
- Fill any whole-number places you removed with zeros, so the size stays the same.
- For a calculator answer, write the full result in scientific notation first, then round the front number. Round only once, at the end.
(i) \(0.03080\)
(ii) \(62.0\)
(iii) \(50\,400\)?
(i) Start at the first non-zero digit, the \(3\). The leading zeros do not count. The middle zero and the zero at the end, after the point, do.
| \(0.0\underline{3080}\) | \(\rightarrow\) | \(\mathbf{4}\) |
(ii) The zero after the point was written on purpose, so it counts.
| \(\underline{62.0}\) | \(\rightarrow\) | \(\mathbf{3}\) |
(iii) The zero between \(5\) and \(4\) counts. The last two zeros only hold the place.
| \(\underline{504}00\) | \(\rightarrow\) | \(\mathbf{3}\) |
(i) \(7284\) to \(2\) significant figures
(ii) \(0.007\,146\,2\) to \(3\) significant figures.
(i) Count two digits and cut. The next digit is \(8\), which is \(5\) or more, so round up. Fill the gap with zeros.
| \(72\,|\,84\) | \(\rightarrow\) | \(73\,|\,00\) |
| \(7284\) | \(\approx\) | \(\mathbf{7300}\) |
(ii) The first significant figure is the \(7\). Count three from there. The next digit is \(6\), so round the \(4\) up.
| \(0.00714\,|\,62\) | \(\rightarrow\) | \(0.00715\) |
| \(0.0071462\) | \(\approx\) | \(\mathbf{0.00715}\) |
The leading zeros stay. They hold the place value.
(i) Round \(29.96\) to \(3\) significant figures.
(ii) A town's population of \(3\,486\,210\) is rounded to the nearest thousand. How many significant figures does the rounded value have?
(i) Cut after three figures. The next digit is \(6\), so the \(9\) rounds up. That carries: \(29.9\) becomes \(30.0\).
| \(29.9\,|\,6\) | \(\rightarrow\) | \(30.0\) |
| \(29.96\) | \(\approx\) | \(\mathbf{30.0}\) |
Keep the zero after the point. It shows the answer has \(3\) significant figures.
(ii) Round first, then count. The digit after the thousands is \(2\), so round down.
| \(3\,486\,210\) | \(\approx\) | \(3\,486\,000\) |
| \(\underline{3486}000\) | \(\rightarrow\) | \(\mathbf{4}\) |
The three zeros at the end are placeholders.
(i) \(48.6^2\)
(ii) \(\dfrac{5.2 \times 10^{-3}}{7 \times 10^{2}}\)
(i) Work it out in full. Write it in scientific notation. Then round the front number.
| \(48.6^2\) | \(=\) | \(2361.96\) |
| \(\) | \(=\) | \(2.36196 \times 10^3\) |
| \(\) | \(\approx\) | \(\mathbf{2.36 \times 10^3}\) |
(ii) Use brackets on the calculator for the whole bottom line.
| \(\dfrac{5.2 \times 10^{-3}}{7 \times 10^{2}}\) | \(=\) | \(0.000\,007\,428\,57\ldots\) |
| \(\) | \(=\) | \(7.42857\ldots \times 10^{-6}\) |
| \(\) | \(\approx\) | \(\mathbf{7.43 \times 10^{-6}}\) |
Three significant figures means three digits in the front number. Round once, at the end.
Common pitfalls
Frequently asked questions
What are significant figures?
Significant figures are the digits that carry meaning in a number, counted from the first non-zero digit. \(0.0045\) has \(2\): the \(4\) and the \(5\).
Do zeros count as significant figures?
Leading zeros never count. Zeros between other digits always count. Zeros at the end count after a decimal point, and are usually placeholders in a whole number.
How do you round to 2 significant figures?
Find the first non-zero digit, count two digits and look at the next one. Round up if it is \(5\) or more, and fill removed whole-number places with zeros: \(7284 \approx 7300\).
What is the difference between decimal places and significant figures?
Decimal places count the digits after the point. Significant figures count from the first non-zero digit. \(0.00412\) has \(5\) decimal places but \(3\) significant figures.
How do you give an answer in scientific notation to 3 significant figures?
Write the full answer in scientific notation, then round the front number to \(3\) digits: \(2361.96 = 2.36196 \times 10^3 \approx 2.36 \times 10^3\).