This note covers rounding to decimal places and significant figures, estimation, percentage error, and calculating with numbers in standard form. These skills sit underneath almost every calculation in the course, because a correct method with sloppy rounding still loses accuracy.
Key ideas
Decimal places (d.p.) count digits after the decimal point. Significant figures (s.f.) count digits from the first non-zero digit. To round, look at the digit just after the one you keep. If it is 5 or more, round up. Otherwise leave the kept digit alone.
- \(7.2846\) is \(7.28\) to 2 d.p. and \(7.28\) to 3 s.f.
- \(0.004052\) is \(0.0041\) to 2 s.f. Leading zeros are never significant.
- \(30\,500\) to 2 s.f. is \(31\,000\) (or \(3.1\times10^{4}\)). Zeros used as place holders stay in the number.
Unless a question says otherwise, give final answers to 3 significant figures. Keep full calculator accuracy in the middle of a calculation and only round at the end. Use the answer memory on your GDC rather than retyping rounded values.
Estimation. Round each quantity to one significant figure before calculating to check that an answer is sensible. For example \(\dfrac{48.7\times 3.9}{0.52}\approx\dfrac{50\times4}{0.5}=400\), so an answer of \(365\) is plausible but \(36.5\) is not.
Percentage error. When you use an approximate value \(v_A\) instead of an exact value \(v_E\), the size of the error compared with the true size is the percentage error:
\[ \varepsilon = \left|\frac{v_A - v_E}{v_E}\right| \times 100\% \]The absolute value bars mean the answer is never negative. The exact value always goes in the denominator.
Standard form is \(a\times10^{k}\) with \(1\le a\lt 10\) and \(k\) an integer. Large numbers have positive \(k\) and small numbers have negative \(k\). To multiply, multiply the mantissas and add the exponents. To divide, divide the mantissas and subtract the exponents. To add or subtract, first give both numbers the same power of ten. After any calculation, check that the mantissa is back between 1 and 10.
On a GDC use the key for "\(\times10^{x}\)" to enter numbers such as \(3.00\times10^{5}\). Check the display, because calculators may show the answer as \(4.98667\mathrm{E}2\), which means \(4.98667\times10^{2}\).
Units are part of the answer. Convert to a common unit before you calculate, for example kilometres to metres, or hours to seconds.
Choosing the accuracy. Some questions ask for a number of decimal places, some for significant figures, and some give no instruction. Money is usually given to 2 decimal places, since the smallest unit is a cent or a penny. Measurements and scientific quantities are normally given to 3 significant figures. If a question says "to the nearest 10" or "to the nearest hundred", round to that place value instead. Whatever you choose, apply it only to the final answer.
Reading percentage error. A small percentage error means the approximation is close to the true value. A percentage error of \(0.249\%\) means the estimate is off by about a quarter of one per cent of the true size. The same absolute difference gives a much larger percentage error when the true value is small, which is why the error is expressed relative to \(v_E\) and not as a plain difference.
Truncating. Truncating means cutting digits off without rounding up, so \(3.789\) truncated to 1 d.p. is \(3.7\), whereas rounded it is \(3.8\). Questions will say which one to use.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Decimal places | Digits after the decimal point. Used for money and fixed-precision measurements. |
| Significant figures | Digits counted from the first non-zero digit. Default accuracy for final answers. |
| \(a\times10^{k}\), \(1\le a\lt10\), \(k\in\mathbb{Z}\) | Standard form of a number. |
| \((a\times10^{m})(b\times10^{n}) = ab\times10^{m+n}\) | Multiplying in standard form. Renormalise if \(ab\ge10\). |
| \(\dfrac{a\times10^{m}}{b\times10^{n}} = \dfrac{a}{b}\times10^{m-n}\) | Dividing in standard form. Renormalise if \(a/b\lt1\). |
| \(\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right|\times100\%\) | Percentage error of an approximate value \(v_A\) against an exact value \(v_E\). |
| Estimate | A quick value found by rounding inputs to 1 s.f. Used to check reasonableness. |
Worked examples
Example 1 Light from the Sun travels \(1.496\times10^{8}\) km to Earth at a speed of \(3.00\times10^{5}\) km s\(^{-1}\). Find the time taken, in seconds, to 3 significant figures and in standard form.
Time \(=\) distance \(\div\) speed.
\(t = \dfrac{1.496\times10^{8}}{3.00\times10^{5}}\)
Divide the mantissas: \(1.496\div3.00 = 0.49867\ldots\) Subtract the exponents: \(8-5=3\).
So \(t = 0.49867\times10^{3}\). The mantissa is less than 1, so write \(0.49867\times10^{3} = 4.9867\times10^{2}\).
Round to 3 s.f.: \(4.99\times10^{2}\).
Answer: \(4.99\times10^{2}\) s.
Example 2 A circular reservoir has radius \(6.4\) m. An engineer estimates its surface area as \(129\) m\(^2\). Find the percentage error in the estimate, to 3 significant figures.
The exact area is \(v_E = \pi\times6.4^{2} = 128.6796\ldots\) m\(^2\) (keep the full value on your GDC).
The approximate value is \(v_A = 129\).
\(\varepsilon = \left|\dfrac{129 - 128.6796}{128.6796}\right|\times100\)
\(= \dfrac{0.3204}{128.6796}\times100 = 0.2490\ldots\)
Answer: \(0.249\%\).
Example 3 A server farm handles \(3.6\times10^{5}\) requests per hour. Each request uses \(2.5\times10^{-3}\) kWh of energy. A storage cell holds \(1.8\times10^{-2}\) kWh. Find how many cells are needed to store one hour of the farm's energy use, that is \(\dfrac{(3.6\times10^{5})(2.5\times10^{-3})}{1.8\times10^{-2}}\), in standard form.
Numerator: \(3.6\times2.5 = 9\) and \(5+(-3) = 2\), so the numerator is \(9\times10^{2}\).
Divide: \(\dfrac{9\times10^{2}}{1.8\times10^{-2}} = \dfrac{9}{1.8}\times10^{2-(-2)} = 5\times10^{4}\).
The mantissa \(5\) is between 1 and 10, so no adjustment is needed.
Answer: \(5\times10^{4}\) cells (that is, \(5.00\times10^{4}\) to 3 s.f.).
Common errors
- Common error: rounding at every step. Early rounding builds up and can change the third significant figure of the final answer.
- Common error: dividing by the approximate value in a percentage error. The exact value always goes on the bottom.
- Common error: leaving an answer such as \(14.4\times10^{2}\), which is not in standard form because the mantissa is bigger than 10.
- Common error: counting leading zeros as significant, so that \(0.00305\) is thought to have 6 s.f. It has 3.
- Common error: adding powers of ten when adding numbers. Powers are added only when multiplying.
Exam tip: write the unrounded value (for example \(128.6796\ldots\)) before you give the rounded answer, and state how many significant figures or decimal places you used.
Check yourself
- Write \(0.000 \,0572\) in standard form and round \(5.0478\) to 3 s.f.
Answer
\(0.0000572 = 5.72\times10^{-5}\). Rounding \(5.0478\) to 3 s.f.: the 4th digit is 7, so round up the 4 to give \(5.05\). - Calculate \((4.2\times10^{6}) + (3.5\times10^{5})\) in standard form.
Answer
Write \(3.5\times10^{5} = 0.35\times10^{6}\). Add: \(4.2+0.35 = 4.55\). Answer: \(4.55\times10^{6}\). - A length is exactly \(2.50\) m but is recorded as \(2.46\) m. Find the percentage error.
Answer
\(\left|\dfrac{2.46-2.50}{2.50}\right|\times100 = \dfrac{0.04}{2.50}\times100 = 1.6\%\) (that is, \(1.60\%\) to 3 s.f.). - Calculate \(\dfrac{6.0\times10^{-4}}{1.5\times10^{3}}\) and give the answer in standard form.
Answer
\(\dfrac{6.0}{1.5}=4\) and \(-4-3=-7\). Answer: \(4\times10^{-7}\).