Approximation: Significant Figures, Decimal Places and Errors

Close Enough, and Knowing How Close

A calculator might tell you that a length is 7.348469228 cm, but no ruler can measure that precisely, and no examiner wants to see all those digits. Measurements are never perfectly exact, and answers must be rounded sensibly. Estimating quickly also helps you catch calculator slips before they cost marks. This tutorial covers approximation as tested in WAEC, NECO, BECE and JAMB mathematics: rounding to the nearest whole number, ten or hundred, decimal places, significant figures, standard form, estimation, truncation, upper and lower bounds, absolute, relative and percentage error, and how errors combine, with worked examples throughout.

Rounding to the Nearest Unit, Ten or Hundred

Look at the digit immediately to the right of the place you are rounding to:

If it is 5 or more, round up (add 1 to the digit you are keeping).
If it is less than 5, round down (keep the digit unchanged).

Worked example 1

Round 46,857 to the nearest (a) ten (b) hundred (c) thousand.

(a) The units digit is 7, so round up: 46,860.

(b) The tens digit is 5, so round up: 46,900.

Decimal Places

Rounding to n decimal places (d.p.) keeps n digits after the decimal point.

Worked example 2

Round 3.14159 to (a) 1 d.p. (b) 2 d.p. (c) 4 d.p.

(a) 3.1 (b) 3.14 (c) 3.1416, because the fifth digit, 9, rounds the 5 up to 6.

Keep any zeros that the rounding requires. For example, 2.398 to 2 d.p. is 2.40, not 2.4, because the final 0 shows that the value is accurate to two decimal places.

Significant Figures

Significant figures (s.f.) count the digits that carry meaning about the size and precision of a number, starting from the first non-zero digit. The rules are:

All non-zero digits are significant.
Zeros between non-zero digits are significant: 4007 has 4 s.f.
Leading zeros are not significant. They only fix the position of the decimal point: 0.0045 has 2 s.f.
Trailing zeros after a decimal point are significant: 2.50 has 3 s.f. Trailing zeros in a whole number, such as 46,900, may or may not be significant. Standard form removes the doubt.

Worked example 3

Round (a) 0.0045678 to 2 s.f. (b) 46,857 to 3 s.f. (c) 7.0049 to 3 s.f.

(a) The first significant digit is 4, so keep 4 and 5. The next digit is 6, so round up: 0.0046.

(b) Keep 4, 6 and 8. The next digit is 5, so round up: 46,900. The zeros hold the place value, and the number is still about forty-seven thousand.

(c) Keep 7, 0 and 0. The next digit is 4, so round down: 7.00. The zeros must be written, to show three significant figures.

Significant figures are usually the best way to round measured or scientific values, because they describe precision, whatever the size of the number.

Standard Form

A number in standard form (scientific notation) is written as A × 10ⁿ, where 1 ≤ A ≤ 9.999… and n is a whole number.

5,600,000 = 5.6 × 10⁶ (the decimal point moves 6 places to the left)
0.000345 = 3.45 × 10⁻⁴ (the decimal point moves 4 places to the right)

Worked example 4

Evaluate (3 × 10⁴) × (5 × 10⁻²), giving the answer in standard form.

3 × 5 = 15, and 10⁴ × 10⁻² = 10², so the product is 15 × 10² = 1.5 × 10³.

The first form, 15 × 10², is not in standard form, because 15 is greater than 10.

Estimation: Approximation for Checking Answers

To estimate an answer, round every number to 1 significant figure, then calculate mentally.

Worked example 5

Estimate (49.7 × 20.3) ÷ 9.8.

≈ (50 × 20) ÷ 10 = 1000 ÷ 10 = 100.

The exact value is about 102.95, so the estimate is close. If your calculator gave 1029.5, the estimate would show immediately that a decimal point had slipped.

Truncation

Truncating is a cruder form of approximation. It means cutting a number off at a given place, without rounding. For example, 3.14159 truncated to 2 d.p. is 3.14, and 2.678 truncated to 1 d.p. is 2.6 (whereas rounding gives 2.7). Computers sometimes truncate when they convert decimals to whole numbers.

Upper and Lower Bounds

Every approximation hides a range of possible values. A measurement rounded to a given accuracy could have been anywhere within half a unit of the stated value, either way.

A length given as 6.5 cm to 1 d.p. lies in the range 6.45 cm ≤ x < 6.55 cm.
A crowd given as 3,000 to the nearest hundred lies between 2,950 and 3,050.

The smallest possible value is the lower bound, and the largest is the upper bound. The upper bound itself is excluded, because 6.55 would round up to 6.6.

Worked example 6: bounds in calculations

A rectangle measures 8 cm by 5 cm, each to the nearest centimetre. Find the greatest and least possible areas.

Length: 7.5 to 8.5 cm. Width: 4.5 to 5.5 cm.

Greatest area = 8.5 × 5.5 = 46.75 cm². Least area = 7.5 × 4.5 = 33.75 cm².

The rule is that, to get the largest result, you use the upper bounds when multiplying or adding, but when dividing or subtracting, you divide the upper bound by the lower bound (or subtract the lower bound from the upper bound).

Errors in Approximation

Absolute error = |measured value − true value|
Relative error = absolute error ÷ true value
Percentage error = (absolute error ÷ true value) × 100%

When the true value is unknown, the absolute error of a rounded measurement is taken as half a unit of the last digit (its maximum possible error).

Worked example 7

A student measures a length as 4.8 cm when the true length is 5.0 cm. Find the absolute and percentage errors.

Absolute error = |4.8 − 5.0| = 0.2 cm. Percentage error = 0.2 ÷ 5.0 × 100% = 4%.

Worked example 8

The mass of a parcel is given as 2.5 kg, to the nearest 0.1 kg. Find the percentage error.

The maximum absolute error is half of 0.1, which is 0.05 kg. Percentage error = 0.05 ÷ 2.5 × 100% = 2%.

Combining errors

When quantities are added or subtracted, their absolute errors add.
When quantities are multiplied or divided, their percentage errors add (approximately).

For example, a 2% error in the length and a 3% error in the width give about a 5% error in the area. The same rules are used in physics practicals.

How Many Figures Should an Answer Have?

Follow the question's instruction exactly, such as "correct to 2 decimal places" or "to 3 significant figures".
If none is given, give the answer to the same number of significant figures as the least precise data value, typically 3 s.f.
Never round in the middle of a calculation. Keep full accuracy (store values in the calculator's memory) and round only the final answer.
For money, give answers to the nearest kobo (2 d.p.) or naira, as appropriate.

Common Mistakes with Approximation

Counting leading zeros as significant. 0.0045 has only 2 s.f.
Dropping place-holding zeros, such as rounding 46,857 to 3 s.f. as 469 instead of 46,900.
Leaving off required zeros: 2.398 to 2 d.p. is 2.40.
Rounding at each step, which builds up errors in the final answer.
Writing A outside the range 1 to 10 in standard form, such as 15 × 10².

Approximation: Practice Questions

Round 0.08562 to 2 significant figures.
Round 5.3967 to 2 decimal places.
Express 0.000782 in standard form.
Estimate 61.8 × 0.49 ÷ 2.9 by rounding each number to 1 s.f.
A length is 12 m to the nearest metre. State its lower and upper bounds.
A school has 1,850 students, which a report gives as 2,000. Find the percentage error.
A square has side 6 cm to the nearest cm. Find the greatest possible area.

Answers: 1. 0.086. 2. 5.40. 3. 7.82 × 10⁻⁴. 4. 60 × 0.5 ÷ 3 = 10. 5. 11.5 m ≤ l < 12.5 m. 6. 150 ÷ 1,850 × 100 ≈ 8.1%. 7. 6.5² = 42.25 cm².

Conclusion

Approximation makes numbers manageable and honest about their precision. Round using the digit to the right of the last place kept: 5 or more rounds up. Decimal places count digits after the point, while significant figures count from the first non-zero digit, ignoring leading zeros. Standard form A × 10ⁿ handles very large and very small numbers, and estimating with 1-significant-figure values catches calculator errors. A rounded value lies within half a unit of its stated value, between its lower and upper bounds, and percentage error = (absolute error ÷ true value) × 100%. Round only the final answer, never in the middle of a calculation.