Approximation of Numbers: Rounding, Estimation and Accuracy Beyond the Basics

Once you know the basic rule for rounding numbers — look at the next digit, then round up or keep the digit — a bigger question appears: how much accuracy does a task actually need, and what happens to your answer when you approximate too early? This article extends the video lesson on approximation of numbers with new methods, worked examples and practice tasks that show you how to control accuracy instead of guessing at it.

Rounding to significant figures — the method scientists use

In school we usually round to a decimal place. In science and engineering, numbers are more often rounded to a chosen number of significant figures (sig figs). Significant figures are counted from the first non-zero digit on the left.

• In 0.004706 the significant figures are 4, 7, 0, 6 — the leading zeros only show the size of the number.
• In 25 300 the digits 2, 5, 3 are clearly significant; the trailing zeros are placeholders.

Worked example. Round 0.004706 to two significant figures. The first two significant digits are 4 and 7; the next digit is 0, which is less than 5, so the 7 stays: 0.0047.

Worked example. Round 25 349 to three significant figures. The third significant digit is 3, the next digit is 4 (less than 5), so the answer is 25 300.

Why this matters: rounding to a decimal place treats a very small and a very large number the same way, while rounding to significant figures keeps the same relative accuracy for both.

Rounding is not the same as truncating

Cutting off digits (truncation) and rounding give different results, and mixing them up is one of the most common mistakes.

• Truncating 7.489 to one decimal place gives 7.4 — you simply delete what follows.
• Rounding 7.489 to one decimal place gives 7.5, because the following digit is 8.

Truncation always pulls the value towards zero, so repeated truncation creates a one-sided, growing error. Rounding pushes values up and down roughly equally, so the errors tend to cancel out. Use truncation only when a task demands it — for example, when you work out how many whole buses are needed and cannot buy 0.7 of a bus.

Round once, at the end

A rule worth memorising: carry out the full calculation first and round only the final answer. Rounding intermediate steps makes small errors grow.

Compare. Calculate the area of a rectangle with sides 4.48 m and 6.52 m.

• Exact: 4.48 × 6.52 = 29.2096 m² → rounded to one decimal place: 29.2 m²
• Rounded first: 4.5 × 6.5 = 29.25 m² → 29.3 m²

The difference looks tiny, but in a long chain of calculations — or when the numbers are multiplied by a large quantity — such shifts add up quickly.

How wrong can an approximation be? Absolute and relative error

Every approximation carries an error, and you can measure it.

• Absolute error = |exact value − approximate value|. It is expressed in the same units as the number.
• Relative error = absolute error ÷ exact value, usually written as a percentage. It tells you how serious the error is compared with the size of the number.

Worked example. A length of 8.36 m is rounded to 8.4 m. The absolute error is |8.36 − 8.4| = 0.04 m. The relative error is 0.04 ÷ 8.36 ≈ 0.0048 = about 0.5 %.

Worked example. A length of 0.36 m is rounded to 0.4 m. The absolute error is again 0.04 m, but the relative error is 0.04 ÷ 0.36 ≈ 11 %. Identical absolute errors can mean completely different levels of accuracy — that is exactly why significant figures exist.

There is also a useful guarantee: if a number is correctly rounded to a given place, the absolute error is never more than half of that place value. Rounding to the nearest hundred means the error is at most 50; rounding to two decimal places means the error is at most 0.005.

Reading an approximation backwards

Approximation also works in reverse. If a measurement is given as 3.8 kg rounded to one decimal place, the true mass x satisfies:

3.75 ≤ x < 3.85

This interval is called the range of possible values. Being able to state it shows that you understand what a rounded number really claims — it is not one value, but a whole set of values that share the same approximation.

Estimation: approximating before you calculate

Approximation is not only a finishing step — it is also a check. Round every number to one significant figure, do the easy calculation in your head, and compare it with your written answer.

• Estimate 6.87 × 41.2 → 7 × 40 = 280. (The exact value is 283.044, so an answer such as 28.3 or 2830 would immediately look wrong.)
• Estimate 1987 ÷ 19 → 2000 ÷ 20 = 100.
• Estimate 9.6 + 30.4 + 19.8 → 10 + 30 + 20 = 60.

This habit catches misplaced decimal points and calculator slips faster than any other technique.

Approximation and scientific notation

Very large and very small numbers are usually approximated in scientific notation: one digit before the decimal point, multiplied by a power of ten. Rounding then happens inside the first factor only.

• The speed of light, 299 792 458 m/s ≈ 3.00 × 10⁸ m/s (three significant figures).
• The mass of an electron ≈ 9.11 × 10⁻³¹ kg.
• The number π = 3.14159265... ≈ 3.14 or 3.1416, depending on the accuracy required.

Rounding up or down, whatever the digit says

Sometimes real life overrides the mathematical rule, because only one direction makes sense.

• Always round up: 43 students travelling in minibuses that hold 8 people each → 43 ÷ 8 = 5.375 → 6 minibuses.
• Always round down: you have 50 euros and each ticket costs 12 euros → 50 ÷ 12 ≈ 4.17 → 4 tickets.

Ask yourself what the number counts before you apply any rule.

Common mistakes to avoid

• Chain rounding. Rounding 4.4489 to 4.449, then 4.45, then 4.5, then 5. Always round in one step, from the original number.
• Looking at the wrong digit. Only the digit immediately after the rounding position decides — not the whole tail.
• Losing the place value. Rounding 4 987 to the nearest hundred gives 5 000, not 50.
• Dropping a final zero. Rounded to two decimal places, 6.198 is 6.20 — the zero shows the accuracy and must stay.
• Writing = instead of ≈. An approximation deserves the ≈ sign.

Practice tasks

1. Round 0.08462 to two significant figures.
2. Round 74 851 to the nearest thousand and to three significant figures.
3. The length 12.7 cm has been rounded to one decimal place. Write the range of possible true values.
4. The value 5.28 is approximated as 5.3. Find the absolute and the relative error.
5. Estimate 8.15 × 29.4 by rounding each factor to one significant figure, then compare with the exact product.
6. A shelf is 200 cm long and each book is 3.5 cm thick. How many books fit? Decide whether to round up or down.

Answers

1. 0.085
2. 75 000 in both cases
3. 12.65 ≤ x < 12.75
4. Absolute error 0.02; relative error 0.02 ÷ 5.28 ≈ 0.38 %
5. 8 × 30 = 240; exact value 239.61 — a very close estimate
6. 200 ÷ 3.5 ≈ 57.1 → round down: 57 books

Summary

Approximation of numbers is far more than a rule about the digit 5. Choose the accuracy that the task requires, round to decimal places or to significant figures depending on the context, keep full precision until the last step, and check the size of your error. Use estimation to test every answer before you accept it. With these habits, approximation stops being a source of mistakes and becomes a tool for making calculations faster, clearer and more reliable.