Rounding & Estimation

Why Round Numbers?

Exact numbers are not always necessary or even useful. Rounding makes numbers easier to work with, communicate, and check. Estimation lets you verify that a calculated answer is in the right ballpark before trusting it.

Examples of rounding in everyday life:

• A population of 4,783,219 is reported as "approximately 4.8 million"
• A measurement of 3.14159... is used as 3.14 (or even just 3)
• A supermarket bill is estimated to avoid overpaying

Key Terms

• Rounding: replacing a number with a nearby, simpler value of appropriate precision
• Decimal places (d.p.): the number of digits after the decimal point
• Significant figures (s.f.): the meaningful digits in a number, starting from the first non-zero digit
• Estimation: using rounded values to quickly approximate an answer
• Truncation: cutting digits off without rounding (different from rounding — 3.9 truncated to a whole number = 3, but rounded = 4)

The Rounding Rule

1. Identify the target place (e.g. tens, hundreds, 1 decimal place, 3 significant figures)
2. Look at the digit immediately to the right of the target place
3. If it is 5 or more → round up (increase the target digit by 1)
4. If it is 4 or less → round down (keep the target digit the same)
5. Replace all digits to the right of the target with zeros (for whole numbers) or drop them (for decimals)

Rounding to the Nearest Whole Number, Ten, Hundred...

Number	Target	Look at	Result
3,748	Nearest hundred	Tens digit: 4	3,700
3,748	Nearest ten	Ones digit: 8	3,750
3,748	Nearest thousand	Hundreds digit: 7	4,000
74,500	2 s.f. (ten-thousands)	Thousands digit: 4	70,000 — wait, look at next digit: 4,500 → 75,000

Rounding Decimals

Round to the specified number of decimal places. Look at the digit one place further right than your target.

1 decimal place (1 d.p.): 4.736 → look at second decimal (3 < 5) → round down → 4.7

2 decimal places (2 d.p.): 6.285 → look at third decimal (5 ≥ 5) → round up → 6.29

3 decimal places (3 d.p.): 2.7346 → look at fourth decimal (6 ≥ 5) → round up → 2.735

Significant Figures

Significant figures start counting from the first non-zero digit (leftmost non-zero digit).

Rules for identifying significant figures:

• All non-zero digits are significant
• Zeros between non-zero digits are significant: 3,048 has 4 s.f.
• Leading zeros (before the first non-zero digit) are NOT significant: 0.0045 — the 4 is the first s.f.
• Trailing zeros after a decimal point are significant: 3.40 has 3 s.f.

Examples:

• 3,748 to 2 s.f. → look at third digit (4 < 5) → 3,700
• 3,748 to 3 s.f. → look at fourth digit (8 ≥ 5) → 3,750
• 0.00452 to 2 s.f. → first s.f. is 4, second is 5; look at third s.f. (2 < 5) → 0.0045
• 74,500 to 2 s.f. → first s.f. is 7, second is 4; look at next digit (5 ≥ 5) → 75,000

Estimation

Use rounding to approximate answers quickly — especially useful for checking calculator results.

Strategy: Round each value to 1 significant figure, then calculate.

48.7 × 3.1 ≈ 50 × 3 = 150 (Actual answer: 150.97 — the estimate is very close ✓)

For division estimates: 597 ÷ 29 ≈ 600 ÷ 30 = 20 (Actual: 20.59 ✓)

Worked Example — Square Root Estimation

Estimate √26 × 9.8:

• √26 ≈ √25 = 5 (choose the nearest perfect square)
• 9.8 ≈ 10
• Estimate: 5 × 10 = 50
• Actual: 5.099 × 9.8 = 49.97 ✓

Rounding Errors and Bounds

When a measurement is rounded, the true value lies within half a unit of the rounded value.

Example: A length is given as 8cm to the nearest cm.

• Lower bound: 7.5cm
• Upper bound: 8.5cm (8.5 rounds up to 9, so the true upper bound is "less than 8.5")

This is important in science, engineering, and any field where measurement precision matters.

Common Mistakes to Avoid

• Rounding from right to left: always identify your target first, look only ONE place to the right, then round — do not chain-round multiple times
• Treating significant figures like decimal places: 0.00452 to 2 s.f. is 0.0045, not 0.00 (leading zeros do not count)
• Forgetting placeholder zeros: 3,748 rounded to the nearest hundred is 3,700 — the zero must be there to maintain the magnitude
• Over-estimating precision: writing 7.00 and 7 both equal seven, but 7.00 implies measurement to 2 decimal places

Tips and Tricks

• When estimating, round to numbers that are easy to compute mentally
• For square roots in estimates, find the nearest perfect square: √50 ≈ √49 = 7
• Always state what you are rounding to — "round to 2" is ambiguous without specifying d.p. or s.f.
• In multi-step calculations, avoid rounding intermediate answers — only round the final result