Learning goals
- Round a number to a stated number of decimal places or significant figures
- Identify which digits are significant, including zeros in different positions
- Avoid overstating precision when measurements or intermediate steps are approximate
Rounding vs significant figures
Rounding shortens a number to a chosen place value — nearest tenth, hundredth, whole number, and so on. Look at the digit one place to the right of the cut: 5 or higher usually rounds up; lower stays the same (half-up is the common school rule).
Significant figures (sig figs) count how many meaningful digits a measurement carries. The number 3.14 has three significant figures; 0.00314 also has three — leading zeros after the decimal only place the value, they do not count as significant.
Use decimal-place rounding when a form asks for “two decimal places” (money often uses this). Use significant figures when the goal is to reflect measurement precision, especially in science and engineering contexts.
Round half up (common school rule)
if next digit ≥ 5, increase the kept digit by 1; else leave it
To round 2.346 to two decimal places, look at the thousandths digit 6 ≥ 5, so 2.35. To round 2.344 to two decimals, the next digit is 4, so 2.34.
Significant figures — quick rules
nonzero digits count; zeros between them count; leading zeros do not; trailing zeros count if a decimal point is shown
Examples: 405 → 3 sig figs; 0.00405 → 3 sig figs; 4050 (ambiguous without a decimal or scientific notation); 4050. → 4 sig figs; 4.050 × 10³ → 4 sig figs.
Example 1 — Round to decimal places
Given
Round 18.2764 to (a) one decimal place and (b) two decimal places.
Steps
- One decimal place: look at the hundredths digit 7 ≥ 5 → 18.3.
- Two decimal places: look at the thousandths digit 6 ≥ 5 → 18.28.
Answer
(a) 18.3; (b) 18.28.
Tip: Money to the cent is “two decimal places,” not “two significant figures.”
Example 2 — Round to three significant figures
Given
Round 0.004562 and 145,700 to three significant figures.
Steps
- 0.004562: first three significant digits are 4, 5, 6; next digit 2 < 5 → 0.00456.
- 145,700: first three digits 1, 4, 5; next digit 7 ≥ 5 → 146,000 (in ordinary form) or preferably 1.46 × 10⁵ to show the figures clearly.
Answer
0.00456 and 1.46 × 10⁵ (146,000 with three sig figs).
Tip: Scientific notation removes ambiguity about which zeros are significant.
Example 3 — Rounding money
Given
A bill subtotal after tax computes to $20.00. What do you charge to the nearest cent?
Steps
- Money uses two decimal places.
- Thousandths digit is 5, so round the cents up: $20.00.
Answer
Charge $20.00.
Tip: Banker’s rounding (half to even) exists in some systems; retail and school math usually use half-up.
Rounding intermediate steps too early
Common mistake
Round every partial result to two decimals, then combine — the final answer drifts from the true value.
Better approach
Keep extra digits through the calculation (or use exact fractions), then round once at the end to the required precision.
Treating leading zeros as significant
Common mistake
Claim 0.0025 has four significant figures because four digits appear after the decimal point starts.
Better approach
Leading zeros are placeholders: 0.0025 has two significant figures (2 and 5).
Check your understanding
1.Round 6.451 to two decimal places.
Answer: 6.45.
2.Round 6.455 to two decimal places (half-up).
Answer: 6.46.
3.How many significant figures does 0.07080 have?
Answer: Four (7, 0, 8, 0).
4.Round 23,450 to three significant figures in scientific notation.
Answer: 2.35 × 10⁴.
Key takeaways
- Decimal-place rounding looks at the next digit and usually rounds half up.
- Significant figures measure precision; leading zeros do not count.
- Prefer scientific notation when trailing zeros would be ambiguous.
- Round at the end of a multi-step calculation unless a method requires intermediate rounding.