Significant Figures: Rules & Rounding
A measured number is a claim about precision: writing 2.51 cm instead of 2.5 cm says the instrument resolves the third digit. Significant figures formalise that claim, and NTA converts it into marks — counting significant figures, rounding by the half-even rule, and truncating products and sums correctly are among the most repeated easy marks in JEE Main and NEET. This page gives the complete rule set with NCERT's own worked numbers, the rounding flowchart, and the log/antilog discipline that pH and decibel questions secretly test. The companion skill of judging how much a measurement can err at all lives on the errors in measurement page.
What are Significant Figures? — Complete Theory
1. Definition and the instrument link
The significant figures of a measured value are all digits known with certainty plus one uncertain digit — the last one, which is the observer's estimate between the smallest scale divisions. A metre scale graduated in mm gives a length like 42.7 mm: the digits 4 and 2 are reliable, the .7 is estimated — three significant figures, and the reading itself carries an uncertainty of about half the least count. This is why the number of significant figures is not cosmetic: it silently states the precision of the measurement. More digits = finer instrument (or more repetitions), nothing more.
2. Rules for counting significant figures
| # | Rule | Example | Count |
|---|---|---|---|
| 1 | All non-zero digits are significant | 346 | 3 |
| 2 | Zeros between two non-zero digits are significant | 605, 40.07 | 3, 4 |
| 3 | Leading zeros (before the first non-zero digit) are never significant | 0.0234, 0.00650 | 3, 3 |
| 4 | Trailing zeros to the right of a decimal point are significant | 4.700, 0.06050 | 4, 4 |
| 5 | Trailing zeros without a decimal point are ambiguous | 4700 (treat as 2 unless stated) | 2 (ambiguous) |
| 6 | In scientific notation, count only the multiplier's digits | 4.700 × 10³, 2.05 × 10⁻³ | 4, 3 |
| 7 | Exact numbers (2 in 2πr, counted objects, 100 in "100 cm = 1 m") have infinite significant figures | 2πr with r = 3.15 | limited by r: 3 |
3. Removing ambiguity: scientific notation
Any measurement can be written as (a number between 1 and 10) × 10n, which makes the precision explicit: the mass of the Earth as 5.97 × 10²⁴ kg (three significant figures) or, to four, 5.972 × 10²⁴ kg. NCERT's convention: report the coefficient with exactly as many significant figures as the measurement justifies. When a problem says "correct to n significant figures", it means the coefficient in scientific notation — not n decimal places.
4. Rules for multiplication and division (least significant figures)
In a ×/÷ chain, the result keeps as many significant figures as the least precise factor. Example: density = mass/volume = 4.237 g ÷ 2.51 cm³. The calculator shows 1.68804…, but 2.51 carries only three significant figures, so the density is reported as 1.69 g cm⁻³. The ½ in ½mv² is exact — it never limits the answer; the 2 in v² = u² + 2as is exact likewise.
5. Rules for addition and subtraction (least decimal places)
In a +/− chain, the result keeps as many decimal places as the term with the fewest decimal places. NCERT's example: 12.11 + 18.0 + 1.012 = 31.122, but 18.0 reaches only one decimal place, so the sum is 31.1. Note the contrast: ×/÷ counts significant figures; +/− counts decimal places. Mixing the two criteria is the classic error — and a deliberate NTA distractor in options like 31.12 vs 31.1.
6. Rounding off: the half-even convention
To round a value, look at the digit immediately after the last digit you keep:
(i) If it is < 5, drop it (2.743 → 2.74). (ii) If it is > 5, or = 5 followed by any non-zero digit, round up (2.746 → 2.75; 2.7452 → 2.75). (iii) If it is exactly 5 with only zeros (or nothing) after, round so the last retained digit becomes even: 2.745 → 2.74 (4 is even), 2.735 → 2.74 (3 rounds up to 4). NCERT uses precisely these two numbers, and NEET has recycled them verbatim.
Figure 1.3 — Rounding decision flowchart (round-half-to-even)
Start at the top with the digit after your last retained digit; follow the branch that first applies.
2.743 → 2.74
2.746 → 2.75 · 2.7452 → 2.75
2.745 → 2.74 · 2.735 → 2.74 · 2.5 → 2 · 3.5 → 4
Exam read-out: the tie-breaker is "even", not "up". Given 12.65 cm rounded to 3 significant figures, the answer is 12.6 cm, not 12.7 — half-even keeps the 6. This exact item has surfaced as a numerical-value question.
7. Rounding in multi-step calculations
Round only once, at the end. Carry one or two guard digits through intermediate steps and round the final result to the precision allowed by the input. Rounding at every stage compounds errors: (13.12 ÷ 5) rounded to 2.62, then multiplied by 3, is not the same as rounding 7.872 to 7.87 — the final digit drifts. JEE Main numerical-answer questions are graded on the exact integer, so premature rounding is a mark-loser, not a style choice.
8. Significant figures in logarithms and antilogarithms (JEE favourite)
Convention: the number of digits in the mantissa (the decimal part of a log) equals the number of significant figures in the original number. Example: [H⁺] = 3.2 × 10⁻³ M (two significant figures) → pH = −log(3.2 × 10⁻³) = 3 − log 3.2 = 3 − 0.5051 = 2.4949 → report 2.49 (two decimal places). Conversely, antilog keeps as many significant figures as the mantissa has decimals. Chemistry's pH questions are where physics students leak marks; the rule is identical.
Visualising Significant Figures
Picture each measured value as a menu price: the significant figures are the digits you actually paid for. Leading zeros cost nothing (0.00650 is "6.50 milli-something"); trailing zeros after a decimal are premium digits (4.700 ≠ 4.7); the ×10ⁿ wrapper is just packaging. The flowchart above is the complete decision tree — in the exam, spend five seconds on it and never lose a rounding mark again.
Solved Examples on Significant Figures (Step-by-Step)
Solved Example 1 — Density with correct precision (NCERT classic)
Q. Mass of a box = 4.237 g; volume = 2.51 cm³. Report density with correct significant figures.
Step 1. Divide: 4.237 ÷ 2.51 = 1.68804…
Step 2. Least precise factor 2.51 → 3 significant figures.
Step 3. Round 1.68804… to 3 significant figures: the digit after "1.68" is 8 (> 5) → round up → 1.69.
ANSWER: ρ = 1.69 g cm⁻³ (3 significant figures)Solved Example 2 — Addition with decimal-place rule
Q. Evaluate 12.11 + 18.0 + 1.012 with correct precision.
Step 1. Raw sum: 12.11 + 18.0 + 1.012 = 31.122.
Step 2. Decimal places of terms: 2, 1, 2 → minimum is 1 (from 18.0).
Step 3. Round 31.122 to one decimal place: digit after is 2 (< 5) → drop → 31.1.
ANSWER: 31.1Practice Questions on Significant Figures (With Solutions)
Practice Q1 (Numerical-value type)
Q. How many significant figures are there in 0.0605?
Solution. Leading zeros (0.0, 0.00) are not significant; the digits 6, 0, 5 are — the middle 0 sits between non-zero digits, so it counts. Count = 3.
ANSWER: 3Practice Q2 (Single-correct MCQ)
Q. 4.32 × 2.0 = ? (a) 8.6 (b) 8.64 (c) 8.7 (d) 8.641
Solution. Product = 8.64. Factors carry 3 and 2 significant figures → result keeps 2. Round 8.64 to 2 significant figures: the digit after 8.6 is 4 (< 5) → drop → 8.6.
ANSWER: (a) 8.6Practice Q3 (Assertion–Reason)
Q. Assertion (A): 0.200 has three significant figures. Reason (R): Trailing zeros to the right of the decimal point are significant.
Solution. The two leading zeros don't count; the 2 and both trailing zeros do — count is 3. R is the governing rule, so it explains A.
ANSWER: (a) Both A and R true; R is the correct explanation of AKey Formulas & Takeaways
| Rule | Statement | Fast example |
|---|---|---|
| Counting | non-zero digits + captive zeros + trailing zeros (decimal present); leading zeros never | 0.0605 → 3; 4.700 → 4 |
| Ambiguity fix | use scientific notation: count the coefficient only | 4700 → 4.7 × 10³ → 2 |
| ×/÷ | result keeps the least number of significant figures | 4.237 ÷ 2.51 → 1.69 |
| +/− | result keeps the least number of decimal places | 12.11 + 18.0 + 1.012 → 31.1 |
| Rounding | <5 drop · >5 up · exactly 5 (zeros after) → even | 2.745 → 2.74; 2.735 → 2.74 |
| Log rule | mantissa digits = significant figures of the number | [H⁺] = 3.2 × 10⁻³ → pH = 2.49 |
| Exact numbers | 2, π in formulas; counted quantities; defined conversions — infinite precision | never limit answers |
FAQs on Significant Figures
What are significant figures?
The number of meaningful digits in a measured value — all digits known reliably plus one uncertain (estimated) digit that reflects the precision of the instrument.
How do you round 2.745 to three significant figures?
The digit to be dropped is exactly 5 with nothing significant after it, so the last retained digit is made even: 2.745 → 2.74. Likewise 2.735 → 2.74 (3 rounds up to even 4).
Why is 4700 said to have ambiguous significant figures?
Written without a decimal point the trailing zeros may or may not be measured; scientific notation removes the ambiguity — 4.7 × 10³ has two significant figures, 4.700 × 10³ has four.
Which rule applies in 12.11 + 18.0 + 1.012?
For addition or subtraction the result keeps the least number of decimal places present in the terms — 18.0 has one decimal place, so the sum 31.122 is rounded to 31.1.
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