JEE/NEET Physics · Units & Measurements series · Part 3 of 6 · All parts →
- A significant figure is a digit you actually measured — writing more claims false precision
- All non-zero digits count; zeros count only if measured or between measured digits
- Multiply/divide → keep the FEWEST sig-figs in the chain; add/subtract → keep the fewest DECIMALS
- Exact numbers (2π, 4 in ‘4 sides’, conversion factors) have infinite sig-figs
- Rounding: 5 or more rounds up; round only at the END of a calculation
A robot says your pizza arrives in 7.2941 minutes. Absurd — not because the maths is wrong, but because the claim is dishonest: nobody can know a delivery time to a tenth of a second. Significant figures are the honesty policy of measurement. Part 3 of the Units & Measurements series.
- Why digits carry honesty
- Counting rules: which zeros count
- The two arithmetic rules
- Exact numbers: the infinite exceptions
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Why Digits Carry Honesty
Write ‘the rod is 12.3 cm’ and you claim three things: the 1 and 2 are certain, the 3 is your best estimate, and you’d be embarrassed to say more. Write 12.30 cm and you additionally claim the 0 was measured. Extra digits aren’t extra information — they’re false advertising. Sig-figs keep you from promising precision your instrument never delivered.
Counting Rules: Which Zeros Count
| Number | Sig-figs | Why |
|---|---|---|
| 204 | 3 | sandwiched zeros always count |
| 0.0204 | 3 | leading zeros NEVER count (they’re place-setting) |
| 2.04 | 3 | same digits, same physics |
| 2.40 | 3 | trailing zero AFTER a decimal point was measured |
| 240 | 2 or 3 | ambiguous — write 2.4×10² or 2.40×10² to say which |
The Two Arithmetic Rules
× and ÷: the answer keeps as many sig-figs as the poorest number in the chain. + and −: the answer keeps as many decimal places as the least precise term. The logic: your answer can never be more honest than your shakiest ingredient.
Exact Numbers: The Infinite Exceptions
Counting numbers (12 eggs), constants in formulas (the ½ in KE = ½mv²), and defined conversions (1 m = 100 cm) have unlimited sig-figs — they were never measured, so they carry no doubt.
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| s.f. | significant figures — digits that carry real information | count from first non-zero digit |
| decimal place | position after the decimal point | governs + and − only |
Solved Examples
Skip leading zeros (0.00): start at the 6.
6, 0, 4, 0 — the middle 0 counts (sandwiched), the last 0 counts (after decimal, trailing).
4 sig-figs. ✔
Answer: 4
Raw: 8.55 m². But 2.5 has only 2 sig-figs — the chain’s weakest link.
Round to 8.6 m² (2 s.f.).
Why: the 5 in 2.5 was already an estimate — an estimate multiplied by precision is still an estimate. ✔
Answer: 8.6 m² (2 s.f.)
Raw: 16.66 g. Least decimals: 0.3 (one decimal place).
Round to 16.7 g (one decimal place).
Note the rule switched: addition cares about DECIMALS, not total digits — 0.3 is shakiest because its uncertainty is ±0.1. ✔
Answer: 16.7 g
- Counting leading zeros. 0.0034 has TWO sig-figs, not five — zeros before the first non-zero digit never count.
- Applying the × rule to + problems. Addition counts decimal places, multiplication counts sig-figs — mixing the two is the classic slip.
- Rounding mid-calculation. Carry extra digits through, round ONCE at the end; early rounding compounds error.
- Treating exact numbers as 1-sig-fig. The 2 in d = 2r and the 100 in 1 m = 100 cm are exact — infinite sig-figs, they never limit the answer.
This Physics in Your Daily Life
- Delivery apps say ’25–35 minutes’, not ‘27.4 min’ — honest uncertainty ranges, the sig-fig philosophy in commerce.
- Cricket win-probability ‘68%’ never shows 68.37% — broadcasters rounding to the precision a model can honestly support.
- Bank interest rounded to the paisa — accounting rules enforce decimal-place discipline identical to the addition rule.
- Medical lab reports (‘Hb 12.3 g/dL’, three sig-figs) — calibrated instruments promise exactly the digits they print, no more; doctors diagnose against those digit budgets.
- Every sports statistic you read — bowling speed 140.2 km/h but batting average 45.6 — silently follows a sig-fig convention.
A tailor with a metre stick measures your sleeve as ’61 cm’, estimating the last digit by eye. If he writes 61.000 cm, he’s claiming a laser lab. Digits beyond your instrument’s power aren’t detail — they’re fiction wearing a suit. Sig-figs are the tailor’s oath: I write only what I saw.
A ruler marked in mm gives ~3 honest sig-figs for a book (21.3 cm); a vernier caliper gives 4 (21.34 cm). Multiply book height × 2.5 (a measured factor): the ruler chain must stop at 3 digits, the vernier chain at 4. Poorer instrument wins — 2.5’s honesty caps everything below it.
Picture each number as a pole whose length is certainty: 4 certain digits = tall pole, 1 estimate = wobbly tip. Chain numbers with ropes (×, ÷): the rope can only be as taut as the shortest pole. Draw the chain — the sag marks where your answer must stop being written.
Practice set (answers hidden — try first)
(NEET-level) Sig-figs in 0.0500:
(JEE Main-level) 4.0 × 3.00 ÷ 6.000 → answer’s s.f.:
(NEET-level) 11.2 + 3.14 (proper sig-figs):
(Concept) In d = 2r, the 2 has how many sig-figs?
(JEE Main-level) Volume = 2.1 × 2.1 × 2.1 (m³):
- sandwiched and trailing-after-decimal zeros count; leading never
- × ÷ → fewest sig-figs; + − → fewest decimals
- exact numbers (counts, 2π, defined conversions) = infinite s.f.
- round once, at the end
- 240 is ambiguous — scientific notation settles it
- 🔁 leading zeros never count
- 🔁 × ÷ → fewest s.f.; + − → fewest decimals
- 🔁 exact numbers carry infinite s.f.
- 🧠 Chant: ‘zeros left, never; zeros between, forever; zeros right (with decimal), measured tight’.
- 🧠 Chain rule: ‘the weakest link writes the last digit’.
- 🏠 Daily: ’25–35 min’ delivery windows = honest uncertainty.
- 🏠 Daily: lab reports print exactly the digits machines can honor.
Quick revision
- A significant figure is a digit you actually measured — writing more claims false precision
- All non-zero digits count; zeros count only if measured or between measured digits
- Multiply/divide → keep the FEWEST sig-figs in the chain; add/subtract → keep the fewest DECIMALS
- Exact numbers (2π, 4 in ‘4 sides’, conversion factors) have infinite sig-figs
- Rounding: 5 or more rounds up; round only at the END of a calculation
- Counting rules: which zeros count
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