Significant Figures: The Complete Guide to Rules & Zeros

Short Answer

Every significant figures rule explained with worked examples — zeros, arithmetic, exceptions, and why AP, IB, and ASTM E29 don't always agree.

Significant figures (sig figs) are the digits in a number that carry real information about how precisely it was measured or calculated. They include every non-zero digit, any zero trapped between non-zero digits, and — depending on whether a decimal point is present — some or all trailing zeros. Counting them correctly tells you how much precision a number actually claims; rounding a result to the correct number of them keeps you from reporting more precision than your data can support.

Most people learn significant figures as a single fixed rulebook. In practice, “3 sig figs” can mean slightly different things depending on whether you’re in a general chemistry class, an AP exam, an IB internal assessment, or a materials-testing lab working from ASTM E29. The five counting rules below are universal. What changes across contexts is how rounding is applied on top of them — and that’s usually where students lose points and where lab reports get flagged. This guide covers both: the rules themselves, and where the conventions built on top of them diverge.

Use the calculator below for instant answers with the rule shown, or read on for the full explanation.

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[Live calculator embeds here] — Enter any number or expression to get its significant figures, with the specific rule that applied shown alongside the answer.

The 5 Rules of Significant Figures

Every significant-figures rule other sources describe is really a restatement of five cases. Learn these five and you can handle anything.

1. All non-zero digits are significant. 1, 2, 3, 4, 5, 6, 7, 8, and 9 always count, no matter where they appear in a number.

2. Zeros between non-zero digits (captive zeros) are always significant. The zero in 205 sits between two non-zero digits, so it counts. 205 has 3 significant figures. This one has no exceptions.

3. Leading zeros are never significant. Zeros that appear before the first non-zero digit only mark the position of the decimal point — they carry no precision information. 0.0056 has 2 significant figures (5 and 6); the three zeros in front of it are not counted.

4. Trailing zeros are significant when a decimal point is present. 25.00 has 4 significant figures. The two zeros after the decimal point are there on purpose — they tell you the measurement was precise enough to confirm those positions are exactly zero, not just unknown. This applies whether the decimal point falls before or after the zeros: 250.0 has 4 significant figures for the same reason.

5. Trailing zeros are ambiguous when no decimal point is present. This is the one genuine weak spot in ordinary decimal notation. Written as 4500, you cannot tell from the digits alone whether this was measured to the nearest thousand (2 sig figs), the nearest hundred (3 sig figs), or the nearest unit with the last two digits happening to be zero (4 sig figs). This isn’t a trick — it’s a real limitation of place-value notation, and it’s the reason scientific notation exists. Writing 4.5 × 10³, 4.50 × 10³, or 4.500 × 10³ states the precision explicitly and removes all doubt.

The exception that sits outside all five rules: exact numbers. Counted items (3 beakers, 12 students) and numbers fixed by definition (60 seconds in a minute; since 1959, exactly 2.54 cm in an inch) are not measurements at all. They carry effectively infinite significant figures and never limit the precision of a calculation they appear in — see Example 6 below.

Significant Figures in Arithmetic

Counting sig figs in a single number is only half the job. The moment you add, subtract, multiply, or divide, a second set of rules takes over — and mixing these two up is the single most common mistake on lab reports (see Common Mistakes below).

Addition and subtraction: match decimal places, not sig fig counts. The result can only be as precise, in absolute terms, as the least precise term you started with. Round the answer to the same number of decimal places as the input with the fewest decimal places — the sig fig counts of the individual numbers don’t matter here at all.

Multiplication and division: match sig fig counts. Here it’s the opposite. Round the result to the same number of significant figures as the input with the fewest significant figures.

Logarithms: match decimal places in the mantissa to sig figs in the original number. This one trips up almost everyone, including some calculators. Because the characteristic (the integer part of a log) only reflects order of magnitude, the sig figs of the original number should show up as decimal places in the mantissa (the part after the decimal point) of the log — not as the total digit count of the log. See Example 7.

In every case: round only once, at the very end. If a calculation has more than one step, carry extra guard digits through the intermediate steps and apply rounding only to the final answer. Rounding after each step compounds error and can change your final digit — sometimes visibly, as Example 5 below shows. This isn’t just a classroom convention; it’s also the explicit guidance in NIST’s own laboratory rounding procedure (GLP 9), which instructs that intermediate values not be rounded before the final uncertainty calculation.

Worked Examples

Example 1 — Counting with leading and trailing zeros

Number: 0.0045600

The leading zeros (0.00) only mark the decimal position and are not significant. Starting from the first non-zero digit, count everything through the end: 4, 5, 6, 0, 0. Because a decimal point is present, those final two zeros count.

Answer: 5 significant figures.

Example 2 — Counting in scientific notation

Number: 7.00 × 10⁶

Scientific notation sidesteps the ambiguous-trailing-zero problem entirely: only digits in the mantissa (7.00) count. That’s 7, 0, 0.

Answer: 3 significant figures. (The exponent never adds or removes sig figs — it just states magnitude.)

Example 3 — Addition

Calculation: 12.11 + 18.0 + 1.013

Raw sum: 31.123. Decimal places in each term: 12.11 has 2, 18.0 has 1, 1.013 has 3. The fewest is 1 (from 18.0), so the answer is limited to 1 decimal place.

Answer: 31.1 — not 31.123, even though that’s what a calculator will display.

Example 4 — Multiplication

Calculation: 6.75 × 3.2

Raw product: 21.6. Sig figs in each term: 6.75 has 3, 3.2 has 2. The fewest is 2, so the answer is rounded to 2 significant figures.

Answer: 22 — the correct rounding of 21.6 to 2 sig figs, not 21.6 itself.

Example 5 — Multi-step calculations and guard digits

Calculation: (12.5 × 3.42) ÷ 1.20

All three numbers have 3 significant figures, so the final answer should too. Here’s what happens with the two approaches:

Correct method — round once, at the end: 12.5 × 3.42 = 42.75 (keep full precision) 42.75 ÷ 1.20 = 35.625 Round to 3 sig figs → 35.6

Common mistake — round after the first step: 12.5 × 3.42 = 42.75 → rounded early to 3 sig figs = 42.8 42.8 ÷ 1.20 = 35.666… Round to 3 sig figs → 35.7

The two methods disagree in the final digit — 35.6 versus 35.7 — purely because of when rounding was applied, not because of any difference in the input data. This is exactly why guard digits matter, and exactly why professional metrology guidance (NIST GLP 9) explicitly prohibits rounding intermediate values before the final result.

Example 6 — Exact numbers

Calculation: 3 identical samples, each weighing 4.52 g. Total mass?

3 × 4.52 g = 13.56 g. The “3” here is a counted, exact number — it has no uncertainty and therefore no sig fig limit of its own. The only real measurement is 4.52 g, with 3 significant figures, so that’s what limits the answer.

Answer: 13.6 g — rounded to match 4.52’s 3 sig figs, not the “1 sig fig” you’d get by mistakenly treating “3” as a measurement.

Example 7 — Logarithms

Calculation: log(2.5 × 10²)

log₁₀(250) = 2.39794. The original number, 2.5 × 10², has 2 significant figures. Because a log’s mantissa (not its total digit count) carries the precision, the mantissa should be rounded to 2 decimal places to match.

Answer: 2.40 — the “2.” is just the order-of-magnitude characteristic and doesn’t count toward precision; only “.40” does, and it correctly reflects 2.5’s 2 sig figs.

Common Exceptions and Gotchas

The five rules cover the ordinary cases. These are the situations where people who know the rules still get tripped up.

The same digits, three different precisions. 100, 100., and 1.00 × 10² all display the digits “1”, “0”, “0” — but they claim 1, 3, and 3 significant figures respectively (100. with an explicit trailing decimal point signals all three zeros are meant to count). A stray decimal point changes the meaning entirely.
A “round” measurement isn’t automatically imprecise. If a graduated cylinder marked in 10 mL increments reads exactly 1200 mL, that trailing-zero ambiguity is a real practical problem, not just an academic one — the person recording it needs to write 1.20 × 10³ mL or add a note, or the next person to use that data won’t know how precise it actually was.
Overline notation exists but is fading out. Some older textbooks put a bar over the last significant digit to resolve ambiguity — 1200 with a bar over the second zero means 3 sig figs are intended. It works on paper but is easy to lose in plain text, copy-paste, or a spreadsheet cell, which is why scientific notation has largely replaced it as the recommended fix.
Not all constants are exact. Defined constants (60 seconds per minute, 2.54 cm per inch) carry infinite precision. Measured physical constants (like the gravitational constant, G) do not — they were determined experimentally, they have a stated uncertainty, and they do limit the precision of any calculation that uses them. Don’t treat “it’s a constant” as automatically meaning “it’s exact.”
Zero, alone, has one significant figure. This almost never matters in practice — a measured value of zero is nearly meaningless without a stated uncertainty (0.0 ± 0.1 is very different information from 0.00 ± 0.01) — but it’s a common trick-question case.
A tool that “shows more decimals” isn’t more accurate. See the note on IEEE 754 below — this is a display/storage artifact, not extra real precision.

How Conventions Differ

The five counting rules and the two arithmetic rules above are universal — nobody disputes them. What differs is how strictly rounding is enforced and what gets layered on top, depending on the setting. These are general tendencies, not official rulebooks — always check your own course’s rubric or your lab’s written procedure for anything graded or compliance-relevant.

Context	What “correct” typically means here	How it differs from the classroom default
General classroom / textbook default	Apply the 5 counting rules plus the decimal-place (addition) and sig-fig-count (multiplication) rules; round only the final answer	This is the baseline every other context modifies
AP Chemistry (College Board)	Free-response graders generally look for a final numeric answer that isn’t reported with wildly more or fewer digits than the given data supports, alongside correct reasoning and units	Tends to weigh shown work and reasoning at least as heavily as an exact digit-count match — but this varies by exam year, so check the specific scoring guidelines for your exam
IB Sciences (Internal Assessment)	Sig figs in both the raw data and any propagated uncertainty are expected to be consistent with each other, and examiners generally want an explicit uncertainty (a ± value), not just a rounded number	Adds a formal uncertainty-propagation expectation on top of basic sig figs — closer to professional metrology than a typical intro chemistry class
ASTM E29 (industrial conformance testing)	A defined procedure for rounding a test result to the same number of digits as a written specification limit before comparing the two, so that different labs reach the same conformance decision	This is a standards-body procedure, not a teaching rule — see our ASTM E29 explainer for the concept; consult the actual standard for compliance-grade work
NIST / GUM (professional metrology)	Sig figs are treated as a rough stand-in for a properly propagated measurement uncertainty. The formal guidance (JCGM 100:2008, the GUM, §7.2.6) is to round the uncertainty to 1–2 significant figures first, then round the reported value to match	This is what sig figs are quietly approximating — see our uncertainty and error propagation guide for the full method

A note on software and IEEE 754: IEEE 754 isn’t a significant-figures convention at all — it’s the binary format almost every calculator, spreadsheet, and programming language uses to store decimal numbers internally. It’s the reason 0.1 + 0.2 can display as 0.30000000000000004 in a browser console, and why a spreadsheet sometimes shows 12 digits after a calculation that should only have 3. A tool that respects sig fig rules has to actively round around IEEE 754’s raw output, not follow it. See our software behavior guide for how this plays out in Excel, Python, and calculators specifically.

Where This Comes From

The classroom rules above are taught consistently across chemistry and physics curricula, but two real standards sit behind the professional versions. ASTM E29 (“Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications”) governs how manufacturers and testing labs round a measured value before checking it against a written spec, using one of two defined approaches: the Absolute Method or the Rounding Method. The GUM (Guide to the Expression of Uncertainty in Measurement, published jointly by the BIPM and adopted by NIST) is the international reference for how measurement uncertainty itself should be calculated, rounded, and reported. Neither document is about “significant figures” as classroom shorthand — both are about doing the same underlying thing with the rigor a lab or a manufacturing line needs. The full comparison, with more worked examples from each standard, is in our Rounding Standards Compared guide.

Common Mistakes

If a sig fig answer got marked wrong and you’re not sure why, it’s almost always one of these six:

Rounding after every intermediate step instead of only at the end. This compounds small rounding errors and can flip your final digit — see Example 5.
Using the multiplication rule (matching sig fig count) on an addition problem, or the addition rule (matching decimal places) on a multiplication problem. These are two different rules for a reason; mixing them up is the single most common error on lab reports.
Treating leading zeros as significant. 0.003 has 1 sig fig, not 4.
Forgetting that a decimal point changes the trailing-zero rule. 250 and 250. are not necessarily the same claim about precision.
Rounding an exact or counted number as if it limited the calculation. “3 trials” is not a 1-sig-fig measurement — it’s an exact count and never limits your answer’s precision.
Reporting an uncertainty with more digits than the measurement supports — for example, 12.7 ± 0.34 cm. Standard practice (and the GUM’s own guidance) is to round the uncertainty first, usually to 1–2 sig figs, then match the measured value to it — giving 12.70 ± 0.34 or, more commonly, 12.7 ± 0.3.

Practice Problems

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Concept: Counting significant figures

Q1. How many significant figures are in 0.03080? A) 2 B) 3 C) 4 D) 5 Answer: C) 4. The leading zeros aren’t counted; starting from the first non-zero digit: 3, 0, 8, 0 — the trailing zero counts because a decimal point is present.

Q2. How many significant figures are in 7.00 × 10⁶? A) 1 B) 3 C) 6 D) 7 Answer: B) 3. Only the mantissa (7.00) counts; the exponent never adds or removes sig figs.

Concept: Zeros

Q3. Which digits in 0.020560 are significant? A) 2 B) 3 C) 4 D) 5 Answer: D) 5. The leading zeros aren’t counted; from the first non-zero digit onward: 2, 0, 5, 6, 0 — all five count.

Q4. A measurement is recorded as 4500 with no decimal point and no other notes. What can you say about its significant figures? A) Exactly 2 B) Exactly 3 C) Exactly 4 D) Ambiguous — could be 2, 3, or 4 Answer: D. Without a decimal point or scientific notation, trailing zeros in a whole number are genuinely ambiguous.

Concept: Addition and subtraction

Q5. What is 12.11 + 18.0 + 1.013, correctly rounded? A) 31.123 B) 31.12 C) 31.1 D) 31 Answer: C) 31.1. The raw sum is 31.123; the fewest decimal places among the terms is 1 (from 18.0), so the answer rounds to 1 decimal place.

Q6. In 154.2 − 3.85, which number limits the decimal places in the answer? A) 154.2 B) 3.85 C) Both equally D) Neither Answer: A) 154.2. It has 1 decimal place versus 3.85’s 2 — the fewest decimal places wins for addition/subtraction.

Concept: Multiplication and division

Q7. What is 6.75 × 3.2, correctly rounded? A) 21.6 B) 22 C) 21 D) 21.60 Answer: B) 22. The raw product is 21.6; 3.2 has the fewest sig figs (2), so the answer rounds to 2 sig figs: 22.

Q8. In 8.314 ÷ 2.0, which number limits the significant figures in the answer? A) 8.314 B) 2.0 C) Both equally D) Neither Answer: B) 2.0. It has 2 sig figs versus 8.314’s 4 — the fewest sig figs wins for multiplication/division.

Significant Figures at a Glance

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0.00[4][5][6][0][0] → bracketed digits are significant → 5 significant figures. The unbracketed “0.00” only marks the decimal position.

[7].[0][0] × 10⁶ → all three mantissa digits are significant → 3 significant figures. The exponent is never bracketed — it carries no precision information of its own.

4500 — same digits, three legitimate readings once rewritten:

4.5 × 10³ → 2 significant figures
4.50 × 10³ → 3 significant figures
4.500 × 10³ → 4 significant figures

Quick Reference

Situation	Rule	Example
Non-zero digit	Always significant	342 → 3 sig figs
Zero between non-zero digits	Always significant	405 → 3 sig figs
Leading zero	Never significant	0.0056 → 2 sig figs
Trailing zero, decimal point present	Always significant	45.00 → 4 sig figs
Trailing zero, no decimal point	Ambiguous	4500 → 2, 3, or 4 sig figs
Exact / counted / defined number	Infinite — never limits precision	12 eggs; 1 in = 2.54 cm
Addition / subtraction	Round to fewest decimal places	12.11 + 18.0 → 1 dp
Multiplication / division	Round to fewest sig figs	6.75 × 3.2 → 2 sf
Logarithm	Mantissa decimal places = original sig figs	log(2.5×10²) → 2.40

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Frequently Asked Questions

Do you count significant figures from the first non-zero digit, or from the decimal point? From the first non-zero digit, always. The decimal point’s position doesn’t affect counting — it only affects whether trailing zeros count (Rules 4 and 5 above).

How many significant figures does the number 0 have? By convention, one. In practice this almost never matters, because a measured zero is only meaningful alongside a stated uncertainty (0.0 ± 0.1 is very different information from 0.00 ± 0.01).

Is 100 one, two, or three significant figures? As written, genuinely ambiguous — it could be any of the three depending on the precision intended. This exact problem is why scientific notation exists.

Do significant figures apply to something like “there are 7 days in a week”? No. Counted and defined quantities are exact numbers with effectively infinite significant figures, and they never limit the precision of a calculation they appear in.

Why does my calculator or spreadsheet show far more digits than my answer should have? Because calculators and spreadsheets don’t track significant figures at all — they compute using whatever binary floating-point precision they have (typically 15–17 digits) and display as many as fit. Rounding to the correct sig figs is a step you (or a sig-fig-aware tool) apply on top of the raw output.

What’s the difference between significant figures and decimal places? Significant figures count all meaningful digits regardless of where the decimal point falls. Decimal places count only digits after the decimal point. 3.14 and 314 both have 3 significant figures but very different decimal-place counts (2 and 0).

Sources and Further Reading

NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results — NIST’s own implementation guide of the international GUM, covering how measurement uncertainty is evaluated and reported. (emtoolbox.nist.gov)
NIST GLP 9 (2019), Rounding — NIST laboratory guidance on when and how to round measurement results and their uncertainties, including the even/odd rounding method and the rule against rounding intermediate values. (nist.gov)
ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications — the industrial standard governing how test results are rounded before being checked against a written specification. (store.astm.org; explainer at astm.org)
A2LA, Figuring Out Significance: What Are Significant Figures — a laboratory accreditation body’s explainer connecting classroom sig fig rules to NIST SP 811 §7.9 and real lab practice. (a2la.org)

Review and Methodology

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Reviewed by: [Pending — reviewer assignment required before publication] Last reviewed: [Pending] Methodology: Rules and worked examples are cross-checked against NIST and ASTM primary sources (see above). All calculator results on this page are generated using an arbitrary-precision decimal engine, not native floating-point math, and are validated against a versioned regression fixture set before any release.

Changelog

v1.0 — Initial draft completed, 2026-08-10.