Significant figures look arbitrary the first time they show up on a general chemistry exam. Why does 12.3 + 4.56 round to 16.9, when a calculator will happily produce 16.86? Why is 3.4 × 1.234 reported as 4.2 instead of 4.1956? The rules are not arbitrary once you see what they are trying to encode: significant figures are a compressed way to carry uncertainty through a calculation without having to write out an explicit error bar at every step. This post covers the five rules, the two combination rules, and the scientific-notation convention that clears up the most common ambiguity.
Why sig figs exist
Every measured quantity has some uncertainty. A ruler marked in millimetres reads to about ± 0.5 mm. A three-digit digital scale reads to about ± 0.005 g. When you feed measured numbers through a calculation, that uncertainty propagates — and no amount of arithmetic can make the answer more precise than the least precise input. If you measure a box as 12.3 cm long and 4.56 cm wide, you cannot honestly claim the area is 56.088 cm². The trailing 8 and 8 imply a precision that the original measurement does not support.
Formal error propagation (using partial derivatives and standard deviations) handles this rigorously but is cumbersome for routine work. Significant figures are a shorthand that approximates the result you would get from full propagation, with rules simple enough to apply in your head. They undercount the uncertainty in some cases and overcount it in others, but for introductory science they are accurate enough and they are what exam rubrics grade.
Rule 1: nonzero digits always count
Every digit from 1 through 9, in any position, is always significant. The number 3.47 has three significant figures. The number 1,234,567 has seven. There is no ambiguity about nonzero digits — they are measured, and they count.
This rule is the easy one. The trouble starts with zeros, which can mean at least three different things depending on where they sit in the number.
Rule 2: captive zeros count
Zeros between nonzero digits — "captive" zeros — are always significant. The number 1,007 has four significant figures. The number 20.06 has four. The number 5.00004 has six. Captive zeros must have been measured; otherwise you would not have known to write them there.
Rule 3: leading zeros never count
Zeros that appear only to position the decimal point are not significant. The number 0.0034 has two significant figures — the 3 and the 4. The leading zero before the decimal and the two zeros after it are there to locate the decimal point, not to report measurement precision. In scientific notation this is written as 3.4 × 10⁻³, which makes the two-sig-fig count unambiguous.
Rule 4: trailing zeros with an explicit decimal count
Zeros at the end of a number that also contains a decimal point are significant. The number 12.300 has five significant figures — someone measured down to the thousandths place and got zeros, which is a real piece of information. The number 100. (with the decimal written explicitly) has three. Without the decimal, as in plain 100, this rule does not apply and you fall to rule 5.
Rule 5: trailing zeros without a decimal are ambiguous
The number 1,500 could have two, three, or four significant figures, and the ambiguity is genuine. If a speedometer read 1,500 rpm rounded to the nearest hundred, that is two sig figs. If it was accurate to the nearest ten, three. If it was accurate to the nearest single rpm, four. The bare notation does not say.
This is where scientific notation earns its keep. Writing 1.5 × 10³ fixes the count at two. Writing 1.50 × 10³ fixes it at three. Writing 1.500 × 10³ fixes it at four. Any time a problem hands you a trailing-zero number with no decimal, the safest assumption is that only the nonzero digits count — but in your own work, write the number in scientific notation so the count is unambiguous. Our scientific notation converter switches between the two forms mechanically.
Combination rule 1: addition and subtraction go by decimal places
When adding or subtracting, the answer has the same number of decimal places as the input with the fewest decimal places — not the fewest sig figs. Worked example:
12.3 + 4.56 = 16.86, reported as 16.9.
The input 12.3 has one decimal place. The input 4.56 has two. The less precise input (12.3) determines the answer's precision, so you round to one decimal place. Note that 16.9 has three significant figures, more than the two-sig-fig input 12.3 — the rule genuinely is about decimal places for addition, not sig figs.
Another example: 150 + 2.34. If we take 150 as having its ones place as the least-precise position (sig-fig count ambiguous, but decimal position clear at the tens place if we assume two sig figs), the answer 152.34 rounds to 150 — or more honestly, we write the result as 1.5 × 10². In practice, this is one of the common places where students get tripped up by ambiguous trailing zeros. When in doubt, move to scientific notation before adding.
Combination rule 2: multiplication and division go by sig figs
When multiplying or dividing, the answer has the same number of significant figures as the input with the fewest sig figs. Worked example:
3.4 × 1.234 = 4.1956, reported as 4.2.
The input 3.4 has two sig figs. The input 1.234 has four. The answer carries two, so we round 4.1956 to 4.2. Another:
6.022 × 10²³ / 18.0 = 3.3456 × 10²², reported as 3.35 × 10²² (three sig figs, limited by 18.0).
Our significant figures calculator handles both combination rules and returns the correctly rounded answer with the inputs flagged so you can see which one constrained the precision.
Exact numbers are infinite-precision
Counted quantities and defined constants do not limit sig figs at all. "Twelve eggs in a carton" is exact: the 12 does not have two sig figs — it has infinitely many. Unit conversion factors that are defined (1 inch is exactly 2.54 cm by international agreement) are also exact. The rule only constrains answers when measured quantities are involved.
This matters when you have an expression like "the half-life of a sample is measured as 14.3 days; how long until exactly one quarter of the sample remains?" The "one quarter" is exact, the 14.3 has three sig figs, and the answer should be reported to three sig figs. Our half-life calculator handles the arithmetic and keeps the sig-fig count consistent.
Intermediate steps: keep extras, round only at the end
When a calculation has multiple steps, the correct practice is to carry at least one extra significant figure through the intermediate arithmetic and round only at the final reported answer. Rounding at every step introduces compounding rounding errors that can shift the final digit by one or two units.
In practice, "carry one extra digit" is the minimum — many textbooks recommend carrying two extra digits for multi-step problems. The rule is: round once, at the end, to the sig-fig count determined by the least precise input. Your scratch paper should look less rounded than your final boxed answer.
What sig figs are not
Sig figs are not the same as decimal places. They do not imply anything about the absolute magnitude of the uncertainty (a measurement of 1.5 m and a measurement of 1.5 km both have two sig figs, but the uncertainties differ by a factor of a thousand). They cannot rescue a poorly designed experiment — if the measurement itself is biased, no amount of sig-fig discipline in the arithmetic will fix it.
What they can do is make your lab report honest. Reporting a result as 4.19567 cm² when your ruler reads to the nearest millimetre is making a claim you cannot support. Reporting it as 4.2 cm² says the same thing you actually know, and not a digit more. That is the entire point of the rules, and it is also why your instructor cares enough to grade on them.