Every sig-fig question on every homework set comes down to five rules. Nonzero digits count. Sandwiched zeros count. Leading zeros never count. Trailing zeros count exactly when a decimal point is present. And scientific notation counts only the coefficient. Everything else is practice. Here are the rules, the traps, and the rounding procedure that follows from them.
The digits 1 through 9 are always significant wherever they appear. 4578 has 4 sig figs. So do 0.4578, 45.78, and 4.578 × 10³. If a digit required a measurement to know, it counts, and nonzero digits always do.
Sandwiched zeros carry measurement information. 3.005 has 4 sig figs. 25007 has 5. The classic trap here is a zero that looks decorative but sits between significant digits: 0.00507 has 3 sig figs, the 5 and the sandwiched 0 and the 7, because the two leading zeros don't count but the trapped one does.
Zeros to the left of the first nonzero digit only position the decimal point. 0.004578 has 4 sig figs. 0.05 has 1. This feels like it penalizes small numbers, but the logic is sound: leading zeros are determined by your choice of units, not by the precision of any instrument. The same measurement is 0.004578 m or 4.578 mm, and the zeros in the first version carry no extra information.
This is the rule that decides 90% of exam disputes. Once a decimal point appears in the number, trailing zeros are claimed digits: 2.000 has 4 sig figs, 25.20 has 4, and 0.050 has 2 (the leading zeros still don't count, the 5 and final 0 do). Without a decimal point, trailing zeros in a whole number are ambiguous placeholders: 100 has 1 sig fig as written, 1500 has 2.
Why the asymmetry? Writing "2.000 g" is a deliberate act. Nobody types three trailing zeros after a decimal by accident. In "1500 g," the zeros might be measured or might be rounding to the nearest hundred, and the notation can't say which.
In 6.02 × 10²³, count the 6, 0, and 2: 3 sig figs. The exponent handles magnitude and never contributes. This is why scientists prefer the notation for big and small numbers; it deletes the ambiguity entirely. 1.0 × 10³ claims 2 sig figs unambiguously while 1000 shrugs.
| Number | Sig figs | Which rule decided it |
|---|---|---|
| 4578 | 4 | All nonzero (Rule 1) |
| 0.004578 | 4 | Leading zeros skipped (Rule 3) |
| 3.005 | 4 | Sandwiched zero counts (Rule 2) |
| 2.000 | 4 | Trailing zeros + decimal (Rule 4) |
| 0.050 | 2 | Leading out, trailing in (Rules 3+4) |
| 2500.0 | 5 | Decimal present, all trailing zeros count |
| 100 | 1 | No decimal, ambiguous placeholder |
| 100. | 3 | Decimal point rescues the zeros |
| 6.02 × 10²³ | 3 | Coefficient only (Rule 5) |
Paste any number, pick N, and see the rounded value, the count, and both notations.
Sig Fig Calculator →Three steps. Find the first significant digit and count to the Nth. Look at digit N+1: 5 or more rounds the Nth digit up, less than 5 leaves it alone. Replace everything after with placeholder zeros (or an exponent).
Example: 0.004578 to 2 sig figs. First two significant digits are 4 and 5; the next digit is 7, so the 5 rounds up to 6: 0.0046. Another: 4578 to 2 sig figs. Keep 4 and 5, the 7 rounds up, giving 4600, which cleanly written is 4.6 × 10³. The zeros in 4600 are placeholders by Rule 4, which is why the scientific form is the honest one.
The rule students mix up most. Multiplication and division follow the least number of sig figs: 4.56 × 1.4 = 6.384, reported as 6.4 because 1.4 has 2 sig figs. Addition and subtraction follow the least number of decimal places: 12.11 + 0.3 = 12.41, reported as 12.4 because 0.3 is only trustworthy to the tenths place.
The intuition: multiplication compounds relative error, so the input with the fewest sig figs dominates. Addition compounds absolute error, so what matters is the coarsest decimal place. In a multi-step problem, don't round intermediates; carry extra digits and round once at the end, applying the rule of the final operation.
Counted quantities (12 eggs), defined conversions (1 m = 100 cm), and formula constants have unlimited sig figs. They never limit your answer. If you time 3 laps on a stopwatch reading 42.6 s each, the 3 is exact and the answer 127.8 s keeps the stopwatch's precision, not "1 sig fig" from the lap count.
One, as written. Trailing zeros without a decimal point are placeholders. Write 100. and it claims 3 sig figs; write 1.00 × 10² and it claims 3 as well; 1.0 × 10² claims 2. Scientists use scientific notation precisely to make this precision explicit.
Because they only set the scale of the number, not its measurement precision. In 0.004578, the significant information starts at the 4. The three leading zeros are determined by where the decimal point sits, not by anything an instrument measured.
Exact numbers, things you count (12 eggs), defined conversions (1 m = 100 cm), and constants in formulas, have unlimited significant figures. They never limit a calculation. Only measured quantities do. That's why multiplying by 60 seconds in a minute doesn't degrade your answer's precision.