Significant Figures
Count significant figures and round to any number of them.
Significant digits: 4050.
Significant figures are the digits in a number that carry real measurement precision. By the standard rules, 0.04050 has 4 significant figures — the leading zeros only set the decimal place and don’t count, while the trailing zero after the decimal point does count because it signals measured precision.
What significant figures are
Significant figures (or “sig figs”) are the digits of a number that actually convey precision — the ones you measured or computed reliably, not the placeholder zeros that merely position the decimal point. They matter because a result is only as precise as its least precise input, so reporting the right number of sig figs is how science and engineering communicate how trustworthy a value is. Writing 2.5 cm claims far less precision than 2.500 cm, even though the numbers are equal.
The significant-figure rules
Apply them left to right across the number.
| Rule | What counts | Example | Sig figs |
|---|---|---|---|
| Non-zero digits | Every non-zero digit is always significant | 472 | 3 |
| Captive zeros | Zeros between non-zero digits always count | 5.06 | 3 |
| Leading zeros | Zeros before the first non-zero digit never count | 0.0072 | 2 |
| Trailing zeros, with a decimal | Trailing zeros count when a decimal point is present | 1.230 | 4 |
| Trailing zeros, no decimal | Trailing zeros in a bare integer are ambiguous | 1200 | 2 (ambiguous) |
Worked example
Count the significant figures in 0.04050:
- 1 Drop the leading zeros. In 0.04050 the “0.0” before the 4 only places the decimal — those zeros are not significant.
- 2 Count from the first non-zero digit. Counting starts at 4. The digits that remain are 4, 0, 5, 0.
- 3 Keep captive and trailing zeros. The 0 between 4 and 5 is captive (counts), and the final 0 is a trailing zero after a decimal point (counts).
- 4 Read the total. 4, 0, 5, 0 → 0.04050 has 4 significant figures.
Rounding, precision, and the 1200 problem
Rounding to N sig figs. To round to N significant figures, keep the first N significant digits and round the rest using the next digit. For example, 3.14159 rounded to 3 sig figs is 3.14, and 0.04050 rounded to 2 sig figs is 0.041. When the kept digits are trailing zeros that fixed notation can’t show honestly, the result is written in scientific notation.
Why sig figs communicate precision. The count tells a reader how finely a quantity was known. A measurement of 4.0 g and one of 4.000 g are numerically equal but claim very different precision — two sig figs versus four. Carrying the correct number through a calculation prevents a result from looking more precise than the data behind it.
The integer ambiguity. A bare integer like 1200 is genuinely ambiguous: the trailing zeros might be measured or might just be placeholders. The common rule, used here, treats them as not significant (so 1200 reads as 2 sig figs). The clean fix is scientific notation — write 1.2 × 10³ for two sig figs, or 1.200 × 10³ for four. That removes all doubt.