How this instrument works
Significant figures count meaningful digits, starting from a number's first nonzero digit, no matter what place value that digit sits in. That is a different question from 'how many decimal places,' and the two rules can pull in opposite directions: 1,234.5 rounded to two decimal places barely moves, but rounded to two significant figures it collapses to 1,200, because only the leading '1' and '2' are being kept.
The formula locates that leading digit with a logarithm. Because log10 of a number counts how many places stand before the decimal point, exp = floor(log10|x|) returns the power of ten the first digit occupies — 3 for any number from 1,000 up to 9,999. Subtracting sigfigs minus 1 from that exponent slides down to the position of the last digit worth keeping, and ordinary rounding does the rest once that position becomes the scale.
Trailing zeros are the classic trap. Written as 1,200, a two-sig-fig answer looks identical to a measurement trusted to four digits, because a bare decimal number cannot show where the significant digits stop — scientists resolve the ambiguity with scientific notation, 1.2 times ten cubed, and this sheet's Rounding scale field carries the same information in plain form. Negative and fractional inputs work the same way: a value like -0.0499 has its leading digit two places after the decimal point, so exp comes out negative and the scale shrinks accordingly.
- Type the measured value into the Number field — positive, negative, whole, or decimal all work.
- Set how many digits to keep in the Significant figures field.
- Order of magnitude shows the power of ten the leading digit occupies, computed automatically.
- Rounding scale shows the place value of the last digit that survives rounding.
- Rounded value gives the final answer, rounded to exactly that many significant figures.
Worked example — 1,234.5 to two significant figures
A lab balance reads a sample mass of 1,234.5 grams, but the balance itself is only trusted to two significant figures. Order of magnitude works out to exp = floor(log10(1234.5)) = floor(3.0915) = 3, since 1,234.5 sits between 1,000 and 9,999. Rounding scale follows as 10^(3 − 2 + 1) = 10² = 100, meaning the last trustworthy digit lands in the hundreds place.
Rounded value divides the reading by that scale, rounds to the nearest whole number, and multiplies back: round(1,234.5 ⁄ 100) × 100 = round(12.345) × 100 = 12 × 100 = 1,200. Reporting 1,200 rather than 1,234.5 is not sloppiness — it is an honest statement that the instrument cannot vouch for the '34.5' part of the reading, only for the leading '1,2'.
Questions
What actually counts as a significant figure?
Every nonzero digit counts, plus any zero trapped between two nonzero digits or used to mark measured precision after a decimal point. Leading zeros never count — they only place the decimal point. In 1,234.5, all five digits are significant; asking for two keeps just the leading '1' and '2' and rounds the rest away.
How is rounding to significant figures different from rounding to a decimal place?
Decimal-place rounding fixes the place value no matter how large the number is — two decimal places always means hundredths. Significant-figure rounding recomputes the place value from the number's own size through exp = floor(log10|x|), so '2 sig figs' rounds 1,234.5 to the hundreds place but rounds 0.012345 to the thousandths place. Same instruction, different scale, because the scale tracks the number rather than sitting fixed.
Why can 1,200 mean two, three, or four significant figures?
Because a plain decimal number gives no way to mark where the significant digits stop — trailing zeros before a decimal point are structurally ambiguous. Scientific notation removes the ambiguity: 1.2 times ten cubed is unmistakably two sig figs, 1.20 times ten cubed is three. This sheet's Rounding scale field states the same fact directly, without needing exponential notation.
What is the most common mistake people make with significant figures?
Treating the digit count as a count of decimal places instead of meaningful digits. Rounding 1,234.5 to two decimal places barely changes it, landing at 1,234.50, while rounding it to two significant figures collapses it to 1,200 — the two instructions sound alike and produce wildly different answers.
Do negative numbers or numbers smaller than one follow the same rule?
Yes — the absolute value inside floor(log10|x|) strips the sign, so magnitude alone decides the exponent. A value like -0.0499 has its leading digit two places after the decimal point, giving exp = -2 and a rounding scale of 0.001, so it rounds to -0.050 at two significant figures, sign preserved.