Sig Fig Calculator

What a significant figure is

Significant figures are the digits of a number that say something about how finely it was measured. Write 12.5 cm from a ruler and all three digits carry information: the 1 and the 2 line up with marks on the scale, and the 5 is your best read of the space between them. Write 12.50000 and you have claimed a reading good to a hundred thousandth of a centimeter, which no household ruler can deliver.

The idea exists because every measurement stops somewhere. Counting significant figures is a compact way of recording where it stopped, so the next person knows how much of the number is measurement and how much is decoration. That is why the concept belongs to measured values rather than to pure arithmetic: the 3 in a textbook exercise has no uncertainty, while 3.0 read off a graduated cylinder does.

This page does two jobs. It counts the significant figures in any number you type, and it rounds a number to however many significant figures you ask for, which is the step that goes wrong most often when it is done by hand.

The four counting rules

Every case reduces to four rules, and the table puts them side by side with a worked number for each.

Read in order, the rules are: nonzero digits are significant; zeros between nonzero digits are significant; leading zeros are never significant; trailing zeros are significant only when a decimal point is present. In scientific notation the same value is 4.500 x 10^-3 = 0.004500, and the mantissa 4.500 shows the four digits at a glance. A number like 0.05070 falls out of those four in one pass: the leading zeros are ignored, the 5 and the 7 are nonzero, and the two zeros after them sit either between significant digits or after a decimal point, so the count is 4.

Only one case needs a judgment call, and it is the fourth row of the table: a whole number that ends in zeros and carries no decimal point. Nothing in the digits themselves says whether the zeros were measured or added to reach the right magnitude, so a convention has to be chosen. This calculator treats them as placeholders.

How to use the calculator

Type a number in the first field, then pick a significant figure count in the second. The default setting is just count, which reports the digit count and leaves the number alone. Choose 1 through 6 and the calculator also rounds to that many significant figures, so the two answers appear together and you can check one against the other.

Both answers come from the digits you typed, not from a floating point approximation. That matters for numbers like 0.004500, where the two trailing zeros are the whole point: the calculator reads the text, keeps those zeros, and reports 4 significant figures with 0.00450 as the value rounded to 3. A routine that converts the input to a number first would silently drop the zeros and report 2.

The counting follows the placeholder convention for whole numbers, so 1200 reports 2 significant figures. Add a decimal point, 1200., and the same digits report 4. The note under the result repeats that convention so the answer is never ambiguous.

Rounding a large whole number can need trailing zeros to hold its magnitude, and a bare 12000 no longer shows how many digits are significant. In that case the second row prints the result in scientific notation, so 12345 rounded to 2 significant figures appears as 1.2 x 10^4 rather than as a string of digits whose count you would have to guess.

If the field holds something the calculator cannot read as a number, it says so instead of guessing. Nothing is sent anywhere: the arithmetic runs in your browser.

Rounding to N significant figures

Rounding to a significant figure count is a three step move. Find the digit that will be the last one kept, look at the digit just after it, and then either leave the kept digit alone or raise it by one. A following digit of 4 or less leaves it, and 5 or more raises it. Digits after the cut are dropped, and any that were to the left of the decimal point become zeros so the magnitude survives.

Run it on 3.14159. To keep 3 significant figures, the last kept digit is the 4 in the hundredths place, and the digit after it is 1, so the answer is 3.14. To keep 4, the last kept digit is that same 4 with 5 behind it, so it rises to 2 and the answer is 3.142.

The last row is where hand work drifts. Keeping 2 digits of 12345 gives 12000, and the zeros are placeholders that hold the size of the number, not measurements. Written as 1.2 x 10^4 the count is visible again, which is exactly what the calculator prints for that input: 1.2 x 10^4 at 2 digits, 1.23 x 10^4 at 3, and 1.235 x 10^4 at 4.

Round once, from the original number. Chained rounding drifts when a step lands on a 5: 0.4449 rounded straight to 1 significant figure is 0.4, but a hand that rounds to 0.45 first and then to 1 digit reports 0.5, which is a different value. One pass from the source number avoids the whole question.

Trailing zeros, whole numbers and scientific notation

The trailing zero convention is the part of this topic that causes the most arguments, because the digits alone cannot settle it. A lab reports a mass as 1500 g. If the balance read to the nearest gram, all four digits are measurements and the count is 4. If the figure was rounded to the nearest hundred, only the 1 and the 5 are measurements and the count is 2. The written number looks identical either way.

Scientific notation removes the ambiguity, since the mantissa holds exactly the significant digits. Write 1.500 x 10^3 and the count is 4. Write 1.5 x 10^3 and the count is 2. The notation and the long form describe one value, so 1.500 x 10^3 = 1,500, with the four digits carried in the mantissa. That is why lab reports, data sheets and published tables switch to notation as soon as a value grows past a few digits.

The calculator applies the same reading, and the note under the result names the convention so a reader who needs the other one can add the decimal point and rerun. Reporting the assumption is more useful than hiding it, since a bare count with no stated convention is an opinion dressed as a number.

For measured work, the safer habit is to write the number in scientific notation as soon as a trailing zero might be mistaken for padding. It costs a few characters and removes a conversation.

Exact numbers and the SI definitions

Not every number in a calculation is a measurement, and significant figure rules only apply to the measured ones. Counted quantities are exact: 12 eggs, 7 days in a week, 100 centimeters in a meter. Conversion factors inside one system are exact by definition, so they never limit the precision of a result.

Some physical constants are exact as well. Since the 2019 revision of the International System of Units, the speed of light in vacuum is defined as exactly 299,792,458 meters per second, a value with no uncertainty at all, as the NIST reference on constants records. The same revision fixed the Planck constant at exactly 6.62607015 x 10^-34 joule seconds and the elementary charge at exactly 1.602176634 x 10^-19 coulombs. Each of those long strings of digits is exact, so trimming one of them to 3 significant figures is a choice about display rather than a claim about precision.

That distinction explains a puzzle that shows up in homework. A student rounds the speed of light to 3.00 x 10^8 m/s, then worries the rounding spoiled the answer. It did not, as long as the final result is reported to the precision of the measured quantities in the problem. The constant was never the limit.

Measured quantities are the other case, and they set the ceiling. A length read as 12.5 cm carries 3 significant figures, and multiplying it by an exact 2 cannot produce more than 3. The rules for multiplication keep the smallest count among the measured inputs, and the rules for addition keep the smallest number of decimal places. The NIST constants reference lists which values are exact and which carry uncertainty, and the 2019 SI brochure gives the definitions in full.

Common mistakes worth catching

Most wrong counts come from one of four habits, and each has a quick test that catches it.

Two more traps are worth naming. The first is confusing significant figures with decimal places: 0.004500 rounded to 2 decimal places is 0.00, which destroys the value, while rounded to 2 significant figures it is 0.0045, which keeps it. The second is carrying a count through an addition: 12.5 cm plus 0.25 cm is limited by the tenths place of the first number, not by its 3 significant figures, so the sum reports 12.8 cm rather than 12.75 cm.

The last habit to build is a sanity read of the output. If a count comes back as 7 for a number that starts with three zeros, or 1 for a value that plainly records a measurement, the convention in play is not the one you expected, and the note under the calculator says which one it used.

The Khan Academy algebra library covers place value and rounding drills if you want practice with the mechanics before trusting a count on a lab report.