A significant figure is a digit you can defend as real. 12.345 has five of them, and 0.000104 has three, because the leading zeros only place the decimal point. When you multiply or divide, the answer keeps the fewest significant figures among the inputs. When you add or subtract, it keeps the fewest decimal places. Those two rules cover most practical rounding.
Key Takeaways
12.345has five significant figures; rounded to three it becomes12.3.- Leading zeros never count:
0.000104has three, and in scientific notation it is1.04 × 10⁻⁴.- In multiply and divide, the result keeps the fewest sig figs:
3.14 × 2.7 = 8.478, rounded to8.5.- In add and subtract, the result keeps the fewest decimal places:
12.34 + 5.6 = 17.9.
What is a significant figure?
A significant figure is a digit that carries information about the size of a number. Writing 2.500 makes a bigger claim than writing 2.5: the first says the value is known to the thousandth place, the second only to the tenth. Significant figures are the agreed way to record how much a number actually knows, and the significant figures guide page lays the rules out in one place.
The counting rules
Four counting rules cover almost every case you will meet:
- Every nonzero digit counts.
12.345has five significant figures. - Zeros between nonzero digits count.
1004has four. - Trailing zeros after the decimal point count.
200.0has four. - Leading zeros before the first nonzero digit do not count.
0.000104has three.
The leading zeros in 0.000104 only fix the position of the decimal point. Write the number in scientific notation and they vanish: 1.04 × 10⁻⁴. Now the count is visible in the numerals themselves, which is why scientific notation and significant figures are frequent companions.
Rounding to a chosen number of significant figures
To round to N significant figures, read the digits from the left, keep N of them, and let the next digit decide. Placeholder zeros stay so the number keeps its size.
Worked example: round 3.14159 to four significant figures. The first four digits are 3.141, and the fifth is 5, so the last kept digit rounds up: 3.142. For comparison, this calculator shows π as 3.14159265359 at its 12-digit display.
Worked example: round 12.345 to three significant figures. Keep 12.3. The next digit is 4, which leaves the 3 alone. The answer is 12.3.
Worked example: round 0.000104 to two significant figures. Keep 1.0 and preserve the place value: 0.00010. Type 0.000104 into the calculator and the display returns 1.04 × 10⁻⁴, which shows the three real digits at a glance.
Significant figures in calculations
This calculator, like most, computes with more digits than it displays and rounds the display to 12 significant digits. You apply the significant figure rules to the result by hand. Two different rules apply, and mixing them up is the most common source of wrong answers.
Multiplying and dividing: the fewest significant figures
Multiply or divide normally, then round to the fewest significant figures among the inputs. With 3.14*2.7 the calculator returns 8.478. The inputs have three and two significant figures, so the honest result keeps two: 8.5.
Adding and subtracting: the fewest decimal places
Addition and subtraction care where the uncertainty sits, not how many digits there are. With 12.34+5.6 the raw result is 17.94, but 5.6 is only known to the tenths place, so the sum can only be trusted to the tenths: 17.9.
Why multiplication and addition differ
In a product, the digit counts combine, and the shortest input limits the answer. In a sum, each input keeps its own last reliable place, and the least precise input fixes the final decimal place. Both rules exist so nobody reads more certainty into a result than the measurements allow.
Why significant figures matter in measurement
Every measured number carries uncertainty, and significant figures are the compact way to record it. A scale that reads 200.0 g claims precision to the tenth of a gram. The same scale reading 200 g claims only the nearest hundred grams. The trailing zero is the difference between the two instruments, and writing it tells the reader which one you used. In money, engineering tolerances, and lab data, the extra zero changes how the number can be used.
Where significant figures meet floating point
Binary calculators store most decimal numbers slightly wrong, because numbers like 0.1 do not exist exactly in base 2. The floating point arithmetic guide explains how results get rounded back for display. The 0.1 plus 0.2 post shows the raw 0.30000000000000004 behind the scenes. Significant figures are the language you use to say which of the stored digits you actually trust, which is precisely why a 12-digit display is safe for everyday work: the real precision is the digits you just counted, not the last few the calculator printed.
Frequently asked questions
Do trailing zeros count as significant figures?
It depends on where the decimal point sits. Zeros after the decimal point count, so 200.0 has four. Plain trailing zeros in a whole number are ambiguous, which is why scientific notation exists: 2.00 × 10⁵ clearly means three significant figures, while 200000 by itself says nothing.
Should I round at every step of a calculation?
No. Round once, at the end. Rounding intermediate results stacks small errors and defeats the purpose of tracking precision.
Why does the rule change between multiplication and addition?
Because the two operations fail in different ways. Multiplying combines digit counts, so the shortest input bounds the product. Adding keeps place values, so the least precise input fixes the final decimal place. The conventions keep the reported result honest for each operation.
Does this calculator round automatically to significant figures?
No. It rounds the display to 12 significant digits, whatever their count. You choose the number of significant figures for the final answer and round the displayed value yourself.