The Significant Figures Addition Rule
Addition and subtraction do not follow the significant figures rule at all: they follow the decimal places rule. The answer is only as precise as the coarsest input, so 12.52 + 3.1 is 15.6, rounded to one decimal place because 3.1 was measured to one. The page opens on the plus sign with that sum in the boxes, or with the numbers from your last visit if you have used the tool before, and every answer names the rule it used and how many places each input carried. Multiplication is the other rule, and it is genuinely different: 12.5 × 3.2 is 40, two significant figures, because the answer there follows figures rather than places.
12.52 + 3.1 is 15.6.
The decimal-places rule. Addition and subtraction keep decimal places, not figures: the answer is only as precise as the coarsest input. 12.52 is good to 2 decimal places and 3.1 to 1 decimal place, so 3.1 sets the limit and the answer is given to 1 decimal place.
Worked cases
Each one is computed by the same code that answers your own numbers, every time this page loads.
The rules
Four rules decide which digits count, and two more decide what survives an operation. The example under each rule is computed by the calculator above, not written out here.
Every non-zero digit counts
1 through 9 are always significant, wherever they sit in the number. This is the rule the other three qualify, and on a number with no zeros in it, it is the only rule you need.
4521 → 4 significant figures
Zeros between non-zero digits count
A zero with a counted digit on each side was measured like any other digit, so it counts. Position makes no difference: a captive zero counts whether or not there is a decimal point anywhere in the number.
1004 → 4 significant figures
Leading zeros never count
Zeros in front of the first non-zero digit only place the decimal point, so they carry no measurement. 0.00452 and 4.52 claim exactly the same three figures, which is why the count does not change when you switch from metres to millimetres.
0.00452 → 3 significant figures
Trailing zeros count past a decimal point
A zero written after the last non-zero digit, with a decimal point in the number, is a measured digit: nobody writes it by accident. Without a decimal point it is genuinely ambiguous, and 1200 is read as 2 figures by the convention. Write 1200. or 1.200 × 10³ when you mean 4.
1.200 → 4 significant figures
Multiplication and division keep significant figures
The answer carries as many significant figures as the input with the fewest. 12.5 has three and 3.2 has two, so the product is given to two, and the extra digits your calculator shows are not knowledge you had.
12.5 × 3.2 → 40
Addition and subtraction keep decimal places
Here it is decimal places, not figures, that carry: the answer is only as precise as the coarsest input. 12.52 is good to two places and 3.1 to one, so the sum is given to one place. Notice that the answer keeps three significant figures, more than 3.1 had, which is exactly why the two rules cannot be swapped.
12.52 + 3.1 → 15.6
Common questions
- What is the addition rule for significant figures?
- That there is not one. Adding and subtracting carry decimal places instead: count the decimal places in each input, take the smallest, and round the answer to that many. 12.52 has two places and 3.1 has one, so the sum 15.62 is written 15.6. The significant figures of the inputs do not enter into it, and the answer can end up with more figures than either input had.
- Why does the answer sometimes have more significant figures than the inputs?
- Because places and figures are different counts. 3.1 carries two significant figures and one decimal place. Add it to 12.52 and the answer 15.6 carries three significant figures, still at one decimal place. Nothing was invented: the sum is known to the nearest tenth because both inputs were, and a number in the fifteens simply needs three digits to say that. This is the clearest sign that the multiplication rule does not apply here.
- How is subtraction different?
- It uses the same rule and can be far more brutal about it. 12.52 minus 3.1 is 9.4, one decimal place. When two close numbers are subtracted, most of the leading digits cancel and the figures that survive are few: 10.01 minus 10.00 leaves 0.01, one significant figure out of four, at the same two decimal places both inputs had. The places rule is what protects you from writing 0.010000 and claiming precision that cancelled out.
- What if one number has no decimal point?
- Then it carries zero decimal places, and the sum is rounded to a whole number: 1200 + 3.14 is 1203. A number written as 1.2 × 10³ is coarser still, good only to the nearest hundred, and the calculator reads it that way. Every answer states how many places each input carried and which one set the limit, so the reason for the rounding is on the page rather than left to you to reconstruct.
- Does this page do multiplication too?
- Yes. Switch the operator to × or ÷ and the answer changes rule: it is then given to as many significant figures as the input with the fewest, and the page says so. The counting and rounding modes are on the same tool, along with all six rules stated in full.
Exact. Counting and rounding to significant figures is a stated convention applied literally, and the page shows which digits it counted and which rule it used. Where a trailing zero is genuinely ambiguous without a decimal point, it says so rather than choosing for you.