How rounding works
Rounding replaces a number with a simpler one that is close to it. You decide how precise the result should be, for example two decimal places, three significant figures or the nearest hundred. Then you keep the digits up to that point and look at the first digit you are about to drop. In everyday rounding, a dropped digit of 5 or more makes the last kept digit go up by one, and 4 or less leaves it unchanged.
Take 2.675 to two decimal places. You keep 2.67 and drop the 5. Since 5 counts as "5 or more", the 7 becomes an 8 and the answer is 2.68. Take 3.14159 to two places instead: the first dropped digit is 1, so the answer is 3.14. The calculator highlights that deciding digit in every result, so you can see exactly where the cut happened.
Only the first dropped digit matters for "up or down", with one exception. When it is a 5, the rule needs to know whether anything nonzero follows. 2.6751 is past the halfway point between 2.67 and 2.68 and rounds up under every nearest rule. 2.675 sits exactly halfway, a tie, and that is where the rules differ.
The seven rounding rules
Four rules go to the nearest neighbor and only differ on exact ties. The other three always move in one direction, no matter how close the number is to either neighbor. Here is what each rule does when rounding to a whole number:
| Rule | 2.5 | 3.5 | 2.4 | 2.6 | −2.5 | −2.6 |
|---|---|---|---|---|---|---|
| Half away from zero | 3 | 4 | 2 | 3 | −3 | −3 |
| Half up | 3 | 4 | 2 | 3 | −2 | −3 |
| Half down | 2 | 3 | 2 | 3 | −3 | −3 |
| Half to even (banker's) | 2 | 4 | 2 | 3 | −2 | −3 |
| Always up (ceiling) | 3 | 4 | 3 | 3 | −2 | −2 |
| Always down (floor) | 2 | 3 | 2 | 2 | −3 | −3 |
| Toward zero (truncate) | 2 | 3 | 2 | 2 | −2 | −2 |
- Half up sends ties to the larger number. For positive numbers it matches the familiar "5 or more rounds up"; for negative numbers, the larger number is the one closer to zero, so −2.5 becomes −2.
- Half down sends ties to the smaller number: 2.5 becomes 2.
- Half to even sends ties to the neighbor whose last digit is even: 2.5 becomes 2, 3.5 becomes 4.
- Half away from zero treats positive and negative numbers alike: 2.5 becomes 3 and −2.5 becomes −3.
- Always up (ceiling) moves to the next value toward positive infinity whenever anything is cut off, and always down (floor) moves toward negative infinity.
- Toward zero (truncate) simply drops the extra digits: 2.9 becomes 2 and −2.9 becomes −2.
Half to even: why some people round 2.5 down
If you always round ties up, a long list of rounded numbers drifts upward. Round 0.5, 1.5, 2.5 and 3.5 half up and you get 1, 2, 3 and 4, which add up to 10, while the original numbers add up to 8. Half to even rounds two of them up and two down, giving 0, 2, 2 and 4, which add up to 8 again. That makes half to even, also called banker's rounding, a good fit wherever many rounded values get added up. Python's built-in round() function, for one, rounds ties toward the even choice, according to its official documentation.
Significant figures
Significant figures count the digits that carry information about precision. Start counting at the first nonzero digit and keep going to the right:
- Leading zeros never count. 0.004567 has four significant figures, 4, 5, 6 and 7.
- Zeros between nonzero digits always count. 1,005 has four.
- Trailing zeros after a decimal point count. 2.50 has three, which says the value was measured to the hundredth.
- Trailing zeros in a whole number are ambiguous. 1,200 could have two, three or four. Scientific notation removes the doubt: 1.2 × 10³ has two, 1.200 × 10³ has four.
To round to significant figures, count off the figures you need and round at that digit like any other rounding. 0.004567 to two figures is 0.0046. A carry can add a digit in front: 9.995 to three figures is 10.0, still three figures. The calculator shows the result in scientific notation as well, with exactly the number of figures you asked for.
Rounding to the nearest 10, 100, 0.05 or 1/16
Rounding to a multiple works in three steps. Divide the number by the step, round that quotient to a whole number, and multiply back. For 7.38 to the nearest 0.05: 7.38 ÷ 0.05 = 147.6, which rounds to 148, and 148 × 0.05 = 7.40. For 1,234,567 to the nearest thousand the step is a power of ten, so you can read the answer straight off the digits: the hundreds digit is 5, so the result is 1,235,000.
Fractions work the same way. To round 5.3 inches to the nearest sixteenth, multiply by 16 to get 84.8, round to 85 and divide by 16, which gives 5 5/16 inches. The result is shown as a mixed number and as a decimal.
Why 2.675 sometimes rounds to 2.67
Programs that work in binary floating point, such as Python with its float type and JavaScript with its numbers, cannot store most decimal fractions exactly; the number base converter shows the repeating binary digits of a fraction like 0.1. The double precision value closest to 2.675 is 2.67499999999999982236431605997495353221893310546875, a hair below the halfway point, so a program that rounds the stored value gets 2.67. Python's documentation describes exactly this case for round(2.675, 2), and MDN shows the same effect for JavaScript, where (2.55).toFixed(1) gives "2.5".
This calculator never turns your number into binary. It keeps the digits you typed as text and rounds them with exact whole-number arithmetic, so 2.675 to two places is 2.68, and 1.005 to two places is 1.01. That makes it a good way to check what a correct decimal answer looks like.
How to use the rounding calculator
- 1Type the number you want to round. Negative numbers, thousands separators and scientific notation like 6.02e23 all work.
- 2Choose what to round to: decimal places, significant figures, a multiple such as 0.05 or 100, or a fraction such as 1/8. Tap a chip or type your own value.
- 3Pick a rounding rule. Half up is the everyday rule; the table under the result shows what every other rule would give.
- 4Read the result and the highlighted deciding digit. To round many numbers with the same settings, paste them into the list box and copy the results or download them as a CSV file.
Frequently asked questions
What is the basic rule for rounding?
Find the last digit you want to keep and look at the digit right after it. If that digit is 5 or more, increase the last kept digit by 1. If it is 4 or less, leave it as it is. Then drop everything after the kept digit. So 3.14159 to 2 decimal places is 3.14, and 2.678 is 2.68.
Does 5 round up or down?
It depends on the rule. In the familiar "5 or more rounds up" rule, a 5 followed by nothing else rounds up, so 2.5 becomes 3. Half to even, also called banker's rounding, sends an exact 5 to the even neighbor instead, so 2.5 becomes 2 and 3.5 becomes 4. If any nonzero digit follows the 5, as in 2.51, every nearest rule rounds up.
How do I round to 2 decimal places?
Keep two digits after the decimal point and look at the third. If it is 5 or more, add 1 to the second digit. 4.736 becomes 4.74 and 4.734 becomes 4.73. Choose Decimal places with 2 in the calculator, and it shows the deciding digit highlighted.
How do you round negative numbers?
The digits work the same, but the direction words change meaning. Half away from zero turns −2.5 into −3, like 2.5 into 3. Half up, which means toward the larger number, turns −2.5 into −2, because −2 is larger than −3. Always down (floor) turns −2.1 into −3, and toward zero turns it into −2.
How do I round to significant figures?
Start counting at the first nonzero digit and keep as many digits as you need, then round at that point. 0.004567 to 2 significant figures is 0.0046, and 1,234 to 2 significant figures is 1,200. Written in scientific notation, 1.2 × 10³ makes clear that only two figures count.
Why does my program round 2.675 to 2.67?
Programs that use binary floating point cannot store 2.675 exactly. The value actually stored is a tiny bit below 2.675, so it rounds down. Python's documentation uses this very example for its round() function. This calculator works with the decimal digits you type, so 2.675 stays exactly 2.675.