About the Rounding Calculator
Rounding looks like the simplest thing in arithmetic and is the source of a surprising amount of trouble. Two people can round the same number correctly and get different answers, because "correctly" depends on which rule they were taught — and the rules genuinely differ.
The school rule sends a trailing five upward. Accountants and statisticians often send it to the nearest even number instead, for a reason that matters when you are adding up thousands of figures. Programmers frequently truncate, which is not rounding at all. Each is right in its own context.
This calculator rounds to any number of decimal places, or to the nearest ten, hundred, thousand or million, under any of five rules — and when the case is genuinely borderline, it shows you what every rule would give, so you can see the disagreement rather than be caught by it.
How to Use the Rounding Calculator
Enter the number.
Choose what to round to. Positive options are decimal places; the negative options round to the nearest power of ten, which is what you want for large figures. Rounding 1,234 to the nearest hundred gives 1,200.
Choose the rule:
- Half up — the usual school rule. A trailing five goes up, away from zero.
- Half to even — banker's rounding. A trailing five goes to whichever
neighbour is even.
- Always up — ceiling. Anything above the mark goes to the next value.
- Always down — floor. Anything below the next mark stays put.
- Cut off — truncate. Simply discard the extra digits.
The rule only matters when the part being dropped is exactly one half. For every other value all five give the same answer.
Press Calculate. You get the rounded figure, the rounding error, and what the other rules would have produced.
How Rounding Is Calculated
Rounding to n decimal places means finding the multiple of 10⁻ⁿ closest to your number. Mechanically:
Shift the decimal point n places to the right, so the digit you care about becomes the units digit.
Round to a whole number using your chosen rule.
Shift back n places to the left.
Rounding to the nearest hundred is the same operation with n = −2: shift left two places, round, shift back.
There is one implementation detail that matters more than it should. The obvious code — multiply by 100, round, divide by 100 — is wrong for some inputs, because computers store decimals in binary and many decimal fractions have no exact binary form. The value 1.005 is actually held as something very slightly below 1.005, so multiplying by 100 gives 100.49999999999999, and rounding that gives 1.00 rather than 1.01.
This calculator shifts the decimal point in the text of the number rather than multiplying the stored float, so the answer matches the digits you typed. It is a small thing that produces visibly wrong answers when ignored, and a great many calculators do ignore it.
Rounding Formula
rounded = round( value x 10^n ) / 10^n
where n is positive for decimal places
and n is negative for tens, hundreds, thousandsThe rules differ only in what round does with an exact half:
half up 2.5 -> 3 3.5 -> 4 -2.5 -> -3
half to even 2.5 -> 2 3.5 -> 4 4.5 -> 4
ceiling 2.1 -> 3 2.9 -> 3 -2.5 -> -2
floor 2.1 -> 2 2.9 -> 2 -2.1 -> -3
truncate 2.9 -> 2 -2.9 -> -2Note how ceiling and floor behave with negatives. They operate on the number line, not on magnitude, so the ceiling of −2.5 is −2 — a smaller magnitude. Truncation always moves toward zero, which makes it match floor for positives and ceiling for negatives.
Step-by-Step Example
Round 3.14159 to two decimal places.
Step 1 — shift. Move the point two places right: 314.159.
Step 2 — round to a whole number. The fractional part is 0.159, which is less than a half, so it rounds down to 314.
Step 3 — shift back. 3.14.
Straightforward, and every rule agrees because nothing was borderline.
Now a case where they do not agree. Round 2.5 to a whole number:
half up -> 3 (the school answer)
half to even -> 2 (2 is even, 3 is not)
always up -> 3
always down -> 2
truncate -> 2Five rules, two different answers, and none of them is a mistake. This is exactly why the calculator shows all of them when it detects a halfway case.
Why does banker's rounding exist? Consider rounding a long column of figures that happen to end in 5 — prices ending in .5 cents, measurements on a half-unit grid. Half up sends every one of them upward, so the total comes out systematically too high. Sending half of them up and half down cancels the bias. Over ten values the difference is negligible; over a million transactions it is not, which is why accounting standards and the IEEE floating-point specification both default to it.
Understanding Your Result
The rounded figure keeps trailing zeros when you asked for decimal places, because 2.50 and 2.5 are not the same statement. The first says the value is known to two places; the second only to one. In a measurement context that difference is information.
The rounding error is how far the rounded value sits from the original. It is negative when you rounded down and positive when you rounded up. Worth a glance when precision matters: rounding 0.0004 to two places gives 0.00, and an error of −0.0004 on a value of 0.0004 has thrown away everything.
Under the other rules lists what the alternatives would give. If they all agree, the case was not borderline and your choice of rule was irrelevant. If they differ, the choice is doing real work, and it is worth being deliberate about it.
When Should You Use This Calculator?
Money. Prices, invoices and interest are almost always two decimal places, but the rule varies: many accounting standards specify half to even, and some countries round cash totals to the nearest five cents.
Measurements and lab work. Reporting a reading to the precision the instrument actually supports rather than the precision the display shows.
Large figures in writing. Populations, budgets and distances read better rounded to the nearest thousand or million, and the negative place options do that directly.
Checking a spreadsheet. If a total does not match the sum of its displayed parts, rounding is usually why. Comparing rules here often finds the culprit.
Programming. Working out what a language's rounding function will actually do, particularly around halves and negatives, where implementations differ.
Common Mistakes
Rounding twice. Rounding 2.45 to one place gives 2.5; rounding that to a whole number gives 3. Rounding 2.45 straight to a whole number gives 2. Always round once, from the original value, to your final precision.
Rounding during a calculation. Round at the end, not in the middle. Each intermediate rounding adds error, and the errors accumulate rather than cancel.
Assuming everyone uses half up. Spreadsheets, programming languages and accounting systems do not agree. If two systems disagree by a cent, the rounding rule is the first thing to check.
Misreading ceiling and floor with negatives. Ceiling of −2.5 is −2, not −3. "Up" means toward positive infinity, not away from zero.
Confusing truncation with rounding down. They match for positive numbers and differ for negatives. Truncating −2.9 gives −2; flooring it gives −3.
Dropping trailing zeros in reported measurements. Writing 2.5 when you measured to two places discards precision information that the reader needs.
Trusting a calculator that multiplies by a power of ten. Try 1.005 to two places. If it gives 1.00, it is doing the arithmetic on a binary approximation rather than on your digits.