Rounding Calculator
Half-up and half-even disagree at exactly the half.
Work out Rounding. Half-up and half-even disagree at exactly the half. States the assumption instead of hiding it.
Negative rounds to tens, hundreds and so on
Rounded to 0 places, half up
3
Half-even gives 2 — the two disagree here
Half-up and half-even disagree only at an exact half, which is precisely where the choice matters. Half-up biases a large set of values upward, because every tie goes the same way. Half-even sends ties to the even neighbour, so they split roughly evenly — which is why financial and scientific systems default to it.
How the Rounding Calculator works
A number under every rounding convention at once — half up, half to even, floor, ceiling and truncation. They agree everywhere except at an exact half, which is precisely where the choice matters.
Also known as: bankers rounding calculator · why does 2.5 round to 2 · round half to even calculator · round to decimal places
Half-up carries a bias that half-even removes
Rounding every tie upward adds a small positive bias to every sum. Over a large number of values that bias accumulates measurably, which is a real problem in accounting and in scientific data reduction.
Half-to-even sends ties to the nearest even neighbour, so 2.5 goes to 2 and 3.5 goes to 4. Across many values the ties split roughly evenly between up and down, and the bias cancels.
That is why it is the default in financial systems and in the IEEE floating point standard. It is called banker's rounding for the first reason and round-half-to-even for the second, and it is the same rule.
Round once, from the original
Rounding twice compounds error. Taking 2.44 to 2.4 and then to 2 gives a different answer from rounding 2.44 straight to 2, because the intermediate rounding moved the value across a boundary it should not have crossed.
The rule is to carry full precision through every step of a calculation and round once at the end, for display. Rounding at each intermediate step accumulates error, and the accumulation grows with the number of steps.
The related problem is totals not reconciling. Rounding each component and rounding the total separately gives two different answers, which is why an invoice can be a penny out — and why some systems deliberately allocate the residual to one line.
Floating point is a separate problem
Binary floating point cannot represent most decimal fractions exactly. One tenth has no finite binary expansion, exactly as one third has no finite decimal one, so 0.1 is stored as a very close approximation.
That is why 0.1 plus 0.2 gives 0.30000000000000004 in most languages. It is not a rounding mode issue; it is the representation itself, and choosing a different rounding rule does not fix it.
The remedy for money is to work in the smallest currency unit as integers — pence rather than pounds — or to use a decimal type that stores base-ten digits. Rounding after floating-point arithmetic hides the symptom rather than removing the cause.
Where to go next
The Rounding question rarely arrives on its own. These are the ones that usually come with it:
- Long Division Calculator — Shows the working, digit by digit.
- Percentage of a Number Calculator — Percentages commute — 4% of 75 is 75% of 4.
- Variance Calculator — Sample divides by n−1, and it matters.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
What is banker's rounding?
Rounding half to even: ties go to the nearest even number, so 2.5 rounds to 2 and 3.5 rounds to 4. It removes the upward bias that always rounding halves up introduces across many values.
Why does 2.5 round to 2 sometimes?
Under half-to-even, because 2 is the even neighbour. Under half-up it rounds to 3. Both are correct conventions and financial and scientific systems generally default to half-to-even.
What is the difference between rounding and truncating?
Truncating discards everything past the cut regardless of value, so 2.9 truncates to 2. Rounding considers what was discarded. Truncation always biases towards zero.
How do I round to significant figures?
Count from the first non-zero digit rather than from the decimal point. 0.004567 to three significant figures is 0.00457, which is five decimal places.
Does rounding twice cause errors?
Yes, and it is a real problem. Rounding 2.44 to 2.4 and then to 2 gives a different answer from rounding 2.44 directly to 2. Always round once, from the original value.
Why do totals sometimes not add up after rounding?
Because each rounded component carries an error and those errors accumulate. Rounding the total separately from the parts is why an invoice can be a penny out, and it is why some systems allocate the residual deliberately.
What is round half away from zero?
Ties round away from zero in both directions, so 2.5 goes to 3 and −2.5 goes to −3. It is what most people mean by half-up, and it is symmetric about zero.
What is stochastic rounding?
Ties are rounded randomly, weighted by how close the value is to each neighbour. It removes bias without the even-neighbour rule, and it appears in machine learning where accumulated bias matters.
Why do computers round strangely?
Because binary floating point cannot represent most decimal fractions exactly. 0.1 + 0.2 gives 0.30000000000000004, which is a representation issue rather than a rounding one.
How should money be rounded?
To the smallest currency unit, using a documented and consistent rule. Financial regulations frequently specify the method, and mixing rules between systems is what makes reconciliations fail.
What is the round-half-up bias?
Always rounding ties upward adds a small positive bias to every total. Over a large number of values it accumulates measurably, which is exactly what half-to-even was designed to prevent.
Should I round intermediate results?
No. Carry full precision through the working and round once at the end. Rounding at each step accumulates error, and the error grows with the number of steps.
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