Rounding Calculator – Decimal Places, Significant Figures and Nearest Values
Round any number exactly the way you mean it — to so many decimal places, so many significant figures, or the nearest ten, hundred, nickel or cent — and see what every other rounding rule would have answered, side by side with yours. The maths here runs on the digits you typed, never on a binary approximation, so 0.045 rounds to 0.05 the way your teacher promised, not to 0.04 the way most apps insist.
Your number and your rule
Round by
Try one of these
The number loaded is 0.045 — the classic troublemaker. As an exact decimal it is a perfect half at two places, so half up gives 0.05. Most apps store it as a binary fraction a hair under the half and answer 0.04. This page never leaves decimal, so the half stays a half.
Advanced options — money and percent presets, rule shortcuts, batch list rounding
For percent presets, type the percent number itself: 12.47 with “Tenth of a percent” rounds to 12.5, and with “Whole percent” it rounds to 12.
Lines that do not read as numbers are flagged and skipped; the rest are rounded with the exact decimal engine, not a binary approximation. Your main answer above is restored after the list runs.
Presets set the grid only — the rounding rule you picked above still decides every tie. Batch rounding uses your current mode and rule, and your main answer is restored afterwards.
🔒 Everything is calculated in your browser — your numbers never leave your device.
The answer
The full picture
The working, digit by digit
No binary detour, no hidden rounding before the rounding — this is the whole journey your number takes, with the cut drawn where it actually falls.
The number, digit for digit: 0.045 — held as exact digits from the moment you type it, never converted into a binary approximation.
The cut for 2 decimal places falls here: 0.04 | 5 — the part that falls off is exactly one half of the last kept place.
Half up: an exact half — and anything over it — goes away from zero. What fell off here is exactly half a step. Result: 0.05.
Check: the neighbours on this grid are 0.04 and 0.05, and 0.045 sits 50% of the way across — a true tie, which is why the rule you choose decides it.
Key takeaways
- Rounding is geography: your number stands between two grid steps, and only an exact half is ever argued about — every other case goes to the nearer step under all six rules.
- Significant figures count meaning, not places: leading zeros are placeholders, captive zeros count, and a trailing zero after the decimal point is a claim about precision.
- Round once, at the end: rounding every line of a calculation and then adding gives a different answer from adding first and rounding once.
Contents
- On the number line — and under every rule
- Keep exploring
- How it works in three steps
- How rounding is worked out
- Going deeper
- Counting significant figures: the rules, with the traps
- Every rule, side by side, in every mode
- Money and percent: the presets, and what they are really rounding
- Round once, at the end: the double-rounding trap
- Rounding a whole list at once
- Frequently asked questions
- Related calculators
On the number line — and under every rule
Rounding is geography: your number stands somewhere between two grid steps, and the rule you pick decides which one it walks to. The diamond is your number, the arrow is the journey — and underneath, the same number rounded all six ways, so a tie can never hide.
Keep exploring
Rounding tidies a number up — these work with the untidy originals.
Percentage Calculator
Parts of a hundred — a percentage of any number, or what percentage one number is of another.
🍕Fraction Calculator
Add, subtract, multiply and divide fractions, with every answer simplified.
🧮Scientific Calculator
Powers, roots, logs and trig for the working behind any formula on the site.
How it works in three steps
Pick the grid
Decimal places for money and measurements, significant figures for science, or a nearest value — the nearest ten, hundred, nickel or quarter. The grid decides which answers are even possible.
Find the cut
Your number is held as the exact digits you typed — all 30 of them if you type 30 — and the cut falls where the grid says. What falls off is compared with exactly one half of a step, no approximations.
Let your rule decide
Over a half, every rule climbs; under a half, every rule settles. Only an exact half is argued about — and your chosen rule settles it, with the other five answers shown underneath for honesty.
How rounding is worked out
The trap, retold: 0.045 at two decimal places stands exactly halfway between 0.04 and 0.05. Half up takes it to 0.05. A binary calculator stores the number as roughly 0.044999999999999999814, sees it as just under halfway, and answers 0.04 — the wrong answer to a question you never asked.
The banker’s pair: at zero decimal places with half even, 2.5 becomes 2 and 3.5 becomes 4. Each half settles on the even neighbour, so over a whole ledger the ups and downs cancel instead of quietly inflating every total.
The nickel: 1.13 on a 0.05 grid is 22.6 steps — more than halfway to step 23 — so it rounds to 23 × 0.05 = 1.15. Cash systems in countries without pennies live on this grid all day.
The big carry: 999 to two significant figures is 1,000: keeping two figures means keeping the 99, the dropped 9 is over half, and bumping 99 rolls the whole number over into a new thousand. The same roll-over turns 9.99 into 10.0 at one decimal place.
Going deeper
Computers count in binary, and most decimal fractions — 0.1, 0.045, 2.675 — cannot be written exactly as a binary fraction, the same way 1/3 cannot be written exactly as a decimal. The machine stores the nearest binary value it can, which for 0.045 is a hair below the true number. Every later calculation inherits that tiny error, and a rounding rule that only sees the stored value faithfully rounds the wrong number.
The fix used here is older than computers: keep the number as decimal digits and do the rounding on the digits themselves, by hand, the school way. It costs a little code and buys exactness — a 5 in the last place is a true 5, a half is a true half, and a 30-digit integer survives with every digit intact, long after binary storage would have blurred its tail into zeros.
Round half a million prices half up, and the halves — which are common in money, because prices love a 5 — all lean the same way: up. The drift is tiny per item and enormous per ledger. Half even removes the lean: an exact half goes to whichever neighbour ends in an even digit, so across many roundings roughly half the halves rise and half settle, and the expected error is zero.
That is why half even — banker’s rounding — is written into standards: it is the default in IEEE 754 floating-point arithmetic and in ASTM E29 for test results, and tax offices use it for the same reason. It feels stranger the first time 2.5 becomes 2; it feels fairer the first time you audit a million rows.
Counting significant figures: the rules, with the traps
Significant figures count the digits that carry meaning, starting from the first non-zero digit. Everything before that first non-zero digit is scaffolding: it tells you how small the number is, but it is not part of the measurement. Everything from that digit onward counts, including zeros, until the precision you were given runs out. The table below states each rule once, with the example that proves it, and every example can be checked in the calculator above by switching to Significant figures mode.
| Rule | Example | Significant figures | Why |
|---|---|---|---|
| Leading zeros are placeholders | 0.00450 | 3 (4, 5, 0) | The zeros only place the decimal point; counting starts at 4 |
| Zeros between non-zero digits count | 1002 | 4 | Captive zeros were measured, not added for show |
| Trailing zeros after a decimal point count | 20.0 | 3 | Writing the zero claims the tenths place was measured |
| Trailing zeros in a whole number are ambiguous | 1200 | 2, 3 or 4 | Without more information nobody can tell which zeros were measured |
| A decimal point removes the ambiguity | 1200. | 4 | The written point marks every digit as significant |
| Scientific notation removes it cleanly | 1.2 × 10³ | 2 | Only the digits written before the power count |
| Counted numbers are exact | 12 eggs | Unlimited | Counting is not measuring; 12 means exactly 12 |
The 1200 row deserves a second look, because it is the trap most often set in exams. A population quoted as 1200 might have been rounded to the nearest hundred, in which case only 1 and 2 are significant, or measured to the nearest ten, in which case 1200 has three significant figures, or counted exactly, in which case all four count. The digits alone cannot settle it, which is exactly why scientific notation exists: 1.2 × 10³ says two significant figures, 1.20 × 10³ says three, and 1.200 × 10³ says four, with no room for argument. When a question gives you a whole number with trailing zeros and no further context, the honest answer states the assumption you are making, and the calculator above follows the digits you type: it treats every digit you enter as given, so 1200 entered as a number keeps all four digits through its working.
The diagram shows 12,345 cut for three significant figures. The kept digits are 1, 2 and 3; what falls off is 45, and half of the hundreds step is 50, so 45 is under half and the kept 3 stays a 3, giving 12,300. The zeros at the end are placeholders that hold the hundreds place, not measurements: 12,300 to three significant figures means the hundreds digit is the last one that counts, and the tens and ones are reported as zero because the grid has no finer step. The same reading works on the small side. Take 0.00456 to two significant figures: the first significant digit is 4 in the thousandths place, the second is 5, and what falls off is 6 in the next place, which is over half of that step, so the 5 rounds up to 6 and the answer is 0.0046. And 0.00450 to two significant figures is 0.0045: the trailing zero in the original was significant, because it claimed the hundred-thousandths place was measured, but it is also the digit that falls off, and dropping a zero changes nothing.
One more distinction keeps the whole system honest. Decimal places count digits after the decimal point only, whatever the size of the number, while significant figures count meaningful digits wherever they fall, so the two measures disagree whenever a number is large or small. The number 0.004509 has six decimal places but only four significant figures, 4, 5, 0 and 9, because the leading zeros are placeholders. The number 123,456 to two decimal places is 123,456.00, which adds claimed precision, while to two significant figures it is 120,000, which throws precision away. Neither is wrong; they answer different questions, and the calculator above keeps them as separate modes precisely so that picking one never silently means the other.
Every rule, side by side, in every mode
The rounding rule you pick is not a decoration on the answer: on an exact tie it is the answer, and on every other number the directional rules still disagree with the nearest rules. The table below runs the same six numbers through all six rules, so the pattern can be read at a glance, and every cell can be reproduced above by setting the mode, the place and the rule. Half up and half down are the school pair: ties go away from zero under half up and toward zero under half down. Half even is the banker’s rule: ties go to the even neighbour. Ceiling, floor and truncate ignore ties entirely and always move one way.
| Number and grid | Half up | Half down | Half even | Ceiling | Floor | Truncate |
|---|---|---|---|---|---|---|
| 2.5, whole number | 3 | 2 | 2 | 3 | 2 | 2 |
| 3.5, whole number | 4 | 3 | 4 | 4 | 3 | 3 |
| −2.5, whole number | −3 | −2 | −2 | −2 | −3 | −2 |
| 0.045, 2 decimal places | 0.05 | 0.04 | 0.04 | 0.05 | 0.04 | 0.04 |
| 1.13, nearest 0.05 | 1.15 | 1.15 | 1.15 | 1.15 | 1.1 | 1.1 |
| 999, 2 significant figures | 1,000 | 1,000 | 1,000 | 1,000 | 990 | 990 |
| 12.47, nearest 1 | 12 | 12 | 12 | 13 | 12 | 12 |
Read the negative row first, because it is where half up surprises people. Half up on this page means ties go away from zero, so −2.5 becomes −3, exactly as the note at the foot of the page says. If your course defines half up as ties toward positive infinity instead, −2.5 would become −2 under that definition, which is the ceiling answer in this table. The two definitions agree on every positive number and differ only on negative ties, so a negative tie is the one case worth checking against your course or office before you commit. Ceiling and floor are defined by direction on the number line, not by distance from zero: ceiling always climbs toward positive infinity, so −2.5 becomes −2, while floor always drops toward negative infinity, so −2.5 becomes −3. Truncate always moves toward zero, cutting digits without looking at them, so −2.5 becomes −2 and 999 to two significant figures becomes 990, the same as floor for positives and the opposite of floor for negatives.
The 1.13 row shows why the rule choice still matters when there is no tie. The number sits at 22.6 nickel steps, past the halfway mark to step 23, so every nearest rule agrees on 1.15, but the directional rules split: ceiling also gives 1.15 because it always climbs, while floor and truncate give 1.1 because they always drop or cut. The 999 row makes the same point from the other side: the dropped 9 is over half of the tens step, so every nearest rule carries the number to 1,000, while floor and truncate hold it at 990. And the 12.47 row is the quiet case: 12.47 is nearer to 12 than to 13 under every nearest rule, and only ceiling, which never looks at distance, disagrees. The practical lesson is short. If you want the nearest answer, pick one of the three half rules and say which one, because only ties separate them. If you want a guarantee in one direction, for stock that must not run short or a charge that must not exceed a cap, pick ceiling or floor and say so, because those rules are promises about direction, not about closeness.
The rule control itself is deliberately mode-independent. The Rule select sits outside the three mode rows, so switching from decimal places to significant figures or to a nearest value never hides it, never resets it and never silently substitutes another rule. The Advanced panel adds the same six choices as buttons, so the rule can be changed from either place without touching the mode, and the table of all six answers under the number line always shows what the other five rules would have said. A rule that changed with the mode would be a trap: the same number, rounded in two modes, would disagree for no reason the user could see, and this page is built so that cannot happen.
Money and percent: the presets, and what they are really rounding
Money is the most common reason anyone rounds, and it is also the case where the grid is chosen by the coins in your pocket rather than by taste. The presets in the Advanced panel set the grid for you: nearest cent, nickel, dime, quarter, dollar and ten dollars, plus whole percent and tenth of a percent for percentage figures. A preset changes only the grid, never your rule, so a nickel preset under half even still settles ties to the even neighbour, and the table below shows what each money grid does to three ordinary prices under the default half-up rule.
| Price | Nearest cent | Nearest nickel | Nearest dime | Nearest quarter | Nearest dollar |
|---|---|---|---|---|---|
| $1.13 | $1.13 | $1.15 | $1.1 | $1.25 | $1 |
| $3.72 | $3.72 | $3.7 | $3.7 | $3.75 | $4 |
| $4.99 | $4.99 | $5 | $5 | $5 | $5 |
Each cell is a division followed by a rounding, and the working is worth seeing once. The price $1.13 on a nickel grid is 22.6 nickels, past halfway to 23, so it rounds to 23 nickels, which is $1.15. The same price on a quarter grid is 4.52 quarters, which rounds to 5 quarters, which is $1.25, a jump of twelve cents caused entirely by the coarser grid. The price $3.72 on a nickel grid is 74.4 nickels, under halfway to 75, so it falls to 74 nickels, which is $3.7, while on a quarter grid it is 14.88 quarters, which rounds up to 15 quarters, which is $3.75. Cash systems in countries without small coins live on exactly these grids all day: the total is rounded once, at the till, and the rounding applies to the total, not to each item, which is why the round-once section below matters to shopkeepers as much as to students.
Percent rounding works the same way, with one extra trap to avoid. To round a percentage figure, type the percent number itself and pick the grid: 12.47 with the whole-percent preset rounds to 12, and with the tenth-of-a-percent preset it rounds to 12.5. The tie case is instructive: 86.5 percent to a whole percent is 87 under half up and 86 under half even, because 6 is even, while 87.5 percent is 88 under both rules, because 8 is even and half up also goes up. The trap is confusing percentage points with percent change. A rate that moves from 12 percent to 15 percent has risen by 3 percentage points, but it has risen by 25 percent relative to where it started, because 3 is a quarter of 12. Rounding does not change that distinction, but it can hide it: 12.4 percent and 15.4 percent rounded to whole percents are 12 and 15, a gap of 3 points either way, while 12.5 percent and 15.5 percent rounded under half up become 13 and 16, still a gap of 3, because both ties moved together. When a rounded gap looks different from an unrounded gap, the rounding, not the data, is usually responsible, and the unrounded figures are the ones to quote.
One caution belongs with money in particular. A rounded price is a display figure, not a settlement figure, unless the rules of the system say otherwise. Tax is normally computed on unrounded amounts and rounded once at the end, and a receipt that rounds each line and then adds will disagree with a receipt that adds and then rounds, by exactly the mechanism of the next section. For money you will actually be taxed or paid on, the note at the foot of this page says it plainly: follow the official rule for your jurisdiction, and check the table of all six answers before you commit, because a tie settled the wrong way on a large total is a real amount of money, not a classroom detail.
Round once, at the end: the double-rounding trap
The most expensive rounding mistake is not picking the wrong rule. It is rounding twice. Take three items priced at 10.005 each. Rounded to two decimal places line by line under half up, each becomes 10.01, and the three rounded lines add to 30.03. Added first, the true total is 30.015, which rounded once to two places under half up is 30.02. The two honest procedures disagree by a cent, and neither arithmetic step is wrong: the error was introduced by the first rounding, and the second rounding could not remove it. The general principle, used by every careful calculator in this family, is to keep guard digits through every intermediate step and round only the final reported value.
The same principle governs significant figures in a longer calculation. When multiplying or dividing, the result carries the smallest significant-figure count among the measured inputs, because a chain is no stronger than its weakest measurement. The product 0.6238 × 6.6 is 4.11708 exactly, but the second factor carries only two significant figures, so the reported result is 4.1. When adding or subtracting, the result carries the coarsest last-significant place among the inputs, not the fewest figures: 2.334 + 0.31 is 2.644 exactly, but the second input stops at hundredths, so the reported result is 2.64. The two rules differ because multiplication scales uncertainty while addition shifts it, and using the wrong one is a genuine error, not a matter of taste. In both cases the rule applies to the final answer only: rounding 0.6238 to two figures before multiplying would give 0.62 × 6.6 = 4.092, which rounds to 4.1 as well here, but in general the early rounding moves the answer by more than the final rounding ever could, and the safe habit is to carry full digits until the end.
Trailing zeros are the visible form of this discipline. The value 15.50 and the value 15.5 are the same number, but they are not the same claim: 15.50 says the tenths place was measured and the hundredths place is zero, while 15.5 says nothing about the hundredths place at all. A measurement reported as 15.50 that was only measured to tenths overstates its precision, and a measurement reported as 15.5 that was measured to hundredths understates it. This is why the calculator above keeps trailing zeros in its decimal-places answers: 2.5 to two decimal places is shown as 2.50, because the grid you asked for has a hundredths place and the answer occupies it. Dropping that zero would still be the same value, but it would no longer show the grid, and a reader could not tell whether the answer was rounded to two places or never rounded at all. When you copy an answer into a report, copy the zeros with it: they are part of the answer.
Rounding a whole list at once
Most real rounding is not one number at a time. A price list, a column of lab readings, a set of exam marks: dozens of numbers, all on the same grid, all under the same rule. The batch panel in Advanced options handles exactly that job. Paste one number per line, press Round list, and each line is rounded with the mode, place and rule currently set above, using the same exact decimal engine as the main calculator, so a line reading 0.045 under two decimal places and half up comes back as 0.05, never as the binary approximation 0.04. Lines that do not read as numbers are flagged and skipped instead of silently becoming zero, and your main answer is restored after the list runs, so the batch never disturbs the number you were working on.
Three habits make batch rounding safe. First, set the rule before you run the list, because a list rounded under half up and the same list rounded under half even differ exactly on the ties, and ties are common in real lists precisely because prices and measurements love a final 5. Second, check the flagged lines rather than deleting them: a line flagged as not a number is usually a stray character, a second decimal point, or a unit written into the number, such as 12.5cm, and the fix is to clean the line and run it again, not to guess its value. Third, remember the round-once rule when the list feeds a total: round the list for display, but add the unrounded values for the total, and round the total once, or the displayed column and the total will disagree by exactly the double-rounding gap of the previous section. Used that way, the batch panel is a display tool with the engine’s exactness behind it, and the list it returns can be pasted straight into a report with its trailing zeros intact.
Frequently asked questions
What is the standard rounding rule for a half?
The school rule is half up: when the part that falls off is exactly half of the last kept place, the number rounds up — away from zero for negative numbers too. So 2.5 becomes 3, 3.5 becomes 4 and 0.045 at two decimal places becomes 0.05. It is the default on this page. It is not the only honest rule, though: banks often use half even instead, which is why this page shows every rule’s answer side by side.
What is banker’s rounding?
Banker’s rounding is half even: an exact half settles on whichever neighbour ends in an even digit. So 2.5 rounds down to 2 (2 is even) while 3.5 rounds up to 4 (4 is even), and 0.045 becomes 0.04 because 4 is even. Spread over thousands of roundings, halves go up as often as they go down, so no systematic bias creeps into totals — which is why banks, tax offices and the IEEE floating-point standard all use it.
How do I round to significant figures?
Count the digits that carry meaning, starting from the first non-zero digit, and keep that many. 12,345 to three significant figures is 12,300 — the 3 in the hundreds place is the last one kept, and everything after it becomes zeros. Small numbers work the same way once you find the first non-zero digit: 0.004567 to two significant figures is 0.0046. Leading zeros are never significant; they only hold the place.
Why do some calculators say 0.045 rounds to 0.04?
Because most calculators and apps store numbers as binary fractions, and 0.045 has no exact binary form. The stored value is about 0.044999999999999999814 — a whisker under the true half — so a rule that looks fair answers 0.04. The maths was fine; the number was swapped before the maths started. This calculator keeps your number as the exact decimal digits you typed, so a half here is a true half.
How do I round money to the nearest nickel or quarter?
Use the Nearest value mode above and pick the step: 0.05 for a nickel, 0.25 for a quarter, 0.01 for a cent. A price of 1.13 is 22.6 nickels, so it rounds to 23 nickels — 1.15. Countries that have withdrawn small coins round cash totals exactly this way, usually with half up, though the legal rule is set country by country.
What is the difference between ceiling, floor and truncate?
They ignore halves entirely and always move one way. Ceiling climbs to the next step up the number line: 2.1 becomes 3, and −2.5 becomes −2. Floor drops to the step below: 2.9 becomes 2, and −2.5 becomes −3. Truncate just cuts, always toward zero: 2.9 becomes 2 and −2.5 becomes −2. For positive numbers floor and truncate agree; for negatives they part ways, which is exactly where the three earn their separate names.
How do I count significant figures in 0.00450?
There are three: 4, 5 and 0. The leading zeros are placeholders that only place the decimal point, so counting starts at 4. The trailing zero counts because it sits after the decimal point and claims the hundred-thousandths place was measured. To two significant figures 0.00450 rounds to 0.0045, because the digit that falls off is that same trailing zero.
How do I round a whole list of numbers at once?
Open Advanced options and use the batch panel: paste one number per line, press Round list, and each line is rounded with the mode, place and rule currently set above, using the same exact decimal engine as the main calculator. Lines that do not read as numbers are flagged and skipped, and your main answer is restored after the list runs.
What is the difference between rounding to decimal places and to significant figures?
Decimal places count digits after the decimal point only, whatever the size of the number, while significant figures count meaningful digits starting from the first non-zero digit. So 0.004509 has six decimal places but only four significant figures (4, 5, 0, 9), and 123,456 to two decimal places is 123,456.00 while to two significant figures it is 120,000. They answer different questions, which is why this page keeps them as separate modes.
Should I round at each step of a calculation, or only at the end?
Only at the end. Three items at 10.005 each become 30.03 if you round every line to two places first, but 30.02 if you add first (30.015) and round once. Keep full digits through every intermediate step — guard digits — and round only the final reported value, or the early rounding error survives into your answer.