What is the remainder, and how is it different from the decimal part?

The remainder is what is left over in units of the dividend; the decimal part expresses that leftover as a fraction of the divisor. 7 ÷ 2 is 3 remainder 1, and the same division is 3.5, because the leftover 1 is one half of the 2 you were dividing by. The two numbers are not interchangeable — writing 3 remainder 1 as 3.1 is only correct when the divisor happens to be 10. Divide the remainder by the divisor to convert it: 1 ÷ 2 = 0.5.

How do you divide by a decimal?

Move both decimal points to the right by the same number of places until the divisor is a whole number, then divide normally. 12.5 ÷ 0.4 becomes 125 ÷ 4. This is legitimate because you are multiplying the top and bottom of a fraction by the same power of ten, which cannot change its value. Only the divisor has to become whole — the dividend can keep a decimal point, and the point in the answer sits directly above it.

Why do some divisions never end?

Because a decimal can only terminate when the divisor divides some power of ten, which means it is built entirely from 2s and 5s. Any other prime factor — a 3, a 7, an 11 — can never divide 10, 100 or 1000, so the remainders never reach zero. Of the 999 divisors between 2 and 1000, only 28 are made purely of 2s and 5s. The other 971 repeat forever. There are only finitely many possible remainders, so one must eventually come back, and once it does the digits after it repeat in the same order.

What does the bar over the digits mean?

It marks the block that repeats forever. 1/3 is written 0.3 with a bar over the 3, meaning threes without end; 1/7 is 0.142857 with a bar across all six. Brackets are the same idea: 0.1(6) means the 6 recurs while the 1 does not. The notation matters because 0.3 with a bar and 0.333 are genuinely different numbers — the second is a rounded version of the first, short by a third of a thousandth.

How many decimal places should I work out?

As many as the question needs, and then one more so you can round rather than cut. Truncating and rounding are different operations: cutting 22 ÷ 7 at four places gives 3.1428, rounding gives 3.1429. Across the repeating divisors under 1000, cutting at six places differs from rounding at six places about 47% of the time, so the choice is not academic. If the expansion repeats, the honest answer is the barred form or the original fraction — both are exact, and no number of decimal places is.

Why is the remainder always smaller than the divisor?

Because if it were not, the divisor would fit at least once more and the quotient digit you wrote was too small. That is the definition doing its job: the quotient digit is the largest whole number of times the divisor fits, so whatever is left cannot contain another one. It is the single most useful check while working — a leftover of 9 when dividing by 4 means the digit above it should have been two larger.

Is −7 ÷ 2 equal to −3 remainder −1 or −4 remainder 1?

Both, depending on the convention, and this is a genuine split rather than a mistake. Rounding toward zero gives −3 remainder −1, and the remainder follows the dividend — that is what hand calculators, C, Java and JavaScript do. Rounding down gives −4 remainder 1, and the remainder follows the divisor and never goes negative — that is what Python and Ruby do. Both satisfy divisor × quotient + remainder = dividend, which is why neither can be called wrong. Over a hundred thousand signed pairs the two conventions gave different quotients 48% of the time.

How do I check a long division answer?

Multiply the divisor by the quotient and add the remainder. If it returns the dividend the answer is right, and if it does not, it is wrong — there is no partial credit in this test. For 125 ÷ 4 = 31 remainder 1, the check is 4 × 31 + 1 = 125. It works for decimals and for negative numbers under either sign convention, which is precisely why both conventions are acceptable. This page runs the check on every answer it gives.

Why can you not divide by zero?

Because the question has no answer to write down. Division asks how many times the divisor fits into the dividend, and zero fits any number of times without ever using anything up — no count is more correct than another. The check equation shows it plainly: 0 × anything + remainder can never come to 5, so nothing qualifies as the answer to 5 ÷ 0. It is not that the answer is infinity; it is that no number satisfies the definition.

Long Division Calculator

Long division is the one piece of school arithmetic that is really an algorithm. Four instructions — divide, multiply, subtract, bring down — repeated once per column until the digits run out, and every intermediate value written down where you can see it. That last part is why it survives the calcu...

A remainder is not a decimal. 7 ÷ 2 is 3 remainder 1, and it is also 3.5 — but the 1 and the 5 are different quantities. The remainder counts leftover units of the dividend; the 5 counts tenths. Read one as the other and 3 r 1 becomes 3.1. Most divisions never resolve into a tidy decimal at all: of the 999 divisors between 2 and 1000, only 28 produce a decimal that ever stops. The rest repeat forever, and this page finds the repeating block exactly rather than cutting the answer off and hoping.

THE DIVISION

DIVIDEND

DIVISOR

DECIMAL PLACES

At that point:

125 ÷ 4 TO 4 PLACES

31.25

The division comes out exactly after 2 decimal places, so nothing is lost.

QUOTIENT AND REMAINDER

31 remainder 1

Check: 4 × 31 + 1 = 125 ✓ back to the dividend

SUBTRACTIONS

steps in the working

REPEATING BLOCK

none

it ends cleanly

BEFORE THE REPEAT

decimal places in total

EXACT AS A FRACTION

125/4

in lowest terms

WHAT IS HAPPENING

Every column asks the same question. How many whole times does the divisor fit into what is standing there? That count is the quotient digit, the multiple comes off, and the next digit drops down beside what is left. The leftover is always smaller than the divisor — if it were not, the digit above it was too small.

Every remainder computed exactly · the tableau and the answer are worked out separately and compared

Created with❤️byeaglecalculator.com

HOW TO USE

1
Type the dividend and the divisor. Decimals are fine — if the divisor is not a whole number the page shifts both by the same power of ten and tells you so, which is exactly what you would do on paper.
2
Read the tableau on the right. The quotient builds above the bar, each subtraction sits under the digits it came from, and the dashed lines show which digit is being brought down into the next step.
3
Check the remainder trail underneath. It shows the leftover at every step, and marks the moment one repeats — which is the reason the decimal repeats, not a coincidence.
4
Set how many decimal places you want and whether to round or cut at that point. The exact answer stays on screen with the repeating block barred, so you can always see what the shortened version threw away.

WORKED EXAMPLE

Take 125 ÷ 4. Four does not fit into 1, so the first quotient digit is 0 and is not written. Bring the 2 across: four fits into 12 exactly three times, so 3 goes above the 2 and 12 comes off, leaving nothing. Bring down the 5. Four fits into 5 once, so 1 goes above the 5, 4 comes off, and 1 is left. The whole-number part is finished: 125 ÷ 4 is 31 remainder 1. Notice that the remainder 1 means one unit of the dividend left over out of the four needed for another whole, not "one tenth". To continue, put a point in the quotient directly above the point in the dividend and start bringing down zeros. Bring one down beside the leftover 1 to make 10. Four fits into 10 twice, 8 comes off, 2 remains. Bring down another zero to make 20. Four fits into 20 exactly five times, 20 comes off, and the remainder is zero. The division is finished: 31.25. It finished because 4 is 2², and a decimal can only terminate when the divisor is built from the factors of ten. Change the divisor to 7 and nothing terminates. Take 22 ÷ 7. Seven fits into 22 three times with 1 left, so the answer starts 3 remainder 1. Bring down a zero: 10 gives 1 with 3 left. Then 30 gives 4 with 2 left, 20 gives 2 with 6 left, 60 gives 8 with 4 left, 40 gives 5 with 5 left, and 50 gives 7 with 1 left. That 1 is the remainder we already had after the first step, so from here every digit must repeat in the same order. The answer is 3.142857 with all six digits recurring, and that is exact — not a rounding, not a truncation. Seven is a full reptend divisor: its block uses every one of the six possible non-zero remainders before returning. Sixty divisors under 1000 do this. Now a decimal divisor: 12.5 ÷ 0.4. Multiply both by ten and it becomes 125 ÷ 4, which we have already done. Scaling the top and bottom of a fraction by the same amount cannot change its value, which is why the trick is legitimate rather than a shortcut. Finally, check it. 4 × 31 + 1 = 125, which is the dividend, so the whole-number answer is right. For the decimal, 4 × 31.25 = 125 exactly. Any long division you can check this way is a long division you can trust.

REFERENCE RULES

Rule	What it says	Why
The four steps	divide · multiply · subtract · bring down	One quotient digit per column, repeated until the digits run out.
The check	divisor × quotient + remainder = dividend	The only test that matters. This page runs it on every answer.
Remainder size	always less than the divisor	A remainder as large as the divisor means the quotient digit was too small.
A decimal divisor	shift both, then divide	12.5 ÷ 0.4 becomes 125 ÷ 4. Moving both points leaves the answer unchanged.
The point in the answer	straight above the point below	Once the divisor is whole, the quotient point sits over the dividend point.
Remainder against decimal	7 ÷ 2 = 3 r 1 = 3.5	The remainder is 1 out of 2, not a tenth. r 1 and .1 are different things.
Going into decimals	bring down zeros	The dividend has an endless supply of them after the point.
When it terminates	divisor is only 2s and 5s	Every other divisor repeats forever. Only 28 divisors under 1000 terminate.
How soon it stops	after max(twos, fives) places	1/40 = 1/(2³×5) stops after three places.
Period length	at most divisor − 1	Because only that many non-zero remainders exist before one must repeat.
Full reptend	period = divisor − 1	1/7 = 0.142857 repeating, using all six. Sixty divisors under 1000 do this.
Writing a repeat	0.1̅6̅ or 0.1(6)	An overbar or brackets on the block. Writing 0.166 instead is a different number.
Negative, toward zero	−7 ÷ 2 = −3 r −1	What calculators, C and JavaScript do. The remainder follows the dividend.
Negative, floored	−7 ÷ 2 = −4 r 1	What Python and Ruby do. The remainder follows the divisor and never goes negative.

FREQUENTLY ASKED QUESTIONS

Four instructions repeated once per column. Divide: how many whole times does the divisor fit into the current working number? That is the next quotient digit. Multiply: that digit times the divisor. Subtract: take it off. Bring down: pull the next digit of the dividend alongside what is left and start again. When the dividend runs out you either have a remainder or, if you keep bringing down zeros, a decimal.

·Divide, multiply, subtract, bring down — once per column
·The quotient digit goes above the digit you just brought down
·What is left must always be smaller than the divisor
·Bring down zeros to carry on past the decimal point

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Last updated: August 5, 2026 · Every remainder exact, no floating point · divisor × quotient + remainder is checked on every answer.