Long Division Calculator with Negative Numbers

Negative-number calculator: enter signed values; sign rules apply to the quotient.

Output: signed quotient and remainder with verification.

The Long Division Calculator with Negative Numbers divides values that may be positive or negative and applies the correct sign rules to the result. It separates the sign from the size: it divides the absolute values with ordinary long division, then decides whether the quotient is positive or negative based on the signs of the dividend and divisor. Two like signs give a positive quotient; two unlike signs give a negative quotient. Enter any combination of positive and negative numbers to see the signed quotient, the remainder, and the step-by-step work with the sign rule explained.

How to use the Long Division Calculator with Negative Numbers

To divide with negatives, follow these 4 steps:

Enter the dividend, using a minus sign if it is negative.
Enter the divisor, using a minus sign if it is negative.
Choose remainder or decimal output.
Click Calculate to read the signed quotient, the remainder, and the sign-rule explanation.

The calculator computes the magnitude first, then attaches the correct sign, so −144 ÷ 12 and 144 ÷ −12 both return −12.

Sign rules for division

Division and multiplication share the same sign rules because they are inverse operations. When the dividend and divisor have the same sign — both positive or both negative — the quotient is positive. When they have opposite signs, the quotient is negative. The size of the answer never depends on the signs; only its direction does. The Long Division Calculator with Negative Numbers keeps these two questions apart: first, how big is the quotient, and second, which sign does it carry. This split makes signed division no harder than ordinary long division.

Division shares a total into equal groups. Long division does this digit by digit.

Divisor (32) — the number you divide by. Place it to the left of the bracket.

How negative numbers work in long division

The calculator produces a signed answer through five internal actions:

Reads the sign of the dividend and the sign of the divisor.
Takes the absolute value of both numbers to remove the signs.
Divides the two positive values with the standard divide–multiply–subtract–bring-down loop.
Applies the sign rule: same signs give a positive quotient, opposite signs give a negative quotient.
Gives the remainder the same sign as the dividend, following the truncated-division convention.

Because the magnitude is computed on positive numbers, the steps look identical to ordinary long division — only the final sign changes.

Reads the sign of the dividend and the sign of the divisor.

Sign rule and identity

The Long Division Calculator with Negative Numbers uses the sign rules (−) ÷ (−) = (+), (+) ÷ (−) = (−), and (−) ÷ (+) = (−), combined with the identity Dividend = Divisor × Quotient + Remainder. The remainder takes the sign of the dividend. For −17 ÷ 5 = −3 R−2, the check is 5 × (−3) + (−2) = −17.

Dividend

Divisor

Quotient

Remainder

-100 = 7 × -14 + 2 → Identity holds

Negative number example problems

These examples show each sign combination.

Example 1 — Negative ÷ positive: −144 ÷ 12

Divide the magnitudes: 144 ÷ 12 = 12.
The signs are opposite, so the quotient is negative.
So −144 ÷ 12 = −12.

Example 2 — Negative ÷ negative: −144 ÷ −12

Divide the magnitudes: 144 ÷ 12 = 12.
The signs are the same, so the quotient is positive.
So −144 ÷ −12 = 12.

Example 3 — Negative with a remainder: −17 ÷ 5

Divide the magnitudes: 17 ÷ 5 = 3 R2.
The signs are opposite, so the quotient is −3 and the remainder takes the dividend's sign.
So −17 ÷ 5 = −3 R−2, and 5 × (−3) + (−2) = −17.

Divide the magnitudes: 144 ÷ 12 = 12.
The signs are opposite, so the quotient is negative.
So −144 ÷ 12 = −12.

Worked negative-number problems

What is 200 ÷ −8?

200 ÷ −8 = −25. Divide the magnitudes: 200 ÷ 8 = 25 with no remainder. The dividend is positive and the divisor is negative, so the signs are opposite and the quotient is negative. Check: −8 × −25 = 200.

2008

How does the remainder get its sign in −23 ÷ 4?

−23 ÷ 4 = −5 R−3. The magnitudes give 23 ÷ 4 = 5 R3. Signs are opposite, so the quotient is −5, and under the truncated-division convention the remainder shares the dividend's sign, making it −3. Check: 4 × (−5) + (−3) = −23.

234

Common negative-number mistakes

Signed division produces 5 frequent errors:

Forgetting that two negatives make a positive quotient.
Attaching a sign to the magnitude before the division instead of after.
Giving the remainder the wrong sign instead of matching the dividend.
Cancelling only one minus sign when both inputs are negative.
Writing the quotient as negative when both signs are actually the same.

The Long Division Calculator with Negative Numbers divides the absolute values first and applies the sign rule last, so the final sign is always correct.

Forgetting that two negatives make a positive quotient.

Attaching a sign to the magnitude before the division instead of after.

Giving the remainder the wrong sign instead of matching the dividend.

Cancelling only one minus sign when both inputs are negative.

Writing the quotient as negative when both signs are actually the same.

Frequently Asked Questions

What is the sign rule for dividing negative numbers?

Like signs give a positive quotient and unlike signs give a negative quotient. So (−) ÷ (−) = (+), while (+) ÷ (−) and (−) ÷ (+) both equal (−).

Is a negative divided by a negative positive?

Yes. Two negatives have the same sign, so their quotient is positive. For example, −144 ÷ −12 = 12.

What sign does the remainder take?

Under the common truncated-division convention the remainder takes the sign of the dividend. For −17 ÷ 5 = −3 R−2, the remainder is negative because the dividend is negative.

How do I divide a positive number by a negative number?

Divide the magnitudes normally, then make the quotient negative because the signs are opposite. For 200 ÷ −8, compute 200 ÷ 8 = 25, then apply the sign to get −25.

Do negative numbers change the long division steps?

No. The divide, multiply, subtract, and bring-down steps run on the absolute values, so they are identical to ordinary long division. Only the final sign is added afterward.

How do I check a signed division answer?

Use divisor × quotient + remainder = dividend, keeping all signs. For −144 ÷ 12 = −12, the check is 12 × (−12) = −144.

Why divide the absolute values first?

Separating size from sign keeps the arithmetic simple: you run standard long division on positive numbers, then apply one sign rule at the end instead of tracking signs through every step.

The identity with signs

The Long Division Calculator with Negative Numbers keeps the identity Dividend = Divisor × Quotient + Remainder valid with signs included, where the quotient's sign follows the sign rule and the remainder matches the dividend's sign. For −17 ÷ 5 = −3 R−2, the identity confirms 5 × (−3) + (−2) = −17.