Complex Number Long Division Calculator

Complex calculator: enter a+bi forms to divide and get real + imaginary parts.

Output: complex quotient (real and imaginary).

The Complex Number Long Division Calculator divides one complex number by another and returns the result in a + bi form. Because you cannot divide directly by an imaginary part, the method multiplies the numerator and denominator by the conjugate of the divisor, which turns the denominator into a real number. The calculator then separates the real and imaginary parts, simplifies each, and shows every step. Enter a dividend and divisor as a + bi to see the conjugate multiplication, the expanded numerator, the real denominator, and the final quotient with its real and imaginary components.

How to use the Complex Number Long Division Calculator

To divide complex numbers, follow these 4 steps:

Enter the dividend as a + bi (for example, 3 + 2i).
Enter the divisor as c + di (for example, 1 − i).
Click Calculate to multiply by the conjugate and simplify.
Read the quotient's real part and imaginary part with each step shown.

The calculator forms the conjugate of the divisor automatically, so you only enter the two complex numbers and read the a + bi answer.

Why complex division uses the conjugate

A complex number a + bi has a real part a and an imaginary part b, where i² = −1. You cannot leave an imaginary number in a denominator, so complex division rationalizes it. The conjugate of c + di is c − di, and multiplying a complex number by its conjugate gives c² + d², a real number, because the imaginary cross-terms cancel. The Complex Number Long Division Calculator uses this to convert a complex divisor into a real one, after which each part of the quotient is a straightforward real 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 complex number division works

The calculator produces the a + bi quotient through five internal actions:

Forms the conjugate of the divisor by flipping the sign of its imaginary part.
Multiplies both numerator and divisor by that conjugate.
Expands the numerator with the FOIL rule, replacing i² with −1.
Simplifies the denominator to the real value c² + d².
Divides the real and imaginary parts of the numerator by that real denominator.

Once the denominator is real, finding each part of the quotient is ordinary division, so the answer resolves cleanly into a + bi form.

Forms the conjugate of the divisor by flipping the sign of its imaginary part.

Complex division formula

The Complex Number Long Division Calculator uses (a + bi) ÷ (c + di) = [(a + bi)(c − di)] / (c² + d²), which expands to [(ac + bd) + (bc − ad)i] / (c² + d²). The denominator c² + d² is the squared modulus of the divisor. For (3 + 2i) ÷ (1 − i), the denominator is 1² + (−1)² = 2.

Dividend

Divisor

Quotient

Remainder

100 = 7 × 14 + 2 → Identity holds

Complex number example problems

These examples show conjugate multiplication and separating real and imaginary parts.

Example 1 — (3 + 2i) ÷ (1 − i)

Multiply top and bottom by the conjugate 1 + i.
Numerator: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 1 + 5i; denominator: 1² + 1² = 2.
So (3 + 2i) ÷ (1 − i) = 1/2 + 5/2 i = 0.5 + 2.5i.

Example 2 — Divide by a real number: (4 + 6i) ÷ 2

The divisor 2 is already real, so no conjugate is needed.
Divide each part by 2: 4/2 = 2 and 6/2 = 3.
So (4 + 6i) ÷ 2 = 2 + 3i.

Example 3 — Divide by a pure imaginary: (2 + 3i) ÷ i

The conjugate of i is −i; multiply top and bottom by −i.
Numerator: (2 + 3i)(−i) = −2i − 3i² = 3 − 2i; denominator: i × −i = 1.
So (2 + 3i) ÷ i = 3 − 2i.

Multiply top and bottom by the conjugate 1 + i.
Numerator: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 1 + 5i; denominator: 1² + 1² = 2.
So (3 + 2i) ÷ (1 − i) = 1/2 + 5/2 i = 0.5 + 2.5i.

Worked complex problems

How do you divide (5 + i) by (2 + 3i)?

(5 + i) ÷ (2 + 3i) = 1 − i. Multiply top and bottom by the conjugate 2 − 3i. Numerator: (5 + i)(2 − 3i) = 10 − 15i + 2i − 3i² = 13 − 13i. Denominator: 2² + 3² = 13. Dividing gives 13/13 − 13/13 i = 1 − i.

1313

Why does the denominator become a real number?

Multiplying c + di by its conjugate c − di gives c² + d², which has no i. The cross terms +cdi and −cdi cancel, and −d²i² becomes +d² because i² = −1. This is why the conjugate is the right multiplier: it clears the imaginary part from the denominator so the quotient can be written as a + bi.

5-2i1-i

Common complex division mistakes

Complex number division produces 5 frequent errors:

Multiplying by the divisor instead of its conjugate.
Forgetting that i² = −1 when expanding the numerator.
Changing the sign of the real part instead of the imaginary part in the conjugate.
Dividing only one part of the numerator by the real denominator.
Leaving an imaginary number in the denominator instead of rationalizing it.

The Complex Number Long Division Calculator forms the correct conjugate and substitutes i² = −1 automatically, so both parts of the quotient are exact.

Multiplying by the divisor instead of its conjugate.

Forgetting that i² = −1 when expanding the numerator.

Changing the sign of the real part instead of the imaginary part in the conjugate.

Dividing only one part of the numerator by the real denominator.

Leaving an imaginary number in the denominator instead of rationalizing it.

Frequently Asked Questions

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the divisor. This makes the denominator real (c² + d²), after which you divide the real and imaginary parts separately to get an a + bi answer.

What is the conjugate of a complex number?

The conjugate of c + di is c − di — the same real part with the sign of the imaginary part flipped. Multiplying a complex number by its conjugate gives the real value c² + d².

Why do you multiply by the conjugate when dividing?

Because it removes the imaginary part from the denominator. The product (c + di)(c − di) = c² + d² is real, which lets you write the quotient in standard a + bi form.

How do you divide a complex number by a real number?

Divide each part separately. For (4 + 6i) ÷ 2, the answer is 4/2 + 6/2 i = 2 + 3i; no conjugate is needed because the divisor is already real.

How do you divide by i?

Multiply top and bottom by −i, the conjugate of i, so the denominator becomes 1. For (2 + 3i) ÷ i, the result is 3 − 2i.

What is the denominator c² + d²?

It is the squared modulus of the divisor. Since it comes from multiplying the divisor by its conjugate, it is always a non-negative real number that scales both parts of the quotient.

How do I check a complex division answer?

Multiply the quotient by the divisor; you should recover the dividend. For (3 + 2i) ÷ (1 − i) = 0.5 + 2.5i, multiplying (0.5 + 2.5i)(1 − i) returns 3 + 2i.

The relationship behind complex division

The Complex Number Long Division Calculator relies on (a + bi) ÷ (c + di) = [(a + bi)(c − di)] / (c² + d²), where the conjugate turns the denominator into the real squared modulus c² + d². As a check, quotient × divisor = dividend must hold: for (5 + i) ÷ (2 + 3i) = 1 − i, multiplying (1 − i)(2 + 3i) returns 5 + i.