Divide Complex Numbers Calculator
Enter the numerator and denominator as complex numbers in rectangular form (real and imaginary parts) or in polar form (modulus and angle). The calculator divides them using the conjugate-multiplication method, shows every step of the working, and gives the quotient in rectangular form, polar form, and exponential form.
Formula
Worked example
Divide (3 + 2i) by (1 - i). Conjugate of denominator is (1 + i). Multiply top and bottom: numerator = (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i^2 = 3 + 5i - 2 = 1 + 5i. Denominator = (1 - i)(1 + i) = 1 + 1 = 2. Result = (1 + 5i) / 2 = 0.5 + 2.5i. Modulus = sqrt(0.5^2 + 2.5^2) = sqrt(6.5) approx 2.55. Argument = atan2(2.5, 0.5) approx 78.69 degrees.
How to divide complex numbers in rectangular form
To divide two complex numbers in rectangular form, you cannot simply divide real and imaginary parts separately. Instead, you multiply both the numerator and denominator by the complex conjugate of the denominator. The conjugate of c + di is c - di (the sign of the imaginary part is flipped). Multiplying the denominator by its own conjugate turns it into a real number, c^2 + d^2, because the cross terms cancel and i^2 = -1. The numerator then expands by FOIL: (a + bi)(c - di) = (ac + bd) + (bc - ad)i. Dividing each part of the numerator by the real denominator gives the final rectangular result: real part = (ac + bd) / (c^2 + d^2), imaginary part = (bc - ad) / (c^2 + d^2). This calculator performs every step automatically and shows the full working in the "Show your work" panel.
Dividing complex numbers in polar form
Division becomes even simpler in polar form. If z1 = r1(cos theta1 + i sin theta1) and z2 = r2(cos theta2 + i sin theta2), then z1 / z2 has modulus r1 / r2 and argument theta1 - theta2. In other words, you divide the magnitudes and subtract the angles. This is one reason polar form is preferred in electrical engineering and signal processing, where impedances and phasors are routinely divided. This calculator accepts polar input directly: enter the modulus and angle (in degrees) for each number, and the result is returned in both polar and rectangular form. Internally, polar inputs are converted to rectangular using a = r cos(theta) and b = r sin(theta) before the conjugate-multiplication method is applied, so the rectangular steps remain exact regardless of input form.
What the modulus and argument of the quotient mean
The modulus of z1 / z2 equals |z1| divided by |z2|, regardless of which form you use. It tells you the magnitude of the quotient in the complex plane. The argument is the angle from the positive real axis, measured counter-clockwise, and it equals arg(z1) minus arg(z2). A purely real quotient (imaginary part = 0) has an argument of 0 or 180 degrees. A purely imaginary quotient (real part = 0) has an argument of 90 or -90 degrees. This calculator reports the principal argument in (-180, 180] degrees using atan2(imaginary, real), which handles all four quadrants correctly. The exponential form r * e^(i * theta) is equivalent to the polar form and is common in advanced mathematics and physics.
When does division by a complex number fail?
Division is undefined when the denominator is zero, meaning both real and imaginary parts are zero (c = 0 and d = 0). This mirrors real-number division by zero. In all other cases, even if the denominator is purely real or purely imaginary, division is well-defined. If the denominator is a pure real number k, the result is simply (a/k) + (b/k)i, and the conjugate step reduces to dividing by k^2 = k * k. If the denominator is the imaginary unit i (c = 0, d = 1), multiplying by the conjugate -i gives (a + bi) / i = (b - ai) / 1 = b - ai. A common error is dividing real and imaginary parts separately, which gives a wrong answer except in the trivial case where the denominator is real.
Complex number division: key formulas
| Form | Formula | Notes |
|---|---|---|
| Rectangular | (a+bi)/(c+di) = [(ac+bd) + (bc-ad)i] / (c^2+d^2) | Multiply by conjugate of denominator |
| Polar | (r1 angle theta1) / (r2 angle theta2) = (r1/r2) angle (theta1-theta2) | Divide magnitudes, subtract angles |
| Exponential | r1*e^(i*theta1) / r2*e^(i*theta2) = (r1/r2)*e^(i*(theta1-theta2)) | Same rule in exponential notation |
| Modulus | |z1/z2| = |z1| / |z2| | Modulus of quotient = ratio of moduli |
| Argument | arg(z1/z2) = arg(z1) - arg(z2) | Argument of quotient = difference of arguments |
Reference formulas for dividing complex numbers in rectangular and polar form.
Frequently asked questions
How do you divide two complex numbers step by step?
Write the division as (a + bi) / (c + di). Multiply the top and bottom by the conjugate of the denominator, which is (c - di). The denominator becomes c^2 + d^2, a real number. The numerator expands to (ac + bd) + (bc - ad)i. Divide each part by c^2 + d^2 to get the real and imaginary parts of the answer.
Why do you multiply by the conjugate when dividing complex numbers?
Multiplying a complex number by its conjugate always produces a real number: (c + di)(c - di) = c^2 + d^2. This removes the imaginary part from the denominator, so you can divide a complex numerator by a plain real number, which is straightforward. Without this trick, the denominator stays complex and you cannot separate real and imaginary parts cleanly.
What is the conjugate of a complex number?
The conjugate of a + bi is a - bi: you keep the real part unchanged and negate the imaginary part. For example, the conjugate of 3 - 4i is 3 + 4i. Multiplying a number by its conjugate gives the square of its modulus: (a + bi)(a - bi) = a^2 + b^2.
Is dividing complex numbers in polar form easier?
Yes, significantly. In polar form, z1 / z2 = (r1 / r2) at angle (theta1 - theta2). You simply divide the two magnitudes and subtract the two angles. There is no conjugate multiplication or FOIL expansion needed. Polar form is preferred for division (and multiplication and powers) for this reason, especially in electrical engineering.
What does it mean if the quotient is a real number?
If the imaginary part of the quotient is exactly zero, the result is a real number. This happens when bc - ad = 0, that is, when the two complex numbers have the same argument (they point in the same direction in the complex plane) or when their arguments differ by exactly 180 degrees.
What is the modulus of the quotient of two complex numbers?
The modulus of z1 / z2 equals |z1| divided by |z2|. For example, if |z1| = 6 and |z2| = 2, then |z1 / z2| = 3. This is a direct consequence of the polar-form rule: divide magnitudes and subtract angles.
Can I divide a complex number by a real number?
Yes. A real number k can be treated as the complex number k + 0i. Its conjugate is itself, and the denominator k^2 + 0 = k^2. The result is simply (a/k) + (b/k)i. Equivalently, you can just divide both the real and imaginary parts by k directly.
What happens when you divide by the imaginary unit i?
To divide (a + bi) by i, multiply top and bottom by -i (the conjugate of i): (a + bi)(-i) / (i)(-i) = (-ai - bi^2) / 1 = b - ai. So 1/i = -i, and (a + bi)/i = b - ai. The real and imaginary parts swap and the new real part gets negated.