How do you multiply complex numbers?

Use the distributive rule (FOIL) on (a + bi)(c + di), then replace every i squared with -1. Collecting the real and imaginary parts gives (ac - bd) + (ad + bc)i. For example, (3 + 4i)(1 + 2i) = -5 + 10i.

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator, c - di. This makes the denominator the real number c squared plus d squared, after which you divide each part. For example, (3 + 4i) / (1 + 2i) = 2.2 - 0.4i.

What is the modulus of a complex number?

The modulus, written |z|, is the distance from the origin to the point a + bi on the complex plane, calculated as the square root of (a squared plus b squared). It is the length of the vector representing the number. For example, the modulus of 3 + 4i is 5.

What is the argument of a complex number?

The argument, written arg z, is the angle the vector a + bi makes with the positive real axis, measured counter-clockwise, and found with atan2(b, a). The modulus and argument together describe the number's polar form. For example, arg(3 + 4i) is about 53.13 degrees.

Complex Numbers Calculator

Formula

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Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

What Is a Complex Number?

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined so that i = √−1 and therefore i² = −1. The value a is called the real part and b is the imaginary part. Complex numbers extend the ordinary real-number line into a two-dimensional complex plane, which is exactly what this calculator draws for you.

z = a + bi, where i = √−1 and i² = −1

Every real number is also a complex number with a zero imaginary part — for example, 7 is the same as 7 + 0i. A number with a zero real part, such as 3i, is called purely imaginary. The real part and imaginary part together pin the number to a single point (a, b) in the plane: the real part is measured along the horizontal (Re) axis and the imaginary part along the vertical (Im) axis.

Is 0 a complex number? Yes. Zero is written 0 + 0i, so both its real and imaginary parts are 0. It sits at the origin of the complex plane and is simultaneously a real number, an imaginary number, and a complex number.

Complex numbers are not just an abstraction. They are how engineers describe alternating current, how signal processing handles frequencies, and how the quadratic formula returns answers when the discriminant b² − 4ac is negative — see the quadratic formula calculator for that connection. You can explore every tool on the site from the free math calculators hub.

How to Use the Complex Numbers Calculator

This calculator works in rectangular form (a + bi). It adds, subtracts, multiplies, and divides two complex numbers, and it finds the modulus and argument of a number — plotting each result as a vector on the complex plane. To use it:

Enter z₁. Type the real part and imaginary part of your first complex number, for example 3 and 4 for 3 + 4i.
Enter z₂ (when needed). For add, subtract, multiply, and divide, type the second number. For modulus and argument you only need one number.
Choose an operation. Pick add, subtract, multiply, divide, or modulus & argument from the operation selector — or tap an example chip to load a ready-made problem.
Read the result. The answer appears in a + bi form, the complex-plane diagram shows the numbers as vectors, and the step-by-step panel walks through the working line by line.

Everything is free with no login, no paywall, and no sign-up wall on the worked steps — a real difference from gated competitors.

Complex Number Operations (with Formulas)

Here is the core of what the calculator does. Each operation below shows the general formula, a worked example, and how it looks on the plane. The table gives a quick reference you can scan first.

Operation	Formula	Example
Add	(a + bi) + (c + di) = (a + c) + (b + d)i	(3 + 4i) + (1 + 2i) = 4 + 6i
Subtract	(a + bi) − (c + di) = (a − c) + (b − d)i	(3 + 4i) − (1 + 2i) = 2 + 2i
Multiply	(a + bi)(c + di) = (ac − bd) + (ad + bc)i	(3 + 4i)(1 + 2i) = −5 + 10i
Divide	(a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)	(3 + 4i) ÷ (1 + 2i) = 2.2 − 0.4i
Modulus	\|z\| = √(a² + b²)	\|3 + 4i\| = 5
Argument	arg z = atan2(b, a)	arg(3 + 4i) ≈ 53.13°

Adding and Subtracting Complex Numbers

To add or subtract, you simply combine like parts: add the real parts together and add the imaginary parts together. Subtraction works the same way, but be careful to apply the minus sign to both parts of the second number.

(a + bi) ± (c + di) = (a ± c) + (b ± d)i

Add the real parts. For (3 + 4i) + (1 + 2i): 3 + 1 = 4.
Add the imaginary parts. 4i + 2i = 6i.
Combine. The result is 4 + 6i.

Geometrically, addition is tip-to-tail: place the vector for z₂ at the end of z₁, and the result reaches the new tip. The diagram in the calculator draws exactly this, so you can see why addition slides one vector along the other.

Multiplying Complex Numbers

Multiplication uses the distributive rule (FOIL), and the key step is remembering that i² = −1. After expanding, the i² term flips sign and merges into the real part.

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

Expand with FOIL. (3 + 4i)(1 + 2i) = 3·1 + 3·2i + 4i·1 + 4i·2i = 3 + 6i + 4i + 8i².
Replace i² with −1. 8i² becomes −8, so the expression is 3 + 6i + 4i − 8.
Collect real and imaginary parts. (3 − 8) + (6 + 4)i = −5 + 10i.

On the plane, multiplying rotates and scales: the lengths (moduli) multiply and the angles (arguments) add. That is the geometric "sense-check" the visual panel highlights.

Dividing Complex Numbers

Division is the trickiest operation because you cannot divide by an imaginary part directly. The standard method is to multiply the top and bottom by the conjugate of the denominator (c − di). This makes the denominator a real number, c² + d², because (c + di)(c − di) = c² + d².

(a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)

Multiply top and bottom by the conjugate of the denominator. For (3 + 4i) ÷ (1 + 2i), multiply both by (1 − 2i).
Expand the numerator. (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 6i + 4i + 8 = 11 − 2i.
Expand the denominator. (1 + 2i)(1 − 2i) = 1² + 2² = 5.
Divide each part. (11 − 2i) ÷ 5 = 2.2 − 0.4i.

Watch the signs: the conjugate flips only the sign of the imaginary part (c − di), and the +8 in the numerator comes from −8i² = +8. A dropped minus sign here is the most common division error.

Modulus and Argument

The modulus |z| is the distance from the origin to the point (a, b) — the length of the vector. The argument arg z is the angle that vector makes with the positive real axis, measured counter-clockwise. Together, modulus and argument describe the same number in polar form: the modulus is the "how far" and the argument is the "which direction."

|z| = √(a² + b²) arg z = atan2(b, a)

Find the modulus. For z = 3 + 4i: |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
Find the argument. arg z = atan2(4, 3) ≈ 0.927 radians ≈ 53.13°.
Read the geometry. On the diagram the arrow has length 5 and tilts about 53° above the horizontal axis.

Degrees vs radians: this calculator reports the argument so you can read it directly off the plane. Using atan2(b, a) rather than a plain arctangent places the angle in the correct quadrant for every sign of a and b, including the negative real axis. The cosine and sine that link polar and rectangular form come from trigonometry — see the trig functions calculator.

The Complex Conjugate

The conjugate of z = a + bi is written z̄ = a − bi: keep the real part, flip the sign of the imaginary part. Geometrically it is the reflection across the real axis. The conjugate is what makes division work, because multiplying a number by its conjugate clears the imaginary part: (a + bi)(a − bi) = a² + b², which equals |z|². Complex numbers also show up as matrix entries in eigenvalue problems — explore those in the matrix calculator.

Common Mistakes to Avoid

Mistake — Forgetting i² = −1When you multiply, every i·i must be replaced with −1, not left as i² or treated as +1. Skipping this step is the single most common multiplication error.

Mistake — Sign errors in subtractionThe minus sign distributes to both parts of the second number: (3 + 4i) − (1 + 2i) = (3 − 1) + (4 − 2)i, not (3 − 1) + (4 + 2)i.

Mistake — Dividing without the conjugateYou cannot divide directly by c + di. Multiply the numerator and denominator by the conjugate (c − di) first so the denominator becomes the real number c² + d².

Mistake — Mixing degrees and radiansThe argument can be given in degrees or radians; 53.13° and 0.927 rad are the same angle. Decide which unit your problem uses and stay consistent throughout.

Mistake — Using plain arctangent for the argumentA bare arctan(b/a) lands in the wrong quadrant when a is negative. Use atan2(b, a) so the angle reflects the true position of the point, as this calculator does.

Frequently Asked Questions

What is a complex number?

A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. The part a is the real part and b is the imaginary part. Complex numbers are plotted as points on the two-dimensional complex plane.
Is 0 a complex number?

Yes. Zero is written 0 + 0i, so both its real and imaginary parts are zero. It is a complex number, a real number, and an imaginary number all at once, sitting at the origin of the complex plane.
How do you multiply complex numbers?

Use the distributive rule (FOIL) on (a + bi)(c + di), then replace every i² with −1. Collecting the real and imaginary parts gives (ac − bd) + (ad + bc)i. For example, (3 + 4i)(1 + 2i) = −5 + 10i.
How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator, c − di. This makes the denominator the real number c² + d², after which you divide each part. For example, (3 + 4i) ÷ (1 + 2i) = 2.2 − 0.4i.
What is the modulus of a complex number?

The modulus, written |z|, is the distance from the origin to the point a + bi on the complex plane, calculated as √(a² + b²). It is the length of the vector representing the number. For example, |3 + 4i| = √(9 + 16) = 5.
What is the argument of a complex number?

The argument, written arg z, is the angle the vector a + bi makes with the positive real axis, measured counter-clockwise, and found with atan2(b, a). The modulus and argument together describe the number's polar form. For example, arg(3 + 4i) ≈ 53.13°.