Complex Number Calculator

Free Online Complex Number Calculator

This complex number calculator solves any operation on complex numbers instantly — addition, subtraction, multiplication, division, modulus, argument, conjugate, powers, and square roots — and shows the complete step-by-step solution alongside a labeled Argand-plane diagram. Type in any complex numbers in the form a + bi, choose an operation, and the calculator substitutes your exact values into the correct formula, works through every line of arithmetic, and plots the result visually so nothing is left as a hidden black box.

Whether you're a student working through algebra or electrical engineering homework, a teacher preparing worked examples, or anyone who simply needs a fast and accurate complex number solver, this tool handles any equation you enter — positive, negative, or decimal real and imaginary parts — and always shows its full working.

What Is a Complex Number? (Definition and Meaning)

A complex number is a number written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = −1. Complex numbers extend the real number line into a two-dimensional plane called the complex plane (or Argand plane), where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Every complex number a + bi corresponds to exactly one point (a, b) on this plane, and can also be represented as a vector pointing from the origin to that point. This graphical view is what makes operations like addition, multiplication, and powers of complex numbers so intuitive once you can see them plotted.

Complex Number Addition and Subtraction Formula

Adding or subtracting complex numbers works exactly like combining like terms in algebra — real parts combine with real parts, and imaginary parts combine with imaginary parts.

• Addition formula: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction formula: (a + bi) − (c + di) = (a − c) + (b − d)i
• Example: (3 + 4i) + (1 − 2i) = (3 + 1) + (4 − 2)i = 4 + 2i

Complex Number Multiplication Formula

Multiplying two complex numbers uses the same distributive expansion (FOIL) as multiplying two binomials, followed by simplifying using the key identity i² = −1.

• Formula: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
• The real part of the product is ac − bd; the imaginary part is ad + bc
• Example: (2 + 3i)(4 − i) = (2×4 − 3×(−1)) + (2×(−1) + 3×4)i = (8 + 3) + (−2 + 12)i = 11 + 10i

Complex Number Division Formula

Dividing complex numbers requires multiplying the numerator and denominator by the conjugate of the denominator, which eliminates i from the denominator and leaves a real number to divide by.

• Formula: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)
• The conjugate of c + di is c − di — flip only the sign of the imaginary part
• Example: (5 + 2i) ÷ (1 + i) = [(5×1 + 2×1) + (2×1 − 5×1)i] / (1² + 1²) = (7 − 3i) / 2 = 3.5 − 1.5i

Modulus and Argument Formula

The modulus (or absolute value) of a complex number is its distance from the origin on the Argand plane, and the argument is the angle its vector makes with the positive real axis.

• Modulus formula: |z| = √(a² + b²)
• Argument formula: θ = atan2(b, a), usually expressed in degrees or radians
• Example: for z = 3 + 4i, |z| = √(9 + 16) = √25 = 5, and θ = atan2(4, 3) ≈ 53.13°

Polar Form and De Moivre's Theorem

Every complex number a + bi can also be written in polar form as r(cos θ + i sin θ), where r is the modulus and θ is the argument. Polar form makes powers and roots of complex numbers dramatically simpler through De Moivre's Theorem.

• De Moivre's Theorem: zⁿ = rⁿ (cos(nθ) + i sin(nθ))
• To raise a complex number to a power, raise its modulus to that power and multiply its argument by that power
• Example: (1 + i)⁴ — first convert to polar (r = √2, θ = 45°), then apply the theorem: (√2)⁴(cos 180° + i sin 180°) = 4(−1 + 0i) = −4

Complex Conjugate and Square Root

The conjugate of a complex number simply flips the sign of the imaginary part, and is essential for division and for finding the modulus. The square root of a complex number uses a direct formula derived from its modulus.

• Conjugate formula: conjugate(a + bi) = a − bi
• Square root formula: √(a + bi) = √((r + a)/2) + sign(b)·√((r − a)/2)·i, where r = |a + bi|
• Example: the conjugate of 3 + 4i is 3 − 4i, and multiplying them gives the real number 25 (which is |z|²)

How to Find Any Complex Number Operation Step by Step

No matter which operation you need, complex number arithmetic always follows the same general sequence:

• Step 1 — Write both numbers in standard form a + bi, identifying the real and imaginary parts.
• Step 2 — Choose the correct formula for the operation: addition/subtraction combine like parts, multiplication expands like binomials, division uses the conjugate trick.
• Step 3 — Substitute your exact real and imaginary values into the formula.
• Step 4 — Carry out the arithmetic, remembering that i² = −1 wherever it appears.
• Step 5 — Combine the results into a final answer in the form a + bi (or convert to polar form if needed).
• Step 6 — Plot the numbers on the Argand plane to visually confirm the result makes sense.

How to Use This Complex Number Calculator

Choose an operation from the dropdown — add, subtract, multiply, divide, modulus & argument, conjugate, power, or square root — then type in the real and imaginary parts of z1 (and z2, when the operation needs a second number). As soon as you enter values, the calculator instantly computes the result in both rectangular (a + bi) and polar (r ∠ θ) form. The Argand-plane diagram below updates live, plotting z1, z2, and the result as vectors from the origin, each labeled directly with its value so the whole calculation is visible at a glance.

Scroll down further for the complete written step-by-step solution, which mirrors exactly what you'd write by hand — formula first, then your real values substituted in, then the arithmetic worked through to the final answer.

Worked Examples

Addition example: (3 + 4i) + (1 − 2i) = (3+1) + (4−2)i = 4 + 2i.

Multiplication example: (2 + 3i)(4 − i) = (8 + 3) + (−2 + 12)i = 11 + 10i.

Division example: (5 + 2i) ÷ (1 + i) = (7 − 3i) / 2 = 3.5 − 1.5i.

Modulus example: |3 + 4i| = √(9 + 16) = 5 — a classic 3-4-5 right triangle on the complex plane.

Key Properties of Complex Numbers

A handful of properties make complex number arithmetic easier to reason about and can help you sanity-check any calculation:

• i² = −1, i³ = −i, and i⁴ = 1 — powers of i cycle every 4 steps.
• Multiplying a complex number by its own conjugate always produces a real number: (a+bi)(a−bi) = a² + b².
• The modulus of a product equals the product of the moduli: |z1·z2| = |z1|·|z2|.
• The argument of a product equals the sum of the arguments: arg(z1·z2) = arg(z1) + arg(z2).
• Every non-zero complex number has exactly two square roots, equal in modulus but opposite in sign.
• A complex number equals zero only when both its real and imaginary parts are zero.

Real-World Applications of Complex Numbers

Complex numbers show up throughout engineering, physics, and applied mathematics wherever rotation, oscillation, or two-dimensional quantities are involved:

• Electrical engineering — representing AC circuit impedance, voltage, and current using phasors.
• Signal processing — Fourier transforms and filter design rely entirely on complex arithmetic.
• Control systems — analyzing system stability using poles and zeros on the complex plane.
• Quantum mechanics — wavefunctions are inherently complex-valued.
• Computer graphics and robotics — 2D rotations can be represented compactly using complex number multiplication.

Frequently Asked Questions

What is i in complex numbers? i is the imaginary unit, defined so that i² = −1. It lets us take the square root of negative numbers, which is impossible using real numbers alone.

How do you divide complex numbers? Multiply the numerator and denominator by the conjugate of the denominator, which turns the denominator into a real number, then divide both parts by that real number.

What is the modulus of a complex number? The modulus |z| = √(a² + b²) is the distance from the origin to the point (a, b) on the complex plane — essentially the 'size' of the complex number.

What is De Moivre's Theorem used for? It provides a fast way to raise a complex number to any power (or find its roots) by converting to polar form, raising the modulus to that power, and multiplying the argument by that power.

Can this calculator handle negative or decimal real and imaginary parts? Yes — every field accepts positive numbers, negative numbers, and decimals, and both the diagram and the step-by-step solution update automatically for whatever values you enter.