Complex numbers (quick, friendly definition)

A complex number is a number that has two parts: a real part and an imaginary part. We usually write it like this: z = a + bi, where a is the real part, b is the imaginary part, and i is the special unit with i² = −1.

Complex numbers aren’t “fake” or “less real” — they’re one of the most useful mathematical tools ever invented. They show up in electrical engineering (AC circuits), signal processing (Fourier transforms), physics (waves and quantum mechanics), control systems, computer graphics, and even in how we describe rotations.

• They complete algebra. Every polynomial has roots in the complex numbers (that’s a huge deal).
• They encode 2D information. A complex number can represent a point (x, y) or a 2D vector.
• They make rotations easy. Multiplying by a complex number can rotate and scale another one.
• They simplify trig. Euler’s formula links exponentials to sin/cos, which powers lots of engineering math.

Instant complex-number operations (no sign mistakes)

This Complex Number Calculator handles the most common tasks students and engineers need: addition, subtraction, multiplication, division, magnitude (modulus), argument (angle), conjugate, rectangular ↔ polar conversion, integer powers, and roots. It formats results cleanly (like 3 − 2i) and shows helpful “extras” such as magnitude and angle.

• Homework checks and exam practice (especially division and polar form).
• Electrical engineering and AC circuit phasors.
• Signal processing / Fourier math (complex exponentials).
• Quick conversions between a + bi and r∠θ.

Pro tip: If you keep getting the sign wrong in division (it’s the most common error), this tool will save you time and frustration.

All the key complex number formulas (explained)

Let z₁ = a + bi and z₂ = c + di. Here are the standard operations. (You’ll see these everywhere, so it’s worth getting comfortable with them.)

You add/subtract complex numbers by combining like parts: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. Think: real with real, imaginary with imaginary.

Multiply using distributive property (FOIL), then simplify with i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The key step is remembering that bi · di = bd i² = −bd.

Division is where most people slip. The trick is: multiply top and bottom by the conjugate of the denominator: \( \frac{a+bi}{c+di} \cdot \frac{c-di}{c-di} \). This turns the denominator into a real number because: (c + di)(c − di) = c² + d². The final result is: \(\frac{a+bi}{c+di} = \frac{(ac + bd) + (bc − ad)i}{c² + d²}\).

The conjugate of z = a + bi is \(\bar{z} = a − bi\). It “flips” the sign of the imaginary part. Conjugates are used in division, magnitude calculations, and simplifying expressions.

The magnitude is the distance from the origin in the complex plane: |z| = \(\sqrt{a² + b²}\). If you treat the complex number as a point (a, b), this is just the Pythagorean theorem.

The argument is the angle a complex number makes with the positive real axis: arg(z) = \(\text{atan2}(b, a)\). We use atan2 (instead of plain arctan) so the quadrant is correct when a or b is negative.

You can write a complex number as z = r(\cos θ + i\sin θ), where r = |z| and θ = arg(z). Engineers often write it as r∠θ (phasor notation).

De Moivre’s theorem makes powers and roots clean in polar form: (r(\cos θ + i\sin θ))ⁿ = rⁿ(\cos(nθ) + i\sin(nθ)). For n-th roots: \(\sqrt[n]{r}\, (\cos\frac{θ+2kπ}{n} + i\sin\frac{θ+2kπ}{n})\), where k = 0, 1, …, n−1.

Shortcut: multiplication/division is often easier in polar form (multiply magnitudes, add angles; divide magnitudes, subtract angles).

Examples you can copy (and test in the calculator)

Add (3 + 2i) + (−5 + 7i). Real parts: 3 + (−5) = −2. Imaginary parts: 2 + 7 = 9. So the result is −2 + 9i.

Multiply (2 − 3i)(4 + i). Use the formula: (ac − bd) + (ad + bc)i. Here a=2, b=−3, c=4, d=1: ac − bd = 8 − (−3)·1 = 11, ad + bc = 2·1 + (−3)·4 = 2 − 12 = −10. Result: 11 − 10i.

Divide (5 + i) ÷ (2 − 3i). Multiply by conjugate: (2 + 3i). Numerator: (5 + i)(2 + 3i) = 10 + 15i + 2i + 3i² = 10 + 17i − 3 = 7 + 17i. Denominator: (2 − 3i)(2 + 3i) = 4 + 9 = 13. Result: \(\frac{7}{13} + \frac{17}{13}i\) ≈ 0.5385 + 1.3077i.

For z = −3 + 3i: |z| = √(9 + 9) = √18 ≈ 4.2426. arg(z) = atan2(3, −3) = 135° (or 3π/4). So in polar: 4.2426 ∠ 135°.

Multiplying by i rotates a point 90° counterclockwise. If z = a + bi, then i·z = i(a + bi) = ai + b i² = −b + ai. That transforms (a, b) into (−b, a) — a perfect 90° rotation. That’s why complex numbers are secretly “rotation machines.”

What happens when you click Calculate

This page runs entirely in your browser (fast + private). When you click Calculate:

• We validate inputs (so blank fields don’t quietly become zero).
• We compute the selected operation using standard formulas (shown above).
• We format the result into friendly math text: a + bi and optional polar form.
• We also compute extras like conjugate, magnitude, and angle.
• You can save your results locally on your device (optional).

Because everything is local, it’s great for quick checks and screenshots. Share buttons generate a clean summary text you can paste into WhatsApp, Telegram, or social media.

• Forgetting i² = −1 during multiplication.
• Wrong conjugate sign when dividing.
• Wrong quadrant for the angle (atan2 fixes that).
• Sign formatting like “a + −bi” (this tool formats it cleanly).

Frequently Asked Questions

• Is the imaginary number i “real”?

  It’s a mathematical object defined by i² = −1. You can’t place i on a standard number line, but complex numbers are completely consistent and incredibly useful. Engineering and physics rely on them daily.

• What’s the difference between modulus and magnitude?

  None — they’re the same concept. Some textbooks say “modulus” |z|, others say “magnitude.” Both mean √(a² + b²).

• Why does multiplying by the conjugate help in division?

  Because (c + di)(c − di) cancels the imaginary part: you get c² + d², a purely real number. That makes it easy to split the final result into a real part + imaginary part.

• Should I use degrees or radians for the angle?

  Both are valid. This calculator shows degrees by default (because it’s easier to read), and also includes radians in the details so you can use whichever your course or work requires.

• Can a complex number be zero?

  Yes — 0 + 0i is the zero complex number. Its magnitude is 0 and its angle is undefined. (The calculator will display the angle as 0 by convention.)

• Does this replace a graphing calculator?

  It’s great for quick operations, checks, and clean formatting. For advanced plotting or symbolic algebra, a full CAS or graphing tool is still useful — but this is faster for everyday complex-number math.