Complex Number Calculator

How to Use This Calculator

1. Enter z1 = a + bi and z2, choose operation — result appears instantly.
2. Use Modulus & Argument tab for polar form r*e^(i*theta).
3. Use Powers tab for De Moivre z^n.
4. Professional shows complex exponential and ln(z).

Formula

Example

Frequently Asked Questions

• How do you multiply two complex numbers? ▼
  To multiply (a + bi)(c + di), use the FOIL method: ac + adi + bci + bdi². Since i² = −1, the last term becomes −bd. Collecting real and imaginary parts: (ac − bd) + (ad + bc)i. Example: (3 + 4i)(2 − i) = 3·2 + 3·(−i) + 4i·2 + 4i·(−i) = 6 − 3i + 8i − 4i² = 6 + 5i − 4(−1) = 6 + 5i + 4 = 10 + 5i. In polar form this is much simpler: multiply the moduli and add the arguments: r₁r₂ at angle (θ₁ + θ₂). A common pitfall is forgetting i² = −1 and leaving the bd term positive, which gives the wrong real part.
• How do you divide complex numbers? ▼
  To divide (a + bi)/(c + di), multiply both numerator and denominator by the complex conjugate of the denominator (c − di). This eliminates the imaginary part in the denominator: (a + bi)(c − di) / [(c + di)(c − di)] = [(ac + bd) + (bc − ad)i] / (c² + d²). Example: (3 + 4i)/(1 + 2i). Conjugate: 1 − 2i. Numerator: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i. Denominator: 1² + 2² = 5. Result: (11 − 2i)/5 = 2.2 − 0.4i. Common pitfall: multiplying by the wrong conjugate (using c + di instead of c − di), which leaves an imaginary denominator.
• What is the modulus of a complex number? ▼
  The modulus (absolute value) of a complex number z = a + bi is |z| = √(a² + b²). It represents the distance from the origin to the point (a, b) in the complex plane, equivalent to the magnitude of a 2D vector. Example: z = 3 + 4i. |z| = √(9 + 16) = √25 = 5. Key properties: |z₁z₂| = |z₁||z₂| (modulus of a product equals the product of moduli); |z₁/z₂| = |z₁|/|z₂|; |z̄| = |z| (conjugate has the same modulus). The argument θ = atan2(b, a) gives the angle, so z can be written as |z|·(cos θ + i·sin θ) = |z|·e^(iθ). Common pitfall: |a + bi| is not |a| + |b| — squaring and taking the root is required.
• What is De Moivre's theorem? ▼
  De Moivre's theorem states that (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any integer n. In polar form, if z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos(nθ) + i sin(nθ)). Example: compute (1 + i)⁶. First convert: |1 + i| = √2, θ = 45°. So (1 + i)⁶ = (√2)⁶ (cos 270° + i sin 270°) = 8(0 − i) = −8i. This theorem also finds nth roots: the n distinct nth roots of z are rⁿ at angles (θ + 360°k)/n for k = 0, 1, …, n−1. Common pitfall: applying the theorem directly in rectangular form — always convert to polar (r, θ) form first to use De Moivre efficiently.
• What is Euler's identity? ▼
  Euler's identity states e^(iπ) + 1 = 0. It is a special case of Euler's formula e^(ix) = cos(x) + i·sin(x), evaluated at x = π: e^(iπ) = cos(π) + i·sin(π) = −1 + 0 = −1, so e^(iπ) + 1 = 0. The formula e^(ix) = cos(x) + i·sin(x) is proved via Taylor series — expanding both sides shows they match term by term. Example: e^(iπ/2) = cos(90°) + i·sin(90°) = 0 + i = i, so the imaginary unit i is literally a 90° rotation in the complex plane. Euler's formula connects five fundamental constants (e, i, π, 1, 0) and is used throughout signal processing, quantum mechanics, and electrical engineering to represent oscillations compactly as e^(iωt).

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