Complex Number Calculator — +-×÷, Conjugate, Modulus, Argument, Roots, Euler Form

Real-world use cases

• Verify a complex-analysis homework answer before turning it in

  You worked out (1 + i)^4 by hand using the binomial expansion and got -4. Type "1+i" into z₁, pick "z₁^n (integer power)" and set n = 4. The calculator returns -4 + 0i in Cartesian, 4∠180° in polar, and 4·e^(i·180°) in Euler form — all three confirm your answer. If you got something different, switch n to 2 and 3 to see the intermediate (1+i)² = 2i, (1+i)³ = -2+2i and pinpoint which step the sign flipped in your hand work.

• See the n-th roots of unity as a regular polygon

  Your textbook claims the sixth roots of unity sit at the vertices of a regular hexagon centered at the origin. Type "1" into z₁, pick "z₁^(1/n) — all n roots" with n = 6, and the calculator returns six numbers: 1, 0.5+0.866i, -0.5+0.866i, -1, -0.5-0.866i, 0.5-0.866i. The complex-plane plot draws them as six diamonds equally spaced 60° apart on the unit circle — connect them in your head and you see the hexagon. Try n = 3, 4, 5, 8 to watch the polygon change.

• Demonstrate Euler's identity to a friend in 10 seconds

  The famous e^(iπ) + 1 = 0 unifies five fundamental constants (e, i, π, 1, 0) in one equation. Click the "Euler's identity" preset — z₁ loads as 0 + 3.141592653589793i, operation = e^z₁ — and the result panel shows -1 + 1.22e-16i (the imaginary part is unavoidable floating-point dust). The "Euler's identity" label appears under the result because the calculator recognizes the pattern. Add 1 mentally and you get 0 — that's why this is many mathematicians' favorite equation.

• Multiply by i to visualize a 90° rotation

  Set z₁ = 3+4i, z₂ = 0+i (just i), and operation = z₁ × z₂. The result is -4 + 3i. Look at the complex-plane plot: z₁ is the vector to (3, 4), and the result is the vector to (-4, 3) — rotated exactly 90° counterclockwise. This is one of the single most useful intuitions in complex analysis: multiplying by i is rotation by 90°, multiplying by e^(iθ) is rotation by θ. Try z₂ = 0.707+0.707i (which is e^(iπ/4)) to see a 45° rotation.

• Convert between Cartesian, polar, and Euler representations

  A signal-processing problem gives you 5∠53.13° and asks for the Cartesian form. Switch input mode to "Polar (r∠θ)", set angle unit to degrees, type "5∠53.13" into z₁ and operation = z̄₁ (conjugate). The result panel shows 3 - 4i in Cartesian (so the original was 3+4i — confirming 3-4-5 right triangle), 5∠-53.13° in polar, and 5·e^(i·-53.13°) in Euler. One look gives you all three forms; no need to mentally compute r·cos(θ) and r·sin(θ) yourself.

• Check that the conjugate-product identity holds for any z

  The identity z · z̄ = |z|² is supposed to hold for every complex number. Pick a random one, say z₁ = 3+4i and z₂ = 3-4i, then multiply: result is 25 + 0i. The modulus of 3+4i is 5, and 5² = 25. Try again with z₁ = -2+5i and z₂ = -2-5i: result is 29 + 0i, and indeed |(-2)+5i|² = 4+25 = 29. The complex-plane plot makes the geometry vivid: z and z̄ are mirror images across the real axis, and their product always lands on the positive real axis at distance |z|².

Common pitfalls

• Treating arg(z) as a continuous function and writing arg(-1) = -π. The principal value is in (-π, π], so the negative real axis is +π by convention. If you cross the branch cut, the argument jumps by 2π — this matters when chaining ln(z) with other operations.

• Computing the n-th root and reporting only the principal value. Every non-zero complex number has exactly n n-th roots, and they are all equally legitimate. The calculator gives all n; in a proof or coursework, name which root you mean (e.g. "the principal cube root of -1 is e^(iπ/3) = 0.5 + 0.866i", not just "-1^(1/3) = -1").

• Believing |sin(z)| ≤ 1 in the complex plane. That bound only holds on the real axis. As soon as z has any imaginary part, sin(z) involves cosh(y) which grows without bound — so sin(i) ≈ 1.175i is correct, not a bug.

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