First value
Edit two rectangular forms, watch modulus and argument react, then tap an operation. The canvas keeps both phasors on the same scale so large homework values do not shove tiny residuals off-screen.
Tap an operation chip. The plane already tracks both inputs.
Red marks z₁, blue marks z₂. Axes rescale from your largest coordinate so both stay visible.
Each field holds a real scalar. Pairing Re with Im rebuilds a point in ℝ², which we treat as the complex plane with the usual identification (a,b) ↦ a+bi.
Telemetry recomputes √(a²+b²) and atan2(b,a) on every keystroke. Those two numbers are the polar backbone most students meet after rectangular form.
Binary chips reuse the same pair of rectangles. Addition and subtraction add or subtract components. Multiplication follows the distributive rule with i² ↦ −1. Division multiplies numerator and denominator by the conjugate of the bottom value so the denominator becomes a nonnegative real. If that real is zero, we stop and show an explicit block message instead of printing NaN soup.
Homework checks. Type the coefficients from a sheet, hit multiply or divide, compare against your hand expansion. The monospace block is easy to screenshot or copy by hand.
Intuition for phasors. Electrical engineering sketches often jump straight to r∠θ. Loading 0.95+0.31i beside 0.6+0.8i, then opening Argument, ties the picture on the canvas to the numeric radians readout.
Preparation for higher dimensions. Once rectangular plus polar feel automatic, jumping to quaternion multiplication hurts less because you already separate magnitude from orientation.
Argument uses atan2, which lands in (−π, π]. If you need a continuous phase unwrap across frequency sweeps, export the radians into a spreadsheet or CAS. This page will not stitch 2π jumps for you.
Floating-point decimals are binary. Patterns like 0.1+0.2 in ordinary JavaScript leak into these formulas. For exam-grade rational results, prefer exact fractions elsewhere, then spot-check here.
Nothing leaves your tab. We recommend rounding mentally to the significant digits your instructor expects. The UI shows six decimals to reduce misleading trailing zeros, not to imply laboratory accuracy.
Need pure radius-angle conversion without a second operand? The polar coordinate converter focuses on (r,θ) ↔ (x,y) pairs. For stacked linear algebra drills, pair this page with the vector calculator when you want ℝ² dot products without invoking i.
We like keeping the canvas tied to both inputs because many mistakes come from plotting only the answer. Watching z₁ and z₂ simultaneously exposes accidental sign flips before you commit ink.
Complex arithmetic punishes tiny notation drift. Below are the repeats we see after grading drafts and office hours.
Representing multiplication as a 2×2 real matrix is another sanity check. Treat z=a+bi as [[a,-b],[b,a]] acting on the column (x,y). Multiplying two such matrices mirrors multiplying the underlying complex numbers. When your handwritten matrix product disagrees with the chip output, the bug is usually a sign on one imaginary entry, not the JavaScript layer. The matrix calculator helps when teachers insist you show the linear map explicitly.
Finally, remember the tool visualizes two static points. Dynamic systems, contour integrals, or conformal maps need a grapher with a parameter, not this page. The mathematical function grapher is the better bridge when you must watch a curve evolve.
Short answers about polar form, division, and what runs locally.
atan2 returns the principal value between −π and π. Crossing that slit moves the angle by 2π in the geometric picture even though the numeric readout snaps to the equivalent principal branch.
Division always means z₁ divided by z₂. If you need the reciprocal of z₁ alone, put 1+0i into z₂ or swap the fields manually.
The polar chip prints r·e^(iθ) with θ in radians. Cosine plus i sine is equivalent, we simply choose the compact exponential string.
No. Parsing, arithmetic, drawing, and telemetry run entirely inside your browser session.
Six digits balance readability with typical homework tolerances. Treat trailing digits as computational noise unless your course states otherwise.
Set Re to 0 and Im to your coefficient, or load the “i vs 1” sample pair and edit from there.
