Plot complex numbers on the Argand plane. Perform arithmetic, convert between rectangular and polar, and visualize operations geometrically.
A complex number a + bi is a point on a plane: real parts run horizontally, imaginary parts vertically. This tool plots the point, shows its position vector, and converts between rectangular (a + bi) and polar (r at angle theta) forms. Arithmetic works geometrically: adding joins vectors tip to tail, multiplying rotates and scales. The surprise for most users: multiplying two complex numbers multiplies their distances from the origin and adds their angles, so (1 + i) squared lands on the imaginary axis even though it looks diagonal. Angles repeat every 360 degrees, so the tool shows one principal value, which may differ from textbook convention.
A concrete input and expected output from the current implementation.
Input
Multiply (1 + i) by itself: (1 + i) x (1 + i)
Expected output
2i, plotted as the point (0, 2) on the imaginary axis. Polar check: the input has modulus sqrt(2) (about 1.414) at 45 degrees; the result has modulus 2 (sqrt(2) x sqrt(2)) at 90 degrees (45 + 45).
Multiplication multiplies moduli and adds arguments, so two 45-degree directions combine into 90 degrees. The square lands straight up the imaginary axis, which is why (1 + i) squared equals 2i rather than a point on the diagonal.
A complex number is a point: in rectangular form a + bi, a fixes the horizontal position, b the vertical. To convert to polar, the modulus r = sqrt(a2 + b2) and the argument theta = atan2(b, a) are computed; atan2 keeps the angle's sign correct in all four quadrants. Converting back, a = r cos(theta) and b = r sin(theta).
Addition is tip-to-tail vector addition: translate one number's vector along the other, and the plot draws the parallelogram. Multiplication and division are the interesting operations. Multiplying (r1, theta1) by (r2, theta2) places the result at distance r1 x r2 and angle theta1 + theta2; dividing reverses both: distances divide, angles subtract. Conjugation, when offered, mirrors the point across the real axis.
Both numbers describe the same direction: adding or subtracting 360 degrees leaves the plotted point unchanged. The tool reports the principal value, normally between -180 and +180 degrees, while textbooks often add whole turns to keep angles positive. The point on the plot is identical either way; only the reported number differs.
No. This tool plots individual numbers and the results of arithmetic between them, not solution sets. The equation z^2 = 1 has the solutions z = 1 and z = -1, and you can plot those two points by entering each one, but the tool will not trace every point satisfying an equation. For loci, curves, and mappings of whole regions, use a dedicated complex function plotter.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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