Polar Form Calculator - a + bi to r cis theta Solver

A polar form calculator turns a complex number a + bi into the same number written as r (cos theta + i sin theta), where r is the modulus and theta is the argument measured counterclockwise from the positive real axis. It applies r = sqrt(a^2 + b^2) and theta = atan2(b, a), so the modulus, the argument in degrees and radians, the cis shorthand, and the quadrant are returned. Use it whenever a problem hands you a complex number in Cartesian form and expects the modulus-argument answer.

• • Convert a complex number to polar form: A number like 3 + 4i becomes r = 5, theta = 53.13 degrees, and 5 cis(53.13 degrees), the standard 3-4-5 right triangle in polar clothing.
• • Read a phasor magnitude and phase: Given the rectangular components of an AC phasor, the calculator returns r and the phase angle theta for a phasor sum or impedance calculation.
• • Compute the modulus and argument: When a problem only needs the magnitude or the angle, the result panel surfaces |z| = r and arg(z) = theta in degrees and radians.

The complex number a + bi and the polar pair (r, theta) describe the same point, just with a different set of rulers. The rectangular form gives the signed coordinates on the axes, while the polar form gives the distance from the origin and the angle from a reference direction. The conversion is exact.

When the same complex number also needs to be added, subtracted, multiplied, or divided, Complex Number Calculator accepts the same a + bi input and returns the result in both rectangular and polar form so the chain stays in one tool.

The calculator takes the real part a and the imaginary part b, builds the modulus r with the Pythagorean theorem, and recovers the argument with atan2(b, a). It multiplies the radian result by 180/pi to get the degree value, picks a quadrant label from the signs of a and b, and assembles both the trigonometric polar form r (cos theta + i sin theta) and the cis shorthand r cis(theta) so the answer can be read at a glance.

• a: Real part of z = a + bi. Positive values are right of the imaginary axis, negative values left.
• b: Imaginary part of z = a + bi, the coefficient of i. Positive values are above the real axis, negative values below.
• r (modulus): Distance from the origin to a + bi, equal to |z|. Always non-negative; computed as the Pythagorean length of a and b.
• theta (argument): Angle from the positive real axis to a + bi, measured counterclockwise. In degrees the result lives in (-180 degrees, 180 degrees]; in radians it lives in (-pi, pi].

Math.atan2 is the standard two-argument inverse tangent in JavaScript, Python, and spreadsheets. Its return value lives in (-pi, pi] (about -180 degrees to 180 degrees), and the four sign combinations of (a, b) place the argument in quadrant I, II, III, or IV.

The same form is described on Wolfram MathWorld: the polar form of a complex number z = a + bi is z = r (cos(theta) + i sin(theta)) where r = sqrt(a^2 + b^2) is the modulus and theta = atan2(b, a) is the angle measured counterclockwise from the positive real axis.

The cis shorthand is the same expression written in one line, and the Wikipedia "Complex number" article makes the equivalence explicit: a complex number z = a + bi can be written in polar form as z = r (cos(theta) + i sin(theta)) = r cis(theta), with r = sqrt(a^2 + b^2) and theta = atan2(b, a) restricted to the half-open interval (-pi, pi].

The argument half of this conversion is exactly what Arcus Tangent Calculator does for a single tangent value, with the same atan2-based quadrant logic on its arctan2 tab.

Four ideas cover every polar form conversion, from a homework problem to a phasor diagram.

The modulus and the argument are two halves of the same point. Knowing r and theta is enough to recover a = r cos(theta) and b = r sin(theta); knowing a and b is enough to recover r and theta. The round trip is exact.

When all you need is the length of the same (a, b) pair without the angle, Vector Magnitude Calculator returns just the modulus and skips the argument row.

Five short steps take any complex number a + bi to its polar form r cis(theta), including pure real, pure imaginary, and zero inputs.

1. 1 Enter a and b: Type the real part and the imaginary part. Positive, negative, zero, and decimals all work, including pure real (b = 0) and pure imaginary (a = 0) inputs.
2. 2 Read the modulus: The top result is r, the distance from the origin to a + bi. It is always non-negative and equals |z|.
3. 3 Read the argument in degrees: The second result is theta in degrees, the angle from the positive real axis to the point, in (-180 degrees, 180 degrees].
4. 4 Read the argument in radians: The third result is the same theta in radians, the unit every programming language and scientific library expects.
5. 5 Read the cis form and check the quadrant: The fourth result is the polar form r cis(theta) in shorthand, and the fifth is the quadrant label (I, II, III, IV, or an axis name, or 'Origin' at 0 + 0i).

An AC voltage source has rectangular phasor V = 60 + 25i volts. Type a = 60 and b = 25 into the calculator. The modulus reads 65.00 V, the argument reads 22.62 degrees, and the polar form reads 65 cis(22.62 deg) V, ready to drop into a phasor sum.

If you need the argument in gradians or DMS instead of the two units already shown, Radians to Degrees Calculator is the next step in the chain without redoing the polar form math.

The polar form calculator handles the four operations you would do by hand, plus the sign and axis cases that trip people up.

• • Both angle units in one step: Degrees and radians appear in the results panel at the same time, so there is no need to remember the 180/pi factor mid-calculation.
• • Cis form rendered automatically: The polar form is returned as r cis(theta) in the results panel, so the phasor-ready answer can be read at a glance.
• • Quadrant label built in: The calculator returns a roman-numeral I, II, III, or IV, or an axis name for pure real and pure imaginary inputs.
• • Atan2 used automatically: The two-argument arctangent picks the correct argument for every (a, b) input, including the four quadrants and the two axes.
• • Domain error handled at the origin: When both a and b are zero, the calculator returns r = 0 and an explicit 'Undefined' argument instead of a silent NaN.

The (a, b) point on the complex plane is the same pair of coordinates an (x, y) point on the rectangular plane, so Cartesian to Polar Calculator is the closest peer for a non-complex context where the result is read without the cis shorthand.

Three things change the answer you should expect, plus two practical caveats about how the polar form is defined.

• • The polar form of a non-zero complex number is not unique: r cis(theta) and r cis(theta + 360 degrees) describe the same number. The calculator reports the principal argument in (-180 degrees, 180 degrees].
• • The argument is rounded to 4 decimal places in degrees and 6 in radians. That is enough for academic work and most AC circuit analysis.

When the modulus r is exactly zero, the argument is mathematically undefined and the calculator says so in plain text. Treat 0 + 0i as a special case rather than zeroing out a denominator downstream.

For the atan2 convention used in the argument, the Wikipedia "Atan2" article confirms the contract this calculator relies on: atan2(b, a) returns the angle whose tangent is b/a and whose sign follows the signs of b and a, with the result in the open interval (-pi, pi] so that every (a, b) pair maps to a unique argument of the complex number a + bi.

When the same idea is applied to a real 2x2 matrix instead of a complex number, Polar Decomposition Calculator factors the matrix into an orthogonal factor U and a symmetric positive definite factor P using a related polar-style split.