Convert a complex number from polar/trigonometric form r(cos θ + i sin θ) to rectangular form a + bi. Shows real part, imaginary part, Euler form, conjugate, and quadrant.
| r | θ | Real (a) | Imaginary (b) |
|---|---|---|---|
| 1 | 0° | 1 | 0 |
| 1 | 30° | √3/2 ≈ 0.8660 | 0.5 |
| 1 | 45° | √2/2 ≈ 0.7071 | √2/2 ≈ 0.7071 |
| 1 | 60° | 0.5 | √3/2 ≈ 0.8660 |
| 1 | 90° | 0 | 1 |
| 1 | 120° | −0.5 | √3/2 ≈ 0.8660 |
| 1 | 135° | −√2/2 | √2/2 |
| 1 | 180° | −1 | 0 |
| 1 | 270° | 0 | −1 |
| 1 | 360° | 1 | 0 |
The n-th roots of unity are n complex numbers evenly spaced on the unit circle.
| n | Root angles | First non-trivial root (rect.) |
|---|---|---|
| 2 | 0°, 180.0° | -1.0000 + 0.0000i |
| 3 | 0°, 120.0°, 240.0° | -0.5000 + 0.8660i |
| 4 | 0°, 90.0°, 180.0°, 270.0° | 0.0000 + 1.0000i |
| 5 | 0°, 72.0°, 144.0°, 216.0°, 288.0° | 0.3090 + 0.9511i |
| 6 | 0°, 60.0°, 120.0°, 180.0°, 240.0°, 300.0° | 0.5000 + 0.8660i |
| 8 | 0°, 45.0°, 90.0°, 135.0°, 180.0°, 225.0°, 270.0°, 315.0° | 0.7071 + 0.7071i |
The **Complex Number to Rectangular Form Calculator** converts any complex number from its polar (trigonometric) representation r·(cos θ + i·sin θ) into the standard rectangular form a + bi. Enter the modulus r and argument θ, and the tool computes the real part a = r·cos θ and the imaginary part b = r·sin θ, along with the Euler exponential form, the complex conjugate, and the quadrant in the complex plane.
Polar-to-rectangular conversion is essential in electrical engineering for converting between phasor and impedance notation, in signal processing for interpreting Fourier coefficients, and in physics for resolving vectors into components. Students encounter this conversion throughout complex analysis, where switching between forms simplifies multiplication, division, and exponentiation.
The calculator supports both degrees and radians for the argument, offers standard (i) and engineering (j) notation, and includes eight presets covering the principal directions on the complex plane. Visual component bars show the relative magnitudes of the real and imaginary parts against the modulus. A reference table of common unit-modulus complex numbers provides quick lookups for standard angles, and an expandable section displays the n-th roots of unity—evenly spaced points that appear in discrete Fourier transforms and polynomial factorization.
Whether you are converting AC circuit phasors, plotting Mandelbrot-set points, or studying Euler's identity e^(iπ) + 1 = 0, the page keeps the rectangular form, Euler form, conjugate, and quadrant together so you can compare representations without reworking the same complex number by hand.
Polar-to-rectangular conversion is usually the point where modulus, angle, and sign conventions have to line up correctly. This calculator keeps the real part, imaginary part, Euler form, and conjugate aligned so you can verify the whole complex-number picture from one input.
It is especially useful when you are switching between geometric and algebraic views of the same complex number. The modulus and argument describe position in polar form, while a + bi exposes the Cartesian components directly.
Given z = r·(cos θ + i·sin θ): Real part a = r·cos θ. Imaginary part b = r·sin θ. Rectangular form: z = a + bi. Euler form: z = r·e^(iθ). Conjugate: z̄ = a − bi. |z|² = a² + b² = r².Result: Computed from the entered values
Using r=1, t=0, the calculator returns Computed from the entered values. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This page is designed for complex-number problems where you need to move from modulus-and-angle form to Cartesian components. It computes a + bi from r and θ, then keeps the Euler form, conjugate, and quadrant information attached to the same result.
Start with the real and imaginary parts, since they are the actual rectangular coordinates. Then compare those values with the modulus and argument to confirm that the sign pattern matches the quadrant implied by the angle.
Convert a standard angle such as 0°, 90°, 180°, or 45° manually first, then compare the result with the calculator. After that, switch between degrees and radians on the same modulus so the angle handling becomes automatic.
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It expresses a complex number as z = r(cos θ + i sin θ), where r is the distance from the origin (modulus) and θ is the angle from the positive real axis (argument).
Multiply: a = r cos θ (real part) and b = r sin θ (imaginary part). The result is z = a + bi.
The Euler form z = r·e^(iθ) uses Euler's identity to write the complex number as an exponential, which is especially convenient for multiplication and powers.
In electrical engineering, "i" is reserved for current. The imaginary unit is written as "j" instead to avoid confusion. Mathematically they are identical.
The conjugate of z = a + bi is z̄ = a − bi. Geometrically it reflects the point across the real axis. Useful identity: z · z̄ = |z|² = r².
No. The modulus r = |z| is always non-negative. A negative sign can be absorbed by adding 180° to the argument.
Convert a complex number from rectangular form a + bi to trigonometric (polar) form r(cos θ + i sin θ). Shows modulus, argument, exponential form, quadrant, reciprocal, and De Moivre powers.
Convert polar (r, θ) coordinates to Cartesian (x, y) and back, with batch mode, coordinate plot, and complete conversion reference.
Enter any angle in degrees or radians to find sin, cos, tan, csc, sec, cot, (x, y) coordinates, quadrant, and reference angle on the unit circle. Common angles table included.