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.
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 instantly 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, this tool gives you every representation in one view.
Complex Number to Rectangular Form Calculator (Polar → a + bi) helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Rectangular Form, Real Part (a), Imaginary Part (b) in one pass.
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 calculator is tailored to complex number to rectangular form calculator (polar → a + bi) workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
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). Use this as a practical reminder before finalizing the result.
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. Keep this note short and outcome-focused for reuse.
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.