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Complex Number to Rectangular Form Calculator

Switching between polar and rectangular representations of complex numbers is essential in engineering, physics, and mathematics. When you have a complex number in exponential form—defined by its magnitude and phase angle—you'll need to extract the real and imaginary components to work with it in standard a + bi notation. This calculator automates that conversion using trigonometric decomposition.

Last updated: May 13, 2026

Creators Jasmine J. Mah, BSc

Reviewers Mateusz Mucha and Anna Szczepanek

4,166 people find this calculator helpful

Understanding Rectangular Form

The rectangular form z = a + bi expresses a complex number using its real and imaginary components. Here, a represents the horizontal distance along the real axis, and b represents the vertical distance along the imaginary axis. When plotted on the complex plane, the number appears as a single point with coordinates (a, b).

This representation is intuitive for addition, subtraction, and many algebraic operations. Unlike polar form, which emphasises magnitude and direction, rectangular form emphasises the individual numerical components that define the complex number's position on a 2D grid.

Conversion Formulas

To convert from polar form z = r × e^iφ (where r is magnitude and φ is phase) to rectangular form, apply these trigonometric identities:

a = r × cos(φ)

b = r × sin(φ)

r — Magnitude of the complex number (distance from origin)
φ — Phase angle, measured counterclockwise from the positive real axis (in radians or degrees)
a — Real part of the rectangular form
b — Imaginary part of the rectangular form

How the Conversion Works

The polar-to-rectangular conversion relies on decomposing the magnitude vector into horizontal and vertical components. If you visualise a complex number as an arrow from the origin at angle φ and length r, the real part is how far right (or left) the arrow points, and the imaginary part is how far up (or down).

Using cosine recovers the horizontal projection; using sine recovers the vertical projection. Both are scaled by the magnitude r to account for the arrow's length. This method works for any angle, including negative phases and those exceeding 360°.

Practical Example

Consider z = 2 × e^iπ/4 (magnitude 2, phase 45°):

Real part: a = 2 × cos(π/4) = 2 × (√2/2) ≈ 1.414
Imaginary part: b = 2 × sin(π/4) = 2 × (√2/2) ≈ 1.414
Rectangular form: z ≈ 1.414 + 1.414i

The calculator performs these trigonometric calculations instantly, eliminating manual computation and rounding errors.

Common Pitfalls and Considerations

Several issues can trip up those converting between complex number forms:

Angle unit mismatch — Always verify whether your phase angle is in radians or degrees before entering it. A phase of 90 degrees (π/2 radians) and 90 radians produce vastly different results. Most scientific contexts use radians, but engineering applications sometimes prefer degrees.
Sign errors with negative angles — Phases can be negative or exceed 360°. A phase of −90° is equivalent to 270°, and both produce valid—but different—real and imaginary components. Ensure your calculator or manual calculation accounts for the quadrant correctly.
Magnitude scaling forgotten — The most frequent mistake is computing cos(φ) and sin(φ) without multiplying by the magnitude <em>r</em>. These trigonometric values alone only give you a + bi when r = 1. Always scale by magnitude to get the correct rectangular components.
Precision loss in approximation — For exact symbolic answers (like √3/2), avoid premature decimal rounding. If your problem requires exact form, keep radicals; if decimals are acceptable, round only at the final step to minimise cumulative errors.

Frequently Asked Questions

Why convert from polar to rectangular form?

Rectangular form (a + bi) is ideal for addition, subtraction, and evaluating polynomial expressions. Polar form (r × e^iφ) excels at multiplication, division, and exponentiation. Many problems mix both forms—you might receive data in polar coordinates but need rectangular form for circuit analysis or control systems. The conversion bridges these two complementary representations.

Can I convert from rectangular back to polar?

Yes, the reverse conversion is straightforward. Given a + bi, compute magnitude as r = √(a² + b²) and phase as φ = arctan(b/a), adjusting for the correct quadrant using the signs of a and b. Some calculators and programming languages offer <code>atan2(b, a)</code> to handle quadrant correction automatically.

What if the magnitude is zero?

When r = 0, the complex number is simply 0 (or 0 + 0i in rectangular form), regardless of the phase angle. The phase is undefined or irrelevant at the origin. A zero-magnitude complex number carries no direction—it's a single point.

Does the phase angle have to be between 0 and 360 degrees?

No. Angles outside this range are valid and equivalent to their reduced counterparts (e.g., 450° equals 90°). Negative angles work too; −45° is the same as 315°. The trigonometric functions handle all values consistently, so you can use whatever phase representation suits your problem.

How accurate do manual calculations need to be?

For engineering and physics, three to four significant figures usually suffice. However, if you're verifying theory or working on proofs, keep more decimal places or use exact symbolic forms (like π/6 or √3/2) throughout. Rounding at each intermediate step compounds errors; it's better to round only at the end.

What's the rectangular form of e^(iπ)?

Using the formulas: a = 1 × cos(π) = −1 and b = 1 × sin(π) = 0, giving −1 + 0i or simply −1. This is a famous result known as Euler's identity, showing that e^(iπ) = −1, a profound connection between exponentials, trigonometry, and the imaginary unit.

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