Complex Number to Rectangular Form Calculator
This calculator converts a complex number from polar form, defined by its modulus and argument, into rectangular form written as a + bi. Complex numbers in polar form, r(cos θ + i sin θ), suit multiplying, dividing and finding powers, but rectangular form is what you need to add or subtract complex numbers, since you work with the real and imaginary parts separately, which matters in electrical engineering, signal processing and other maths or physics involving the complex plane. To use it, enter the modulus r, the distance of the point from the origin, which must be zero or greater, and the argument θ, the angle from the positive real axis, then choose whether that angle is in degrees or radians. The calculator instantly returns the real part a, found from r cos θ, and the imaginary part b, found from r sin θ, combines them into the full rectangular form a + bi, and shows step-by-step workings with the angle converted to radians and the cosine and sine values used. A worked example using the classic 3-4-5 triangle, r = 5 and θ = 53.13°, is included so you can check the method. Results are rounded to three decimal places, and values that are mathematically zero, such as the imaginary part at 0° or 180°, may appear as very small numbers due to floating-point rounding.
Polar Form Input
Rectangular Form Result
The rectangular form a + bi where:
Step-by-Step Workings
Worked Example (Default Values)
Convert the complex number with modulus r = 5 and argument θ = 53.13° to rectangular form.
Step 1 - Find the real part: a = r cos(θ) = 5 × cos(53.13°) = 5 × 0.6000 = 3.000
Step 2 - Find the imaginary part: b = r sin(θ) = 5 × sin(53.13°) = 5 × 0.8000 = 4.000
Result: The rectangular form is 3.000 + 4.000i
This is the classic 3-4-5 right triangle example: the complex number 3 + 4i has modulus 5 (since √(3² + 4²) = 5) and argument arctan(4/3) = 53.13°.
How to Convert Polar Form to Rectangular Form
A complex number can be written in two equivalent ways. In polar form it is expressed as r(cos θ + i sin θ), or more compactly as r∠θ, where r is the modulus (the distance from the origin on the complex plane) and θ is the argument (the angle measured anticlockwise from the positive real axis). In rectangular form the same number is written as a + bi, where a is the real part and b is the imaginary part.
The conversion formulas come directly from trigonometry. If you draw the complex number on the Argand diagram, it forms a right triangle with the real axis, so:
| Component | Formula | Description |
|---|---|---|
| Real part (a) | a = r cos(θ) | Horizontal component |
| Imaginary part (b) | b = r sin(θ) | Vertical component |
The angle θ can be given in degrees or radians. To convert from degrees to radians, multiply by π/180. This calculator accepts both.
Common Conversion Examples
| Polar Form (r, θ in degrees) | Real Part (a) | Imaginary Part (b) | Rectangular Form |
|---|---|---|---|
| r = 1, θ = 0° | 1.000 | 0.000 | 1 + 0i |
| r = 1, θ = 90° | 0.000 | 1.000 | 0 + 1i |
| r = 1, θ = 180° | -1.000 | 0.000 | -1 + 0i |
| r = 1, θ = 270° | 0.000 | -1.000 | 0 - 1i |
| r = 5, θ = 53.13° | 3.000 | 4.000 | 3 + 4i |
| r = 2, θ = 45° | 1.414 | 1.414 | 1.414 + 1.414i |
| r = 3, θ = 30° | 2.598 | 1.500 | 2.598 + 1.500i |
When to Use Each Form
Rectangular form (a + bi) is most convenient for addition and subtraction of complex numbers: you simply add or subtract the real parts and the imaginary parts separately. For example, (3 + 4i) + (1 + 2i) = 4 + 6i.
Polar form (r, θ) is most convenient for multiplication and division. To multiply two complex numbers in polar form, multiply their moduli and add their arguments: r1 * r2 at angle θ1 + θ2. To divide, divide the moduli and subtract the arguments.
Powers and roots are also straightforward in polar form using De Moivre's theorem: (r∠θ)^n = r^n ∠ nθ.
The Euler and Cis Notation
A complex number in polar form can also be written using Euler's formula: r e^(iθ) = r(cos θ + i sin θ). The shorthand cis(θ) means cos(θ) + i sin(θ), so r∠θ = r cis(θ) = r e^(iθ). All three notations describe the same complex number, and the rectangular form a + bi is obtained from any of them using a = r cos θ and b = r sin θ.
Precision Note
Due to floating-point arithmetic, values that are mathematically zero (such as the imaginary part at θ = 0° or 180°) may appear as very small numbers like 0.000 or -0.000. This calculator rounds results to three decimal places to keep the output clean. For exact symbolic results, treat any value below 1×10⁻¹⁰ as zero.
Related Calculators
- Essential Calculators: browse all converter and utility tools.
- Complex Number Calculator: add, subtract, multiply and divide complex numbers.
- Quadratic Formula Solver: solve ax² + bx + c = 0, including complex roots.
- Equation of a Line Calculator: find the equation of a straight line from two points or slope-intercept form.
- Scientific to Decimal Notation Converter: switch between scientific and standard notation.
Sources and method: Conversion formulas from standard complex analysis: a = r cos(θ), b = r sin(θ), derived from Euler's formula and the definition of polar coordinates on the complex plane. See Kreyszig, Advanced Engineering Mathematics, 10th ed., Chapter 13.
This calculator computes results to floating-point precision and rounds to three decimal places. For angles that are exact multiples of 30° or 45°, the results are exact to many more decimal places than displayed. Always verify critical computations independently.