Complex Number to Rectangular Form Calculator
Enter the magnitude r and phase angle phi of a complex number in polar form to get its rectangular form a + bi. Switch to rectangular-to-polar mode to go the other way. Supports both degrees and radians, shows each arithmetic step, and displays the result as a formatted complex number. All calculations happen in your browser and update as you type.
Formula
Worked example
Convert z = 5 at 53.13 degrees to rectangular form. First, convert the angle: 53.13 deg * (pi/180) = 0.927296 rad. Then a = 5 * cos(0.927296) = 5 * 0.600000 = 3.000, and b = 5 * sin(0.927296) = 5 * 0.800000 = 4.000. Rectangular form: 3 + 4i. Verification: sqrt(3^2 + 4^2) = sqrt(25) = 5 (equals r).
What is rectangular form of a complex number?
A complex number z can be written in rectangular (also called Cartesian) form as z = a + bi, where a is the real part and b is the imaginary part. The real part a is the horizontal coordinate on the complex plane and the imaginary part b is the vertical coordinate. This form makes addition and subtraction straightforward: you simply add or subtract the real parts and the imaginary parts separately. For example, (3 + 4i) + (1 - 2i) = 4 + 2i.
Polar form and how to convert to rectangular
Polar form expresses the same complex number using its distance from the origin (the modulus r) and the angle it makes with the positive real axis (the argument phi). The connection between the two forms comes from basic trigonometry applied to the right triangle formed by a, b, and r:
- Real part: a = r * cos(phi)
- Imaginary part: b = r * sin(phi)
- Modulus (reverse): r = sqrt(a^2 + b^2)
- Argument (reverse): phi = atan2(b, a)
Degrees vs radians: which angle unit to use?
Angles in complex number problems appear in both degrees and radians depending on the context. Physics and engineering courses often work in degrees (0 to 360) for phasors and AC circuit analysis, while pure mathematics uses radians (0 to 2*pi) because many identities simplify cleanly. This calculator accepts either unit and converts internally before applying the trigonometric formulas. To convert manually: degrees = radians * (180 / pi), and radians = degrees * (pi / 180). The table below lists exact values at the most common angles on the unit circle.
Rectangular vs polar: when to use each form
The rectangular form a + bi is best for addition and subtraction. The polar form r*e^(i*phi) (or equivalently r angle phi) is best for multiplication, division, and exponentiation. When multiplying two complex numbers in polar form you multiply their moduli and add their arguments: |z1*z2| = r1*r2 and arg(z1*z2) = phi1 + phi2. When raising to a power (De Moivre's theorem): z^n = r^n * (cos(n*phi) + i*sin(n*phi)). Converting between forms is a routine step whenever the operation at hand favors the other representation.
Common polar-to-rectangular conversions
| Angle (deg) | Angle (rad) | Real part (a = cos phi) | Imaginary part (b = sin phi) | Rectangular form |
|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 + 0i |
| 30 | pi/6 | sqrt(3)/2 ~ 0.866025 | 1/2 = 0.5 | 0.866025 + 0.5i |
| 45 | pi/4 | 1/sqrt(2) ~ 0.707107 | 1/sqrt(2) ~ 0.707107 | 0.707107 + 0.707107i |
| 60 | pi/3 | 1/2 = 0.5 | sqrt(3)/2 ~ 0.866025 | 0.5 + 0.866025i |
| 90 | pi/2 | 0 | 1 | 0 + 1i |
| 120 | 2*pi/3 | -0.5 | 0.866025 | -0.5 + 0.866025i |
| 135 | 3*pi/4 | -0.707107 | 0.707107 | -0.707107 + 0.707107i |
| 150 | 5*pi/6 | -0.866025 | 0.5 | -0.866025 + 0.5i |
| 180 | pi | -1 | 0 | -1 + 0i |
| 270 | 3*pi/2 | 0 | -1 | 0 - 1i |
| 360 | 2*pi | 1 | 0 | 1 + 0i |
Exact values for frequently used angles. r = 1 for all entries (unit circle).
Frequently asked questions
How do I convert a complex number from polar to rectangular form?
Use the formulas a = r * cos(phi) and b = r * sin(phi), where r is the modulus and phi is the argument. The result is the rectangular form a + bi. For example, 5 at 53.13 degrees gives a = 5 * cos(53.13 deg) = 3 and b = 5 * sin(53.13 deg) = 4, so the rectangular form is 3 + 4i.
What is the difference between polar form and rectangular form?
Rectangular form z = a + bi describes a complex number by its real component (a, horizontal) and imaginary component (b, vertical). Polar form z = r angle phi (or r*e^(i*phi)) describes it by its distance from the origin r and the angle phi it makes with the positive real axis. Both representations encode exactly the same point on the complex plane.
Can I convert from rectangular back to polar with this calculator?
Yes. Switch the "Conversion direction" selector to "Rectangular to polar" and enter the real part a and imaginary part b. The calculator computes the modulus r = sqrt(a^2 + b^2) and the argument phi = atan2(b, a), and displays the polar form in your chosen angle unit.
Why do some angles give exact values (like 0, 1, 0.5, sqrt(3)/2)?
At certain special angles (multiples of 30 and 45 degrees, which correspond to pi/6, pi/4, pi/3, pi/2, and pi radians) the sine and cosine take values expressible as simple fractions or square roots: 0, 1/2, 1/sqrt(2), sqrt(3)/2, 1. These are the values on the unit circle (r = 1) memorized in trigonometry. The reference table on this page lists them for quick lookup.
What is the modulus of a complex number?
The modulus (also called magnitude or absolute value) of a complex number z = a + bi is the distance from the origin to the point (a, b) on the complex plane. It equals sqrt(a^2 + b^2) and is always non-negative. The modulus is the r in the polar form r angle phi.
What is the argument of a complex number?
The argument (or phase) is the angle phi that the line from the origin to the point (a, b) makes with the positive real axis. It is computed as atan2(b, a), which correctly handles all four quadrants. The principal argument lies in the range (-180, 180] degrees or (-pi, pi] radians. This calculator returns the principal argument.
How does this calculator handle purely real or purely imaginary numbers?
A purely real number (b = 0) has an argument of 0 degrees (or 180 degrees if negative). A purely imaginary number (a = 0) has an argument of 90 degrees (b > 0) or -90 degrees (b < 0). The calculator handles these cases correctly: cos(0) = 1 and sin(0) = 0 give the real number r + 0i, while cos(90 deg) = 0 and sin(90 deg) = 1 give the imaginary number 0 + ri.