Polar Form
Convert complex numbers a+bi to polar form r·e^(itheta) with charts. Enter real and imaginary parts to find magnitude r = sqrt(a^2+b^2) and angle theta in degrees and radians.
About This Calculator
What is Polar Form?
The polar form of a complex number expresses it as r·e^(itheta) or r(cos theta + i·sin theta), where r is the magnitude (or modulus) and theta is the angle (or argument). Every complex number z = a + bi in rectangular form has an equivalent polar representation, making certain mathematical operations significantly easier.
How the Conversion Works
Converting a complex number from rectangular to polar form uses two formulas:
- r = sqrt(a^2 + b^2) -- the magnitude, representing the distance from the point (a, b) to the origin (0, 0)
- theta = atan2(b, a) -- the angle in radians, measuring the counterclockwise rotation from the positive real axis
The atan2 function automatically determines the correct quadrant for the angle, avoiding the common pitfalls of using atan(b/a) alone. Results are provided in both degrees and radians for your convenience.
Example
For the complex number 3 + 4i: r = sqrt(9 + 16) = 5, theta = atan2(4, 3) = 53.13 deg (0.927 rad). The polar form is 5·e^(i·53.13 deg) or 5(cos 53.13 deg + i·sin 53.13 deg).
Regional Notes
Polar form conversion uses universal mathematical conventions that are identical across all regions including India, the United States, and the United Kingdom. The formulas r = sqrt(a^2+b^2) and theta = atan2(b, a) are standard worldwide.
Frequently Asked Questions
What is the polar form of a complex number?
The polar form represents a complex number as r·e^(itheta), where r is the magnitude (distance from the origin) and theta is the angle from the positive real axis. It is also written as r(cos theta + i·sin theta).
How do you convert a+bi to polar form?
For a complex number a+bi, calculate magnitude r = sqrt(a^2+b^2) and angle theta = atan2(b, a). The polar form is r·e^(itheta). For example, 3+4i becomes 5·e^(i·53.13 deg).
What is the magnitude r in polar form?
The magnitude r (also called modulus) is the distance from the origin to the point (a, b) in the complex plane. It is always non-negative and equals sqrt(a^2+b^2).
What is the angle theta in polar form?
The angle theta (also called argument or phase) is measured counterclockwise from the positive real axis. It can be expressed in degrees (0 deg to 360 deg) or radians (0 to 2pi). Negative angles indicate clockwise rotation.
How does atan2 handle negative values?
The atan2(b, a) function returns the correct angle for all four quadrants. Unlike atan(b/a), it correctly handles cases where a is negative and avoids division-by-zero when a = 0, returning the proper quadrant-adjusted angle.
Why use polar form instead of rectangular form?
Polar form simplifies multiplication, division, and exponentiation of complex numbers. While rectangular form a+bi is best for addition and subtraction, polar form r·e^(itheta) makes multiplication (multiply r, add angles) and powers (De Moivre theorem) much easier.
What are real-world applications of polar form?
Polar form is used extensively in electrical engineering for AC circuit analysis (phasors), signal processing (Fourier transforms), control systems, quantum mechanics, and computer graphics for rotations and transformations.
Can a real number be expressed in polar form?
Yes, any real number a can be written in polar form. If a > 0, then r = a and theta = 0. If a < 0, then r = |a| and theta = pi (or 180 deg). For zero, r = 0 and theta is undefined.