Complex Number To Polar Form
Convert complex numbers from rectangular a+bi to polar form r·e^(itheta) instantly. Enter real and imaginary parts to find magnitude r = sqrt(a^2+b^2) and angle theta in degrees and radians.
About This Calculator
Complex Number to Polar Form Conversion
Every complex number z = a + bi (rectangular form) can also be expressed in polar form as r·e^(itheta) or r(cos theta + i·sin theta).
The conversion formulas are:
- r = sqrt(a^2 + b^2) -- magnitude (modulus)
- theta = atan2(b, a) -- angle (argument) in degrees or radians
Example
The complex number 3 + 4i converts to polar form as 5·e^(i·53.13 deg) because:
r = sqrt(3^2 + 4^2) = 5, theta = arctan(4/3) = 53.13 deg
Polar form is especially useful for multiplying complex numbers (multiply magnitudes, add angles) and for representing oscillations and waves in physics and engineering.
Frequently Asked Questions
How do you convert a complex number to polar form?
For a complex number a+bi, calculate magnitude r = sqrt(a^2+b^2) and angle theta = arctan(b/a). The polar form is r·e^(itheta) or r(cos theta + i·sin theta). For example, 3+4i becomes 5·e^(i·53.13 deg).
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 origin) and theta is the angle from the positive real axis. It is also written as r(cos theta + i·sin theta).
Is this complex number to polar form calculator free?
Yes, this complex number to polar form converter is completely free to use with no registration or usage limits.
What if my imaginary part is negative?
Negative imaginary parts work correctly. The calculator uses atan2(b, a) which handles all four quadrants and returns the correct angle between -180 deg and 180 deg.
Why use polar form for complex numbers?
Polar form simplifies multiplication, division, and exponentiation of complex numbers. It is essential in electrical engineering (AC circuit analysis), quantum mechanics, signal processing, and control systems.
What is the difference between polar form and rectangular form?
Rectangular form a+bi expresses a complex number by its horizontal (real) and vertical (imaginary) coordinates. Polar form r·e^(itheta) expresses it by its distance from the origin r and angle theta from the positive real axis. Rectangular form is best for addition and subtraction; polar form excels at multiplication, division, and powers.
How do you convert polar form back to rectangular form?
Given polar form r·e^(itheta), the rectangular form a+bi is a = r·cos(theta) and b = r·sin(theta), where theta is in radians. For example, 5·e^(i·0.927 rad) converts to 5·cos(0.927) + i·5·sin(0.927) = 3 + 4i.