Complex Number to Polar Form
a+bi → r∠θ.
Polar form r∠θ.
How the Math Works
The Complex Number to Polar Form calculator converts a complex number expressed as a+bi into its polar representation r∠θ. The magnitude r is calculated using the formula r = √(a² + b²), which represents the distance from the origin to the point (a,b) in the complex plane. The angle θ (argument) is found using θ = arctan(b/a), adjusted by 180° or 270° when necessary to account for the correct quadrant, giving the direction of the vector from the origin.
Practical Applications
To use this calculator, simply enter the real part (a) and imaginary part (b) of your complex number. The tool instantly computes the polar form, displaying both the magnitude (r) and angle (θ) in degrees or radians. This is particularly useful when multiplying or dividing complex numbers, as polar form simplifies these operations by allowing you to multiply/divide magnitudes and add/subtract angles, respectively.
Day-to-Day Use
While complex numbers may seem abstract, converting them to polar form has practical applications in everyday technology. Electrical engineers use it to analyze alternating current (AC) circuits, where impedance is represented as a complex number. Signal processing in your smartphone, GPS systems, and even music streaming services rely on these mathematical concepts to filter and process signals efficiently.
FAQ
Radians?
θ in rad = atan2(b,a).