Polar Form Calculator

A complex number in rectangular form is expressed as z = x + iy, where x represents the real part and y represents the imaginary part. To convert this into polar form, r * e^(iθ) or r(&cos; θ + i &sin; θ), the calculator determines two primary parameters: magnitude (r) and phase angle (θ). The magnitude is calculated using the Pythagorean theorem as r = √(x^2 + y^2). The phase angle is determined using the two-argument arctangent function, θ = atan2(y, x), ensuring the correct quadrant is always applied for any combination of positive or negative real and imaginary components.

Let us convert the rectangular complex number z = 3 + 4i into polar form. First, identify the real part (x = 3) and the imaginary part (y = 4). Next, calculate the magnitude by squaring both components, summing them, and taking the square root: r = √(3^2 + 4^2) = √(9 + 16) = √(25) = 5. Then, calculate the phase angle using the arctangent function: θ = atan2(4, 3) ≈ 0.927 radians, or approximately 53.13 degrees. Thus, the polar form is expressed as a magnitude of 5 and a phase angle of 0.927 radians.

Always pay close attention to your angle units, as mixing degrees and radians is a frequent source of calculation errors. When dealing with negative real numbers, ensure your arctangent function accounts for the correct quadrant by using an atan2 function rather than a standard inverse tangent. Finally, remember that multiplication and division become significantly simpler when numbers are expressed in polar form, while addition and subtraction are easiest in rectangular form.