Complex Numbers in Polar Form: Modulus and Argument

The polar form of complex numbers is the single most useful tool you will meet when a complex number stops being a point on a page and starts being a rotation or a wave. Every complex number a + bi sits at a distance from the origin and at an angle from the positive real axis; those two numbers, the modulus and the argument, are the whole story. The conversion between rectangular, polar and exponential forms, the arithmetic of multiplication and division in that system, and the one mistake that sends students and engineers to the wrong answer, choosing the wrong quadrant for the argument, are all covered. If you can place the point and measure the angle, you can convert anything.

Rectangular to Polar: Worked Examples in All Four Quadrants

Rectangular to polar conversion means taking the pair (a, b) and producing the pair (r, θ). The modulus of complex number z = a + bi is r = √(a² + b²), and the argument is θ = arctan(b/a), but only after you check which quadrant the point is in. The arctangent on a calculator returns a value between −π/2 and π/2, which is only correct for the first and fourth quadrants. For the other two, you must add or subtract π.

Four Quadrants, Four Angles

Work the four quadrants with real numbers. First quadrant: z = 1 + i. r = √(1 + 1) = √2, θ = arctan(1) = π/4. Second quadrant: z = −1 + i. r = √2 still, but θ = 3π/4, not −π/4. Third: z = −1 − i gives θ = −3π/4 or 5π/4, your choice of range. Fourth: z = 1 − i gives θ = −π/4. The modulus never changes; the argument does, and that is where every conversion fails. Draw the point, read the quadrant, then adjust.

Frequently Asked Questions

How do I calculate the modulus of a complex number?

The modulus r of a complex number z = a + bi is r = √(a² + b²). For example, z = 1 + i gives r = √(1 + 1) = √2.

What is the argument for a complex number in the second quadrant?

For z = −1 + i in the second quadrant, θ = 3π/4. The calculator arctan(b/a) would return −π/4, so you must add π to get the correct quadrant angle.

Can the argument be expressed in more than one way?

Yes, for z = −1 − i in the third quadrant, θ can be −3π/4 or 5π/4, depending on your chosen range. Both represent the same angle.

What is the modulus of a complex number in the fourth quadrant?

The modulus is always r = √(a² + b²) regardless of quadrant. For z = 1 − i, r = √2, the same as for 1 + i, −1 + i, and −1 − i.

What is the argument of a complex number in the first quadrant?

For z = 1 + i in the first quadrant, θ = arctan(1) = π/4. No adjustment is needed because the calculator arctan returns a value in the correct range.

What common mistake causes conversion errors?

Choosing the wrong quadrant for the argument is the one mistake that sends students and engineers to the wrong answer. The calculator arctan only returns values between −π/2 and π/2, which is correct only for the first and fourth quadrants.