(1+i)/(1-i) Polar Form

3 min read Jun 16, 2024

Finding the Polar Form of (1 + i) / (1 - i)

This article will guide you through the process of finding the polar form of the complex number (1 + i) / (1 - i).

1. Simplifying the Complex Number

First, we simplify the given complex number by multiplying the numerator and denominator by the conjugate of the denominator:

(1 + i) / (1 - i) * (1 + i) / (1 + i) = (1 + 2i + i²) / (1 - i²)

Since i² = -1, we can simplify further:

(1 + 2i - 1) / (1 + 1) = 2i / 2 = i

2. Finding the Magnitude (r)

The magnitude of a complex number z = a + bi is given by |z| = √(a² + b²).

In our case, z = i = 0 + 1i. Therefore, the magnitude is:

|z| = √(0² + 1²) = √1 = 1

So, r = 1.

3. Finding the Argument (θ)

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. We can find the argument using the following formula:

θ = arctan(b / a)

In our case, a = 0 and b = 1. Therefore:

θ = arctan(1 / 0) = π/2

Since the complex number lies on the positive imaginary axis, its argument is π/2.

4. Expressing in Polar Form

The polar form of a complex number is given by:

z = r(cos θ + i sin θ)

Substituting the values of r and θ that we found:

z = 1(cos(π/2) + i sin(π/2))

Therefore, the polar form of (1 + i) / (1 - i) is 1(cos(π/2) + i sin(π/2)).

Note: The polar form of a complex number can also be expressed using Euler's formula:

z = re^(iθ)

In this case, the polar form would be:

z = 1 * e^(iπ/2)