How to Convert a Complex Number to Polar Form: A Case Study of z 1/(1 i√3^5)

Understanding the polar form of a complex number is a fundamental concept in advanced mathematics, particularly in electrical engineering and physics. This article will walk you through the process of converting z 1/(1 i√3^5) into its polar form. We will break down each step, from rationalizing the denominator to using the polar form of a complex number. By the end of this guide, you will have a clear understanding of how to tackle similar problems in the future.

Step 1: Rationalize the Denominator

If we start with the complex number z 1/(1 i√3^5), the first step is to rationalize the denominator. This involves eliminating the imaginary unit from the denominator. We can use the conjugate of the denominator to achieve this. The conjugate of 1 i√3^5 is simply 1 - i√3^5. Multiplying both the numerator and the denominator by this conjugate, we get:

z (1 - i√3^5) / [(1 i√3^5)(1 - i√3^5)]

Step 2: Simplify the Denominator

The denominator can be simplified using the difference of squares formula, which states that (a b)(a - b) a^2 - b^2. Applying this formula, we get:

(1 i√3^5)(1 - i√3^5) 1^2 - (i√3^5)^2 1 - (-1)^5(3^5)

Since (-1)^5 -1 and 3^5 243, the denominator simplifies to:

1 - (-1)(243) 1 243 244

Step 3: Simplify the Expression

Now, we simplify the entire expression:

z (1 - i√3^5) / 244

We can now separate the real and imaginary parts:

z 1/244 - i(√3^5)/244

Let's denote M 1/244 and A -arc(√3^5/244). Here, arc denotes the argument (angle) of the complex number. The polar form of a complex number is given by z M e^(iA).

Step 4: General Case

To consider the general case, let's denote z 1/x i y^n. The polar form of this complex number can be written as:

z M e^(-inA)

Here, M modulus and A argument, and if x iy Me^(iA).

Step 5: Polar Form of the Given Equation

Let's apply this to the given equation z 1/(1 i√3^5). From the previous steps, we found:

M 1/244

A -arc(√3^5/244)

So, the polar form of the given complex number is:

z 1/244 e^[-i arc(√3^5/244)]

For a more general complex number z 1/x i y^n, the polar form can be written as:

z M e^(-inA), where M √(x^2 y^2) and A arg(x iy).

Conclusion

In conclusion, understanding the polar form of a complex number is essential in many advanced mathematical and scientific applications. The key steps include rationalizing the denominator, simplifying, and applying the general formula for the polar form. By following the steps outlined in this article, you can confidently convert any complex number to its polar form, making it easier to analyze and manipulate.