Understanding the Complex Number Z in its Arg and Polar Forms

In the realm of complex analysis, understanding the representation of complex numbers in their polar forms and arguments is essential. This article will delve into a specific case where Z √6 √6i/1√3i. We will break down the problem to find the argument of Z (Arg Z) and its polar form.

Step-by-Step Solution

Defining the Complex Numbers

Let z1 √6 √6i and z2 1√3i.

Converting to Polar Form

To convert these complex numbers to their polar forms, we will first simplify them:

z1 can be written as:

z1 √6 √6i √6 · √2 · (1/√2i - 1/√2i) 2√3 (cos(π/4) i sin(π/4)) 2√3 eπi/4

z2 can be written as:

z2 1√3i 2 · (1/2i √3/2i) 2 (cos(π/3) i sin(π/3)) 2 eπi/3

Dividing the Complex Numbers

Now, we need to find Z, which is the division of z1 by z2:

Z z1 / z2 2√3 eπi/4 / 2 eπi/3 √3 eπi/4 - πi/3 √3 e-πi/12

Finding the Argument of Z

From the polar form of Z, we can find its argument:

Arg Z -π/12 2kπ, where k is an integer.

Polar Form of Z

The polar form of Z is:

Z √3 e-πi/12

Conclusion

By understanding how to convert complex numbers from their standard form to their polar form and how to find the argument of a complex number, we can effectively manipulate and analyze complex functions in various applications, from electrical engineering to physics.

For more information on complex numbers and their applications, consider exploring the following:

Further reading on complex number theory Polar form conversion in complex analysis Applications of complex numbers in real-world scenarios

If you have any questions or need further assistance with complex number problems, feel free to ask in the comments below.