Complex Analysis: Finding Arg(z)

In summary: Arg(z) for the complex number z = 6 / (1 + 4i). He had already calculated the absolute value of z, but was unsure how to find the argument. Another person in the conversation provided a definition of Arg(z) as the angle between the positive x-axis and the complex number on an Argand diagram, denoted as arg(z). They also mentioned that Arg(z) is case sensitive, unlike other complex functions. Hurkyl confirmed Yang's guess that Arg(z) in this case is equal to arctan(-4). Another person thanked Hurkyl for their help and shared a link to a website that provided further explanation on the concept. In summary, the conversation
  • #1
Iron Eagle
5
0
Hello everyone,

I am trying to solve this follow problem, but don't quite know how to go about getting Arg(z).

z = 6 / (1 + 4i)

I got that lzl is sqrt((6/17)^2+(-24/17)^2) but am stuck with finding Arg(z). It told me to recall that -pi < Arg(z) <= pi

Can you guys teach me how to go about finding arrrggg... this Arg(z)?

TIA.
 
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  • #2
Is there any sort of number for which you know how to compute Arg?
 
  • #3
This Arg(z) thing is new to me, so no. I'm trying to find out what it is. Thanks.
 
  • #4
I found this on Google:

The argument of a complex number is the angle between the positive x-axis and the line representing the complex number on an Argand diagram. It is denoted arg (z)

So Arg(z) is pretty much just the angle theta that r makes with the x-axis?

Assuming that arg(z) is not case sensitive, then is it simply arctan(-4) in this case?
 
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  • #5
In this case, yes, Arg(z) = Arctan(-4). (but be wary; for some z, Arg(z) cannot equal the Arctangent of anything at all! You have to make sure you take the right value for the arctangent)

Arg/arg is case sensitive, just like other complex functions. arg is a "multi-valued function", whereas Arg picks out a specific value.
 
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  • #6
Thank you so much Hurkyl... your help is much appreciated! In the meantime, I also found http://scholar.hw.ac.uk/site/maths/topic11.asp?outline=no" page, which led me to believe that I was right. Your reply, however, gave a positive confirmation of my guess. Thanks again - now I have a much better grasp of this concept.

Yang
 
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1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It combines concepts from both calculus and algebra to analyze and understand the behavior of these functions.

2. What is the argument of a complex number?

The argument of a complex number is the angle formed between the positive real axis and the line connecting the complex number to the origin in the complex plane. It is often denoted by Arg(z) or θ and is measured in radians.

3. How do you find the argument of a complex number?

To find the argument of a complex number, you can use the formula Arg(z) = arctan(b/a), where a is the real part of the complex number and b is the imaginary part. You can also use the geometric method by plotting the complex number on the complex plane and measuring the angle formed.

4. What is the principal argument of a complex number?

The principal argument of a complex number is the argument that falls within the range (-π, π]. It is considered the primary value of the argument and is often denoted by arg(z) or φ.

5. Can the argument of a complex number be negative?

Yes, the argument of a complex number can be negative. The argument of a complex number is considered negative when it falls in the range (-π, 0). This means that the complex number is located in the lower half of the complex plane.

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