Undefined argument for a complex number

In summary: Thanks. I got those two equations, and just noticed that I'd missed copying a negative. OK yay! I got the answer.But yes, the solution does say "arg z = -pi/2 which is undefined ". After that it does go to say that tan(arg(z)) is undefined and solves a quadratic equation with two sets of solutions, the other being the one I got originally because of a copying error. When I saw the solution I was worried I'd completely missed out on the two sets of solutions and gotten the wrong one. Thanks again!Thanks. I got those two equations, and just noticed that I'd missed copying a negative. OK yay! I got the answer.But yes
  • #1
Anielka
2
0
z is a complex number such that z = [itex]\frac{a}{1+i}[/itex] + [itex]\frac{b}{1-3i}[/itex]
where a and b are real. If arg(z) = -[itex]\frac{\pi}{2}[/itex] and |z|= 4, find the values of a and b.I got as far as

z = ([itex]\frac{a}{2}[/itex] + [itex]\frac{b}{10}[/itex]) + i([itex]\frac{3b}{10}[/itex] - [itex]\frac{a}{2}[/itex])

by simplifying the original expression. Then I expressed z in the exponential form.
and

z = 4e[itex]^{-i({\pi}/2)}[/itex]

cos[itex]\frac{{\pi}}{2}[/itex] = [itex]\frac{x}{4}[/itex]
x= 0, x would be the real part of z.

From the geometric representation of the complex number it seemed to me that the argument -[itex]\pi[/itex]/2 was reasonable as the complex number would simply lie on the imaginary axis i.e. at (0, -4)

After that I compared real and imaginary parts as z = -4i
and got b = 10 and a = -2. This is apparently wrong. The answer given is b = -10 and a = 2

The solution states that an argument of -pi/2 is undefined. Could someone please explain why it is undefined? And what is the argument of a complex number that lies only on the imaginary axis? Could someone also explain where I went wrong/ why the given answer and my answer just have a sign difference?

Many thanks and if there are problems with formatting, I apologise in advance. It's my first time posting.
 
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  • #2
Anielka said:
z is a complex number such that z = [itex]\frac{a}{1+i}[/itex] + [itex]\frac{b}{1-3i}[/itex]
where a and b are real. If arg(z) = -[itex]\frac{\pi}{2}[/itex] and |z|= 4, find the values of a and b.I got as far as

z = ([itex]\frac{a}{2}[/itex] + [itex]\frac{b}{10}[/itex]) + i([itex]\frac{3b}{10}[/itex] - [itex]\frac{a}{2}[/itex])

by simplifying the original expression. Then I expressed z in the exponential form.
and

z = 4e[itex]^{-i({\pi}/2)}[/itex]

cos[itex]\frac{{\pi}}{2}[/itex] = [itex]\frac{x}{4}[/itex]
x= 0, x would be the real part of z.

From the geometric representation of the complex number it seemed to me that the argument -[itex]\pi[/itex]/2 was reasonable as the complex number would simply lie on the imaginary axis i.e. at (0, -4)

After that I compared real and imaginary parts as z = -4i
and got b = 10 and a = -2. This is apparently wrong. The answer given is b = -10 and a = 2

The solution states that an argument of -pi/2 is undefined. Could someone please explain why it is undefined? And what is the argument of a complex number that lies only on the imaginary axis? Could someone also explain where I went wrong/ why the given answer and my answer just have a sign difference?

Many thanks and if there are problems with formatting, I apologise in advance. It's my first time posting.
Hello Anielka. Welcome to PF !Are you sure it doesn't say that tan(arg(z)) is undefined ?

Solve the following:
[itex] \displaystyle
\frac{a}{2}+\frac{b}{10} = 0[/itex]

[itex] \displaystyle
\frac{3b}{10}-\frac{a}{2}=-4[/itex]​
You made a simple error.
 
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  • #3
SammyS said:
Hello Anielka. Welcome to PF !


Are you sure it doesn't say that tan(arg(z)) is undefined ?

Solve the following:
[itex] \displaystyle
\frac{a}{2}+\frac{b}{10} = 0[/itex]

[itex] \displaystyle
\frac{3b}{10}-\frac{a}{2}=-4[/itex]​
You made a simple error.

Thanks. I got those two equations, and just noticed that I'd missed copying a negative. OK yay! I got the answer.

But yes, the solution does say "arg z = -pi/2 which is undefined ". After that it does go to say that tan(arg(z)) is undefined and solves a quadratic equation with two sets of solutions, the other being the one I got originally because of a copying error. When I saw the solution I was worried I'd completely missed out on the two sets of solutions and gotten the wrong one. Thanks again!
 
  • #4
Anielka said:
Thanks. I got those two equations, and just noticed that I'd missed copying a negative. OK yay! I got the answer.

But yes, the solution does say "arg z = -pi/2 which is undefined ".
That's very peculiar grammer! It gives a specific value and then tells you it is "undefined'? those two statements cannot possibly both be true. Go back and read it carefully. It it really does say, that, laugh and go on. And if you misread it, laugh and go on anyway!

After that it does go to say that tan(arg(z)) is undefined and solves a quadratic equation with two sets of solutions, the other being the one I got originally because of a copying error. When I saw the solution I was worried I'd completely missed out on the two sets of solutions and gotten the wrong one. Thanks again!
 

FAQ: Undefined argument for a complex number

1. What is an undefined argument for a complex number?

An undefined argument for a complex number refers to a complex number that does not have a defined angle or direction in the complex plane. This can occur when the complex number has a real part of zero, making it a purely imaginary number, or when the complex number is equal to zero.

2. Why can't we find the argument of a complex number with a real part of zero?

The argument of a complex number is defined as the angle between the positive real axis and the complex number in the complex plane. However, when the real part of the complex number is zero, the complex number lies on the imaginary axis and does not have a defined angle with the real axis. Therefore, the argument is undefined.

3. Can we still perform operations on a complex number with an undefined argument?

Yes, we can still perform operations on a complex number with an undefined argument. However, some properties and formulas related to complex numbers, such as De Moivre's formula, may not be applicable in this case.

4. Is an undefined argument for a complex number the same as having no argument at all?

No, an undefined argument for a complex number is not the same as having no argument at all. Having no argument means that the complex number is equal to zero and has no direction in the complex plane. An undefined argument, on the other hand, occurs when the complex number has a real part of zero, making it a purely imaginary number, but it still has a direction in the complex plane.

5. How can we represent a complex number with an undefined argument?

We can represent a complex number with an undefined argument in polar form, where the modulus (absolute value) is non-zero and the argument is undefined. For example, the complex number 0 + 3i can be represented as 3(cosθ + isinθ), where θ is undefined. Alternatively, we can also represent it in rectangular form as 0 + 3i, where the real part is zero and the imaginary part is non-zero.

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