Undefined argument for a complex number

z is a complex number such that z = $\frac{a}{1+i}$ + $\frac{b}{1-3i}$
where a and b are real. If arg(z) = -$\frac{\pi}{2}$ and |z|= 4, find the values of a and b.

I got as far as

z = ($\frac{a}{2}$ + $\frac{b}{10}$) + i($\frac{3b}{10}$ - $\frac{a}{2}$)

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

z = 4e$^{-i({\pi}/2)}$

cos$\frac{{\pi}}{2}$ = $\frac{x}{4}$
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 -$\pi$/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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 Quote by Anielka z is a complex number such that z = $\frac{a}{1+i}$ + $\frac{b}{1-3i}$ where a and b are real. If arg(z) = -$\frac{\pi}{2}$ and |z|= 4, find the values of a and b. I got as far as z = ($\frac{a}{2}$ + $\frac{b}{10}$) + i($\frac{3b}{10}$ - $\frac{a}{2}$) by simplifying the original expression. Then I expressed z in the exponential form. and z = 4e$^{-i({\pi}/2)}$ cos$\frac{{\pi}}{2}$ = $\frac{x}{4}$ 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 -$\pi$/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:
$\displaystyle \frac{a}{2}+\frac{b}{10} = 0$

$\displaystyle \frac{3b}{10}-\frac{a}{2}=-4$

 Quote by SammyS Hello Anielka. Welcome to PF ! Are you sure it doesn't say that tan(arg(z)) is undefined ? Solve the following:$\displaystyle \frac{a}{2}+\frac{b}{10} = 0$ $\displaystyle \frac{3b}{10}-\frac{a}{2}=-4$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!

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