Undefined argument for a complex number

Anielka

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.

SammyS

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.

Anielka

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!

HallsofIvy

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!