# Complex numbers question.

1. Apr 16, 2007

### jernobyl

1. The problem statement, all variables and given/known data

Express -i in polar form, using the principal value of the argument.

2. Relevant equations

modulus = $$\sqrt{a^2 + b^2}$$

$$\theta$$ = arg(0 - i)

3. The attempt at a solution

Well, the complex number is 0 -i. a = 0, b = -1 so:

$$r = \sqrt{0^2 + (-1)^2}$$ which comes out to be 1.

But for the argument, $$\theta$$ comes out to be:

$$\tan\theta = \frac{-1}{0}$$

Ummm...where do we go from here?! Also, err, what IS the principal argument of the argument? I mean, it seems to be that the value of $$\theta$$ changes depending not where on the CAST diagram it is, but on where in the Argand diagram the complex number turns out to be...but, err, not always. Like, if it lies in the fourth quadrant, you don't do 360 - $$\theta$$...

Last edited: Apr 16, 2007
2. Apr 16, 2007

### cristo

Staff Emeritus
The argument of a complex number x+iy is only tan(y/x) if neither x nor y is zero.

You can find the argument of -i by simply considering the argand diagram. Where does -i lie on the argand diagram? When you plot this, it should be obvious what the argument is. Note that the principal argument, is a value between -pi and pi. (or -180 and 180 degees)

Last edited: Apr 16, 2007
3. Apr 16, 2007

### jernobyl

Thanks for this. The exercises I have been doing on complex numbers have all asked for the principal argument, and some of the answers are not between $$\pi$$ and -$$\pi$$, such as 239.04 degrees.

Okay, err, I've plotted -i on the argand diagram...-i is at "-1"...I'm really not getting it here...

4. Apr 16, 2007

### cristo

Staff Emeritus
you may have a different definition of the principal argument then; although i thought it was always between pi and -pi

It's at -1 on the imaginary axis. Now, what is the angle between the positive real axis and the negative imaginary axis? This will give you the argument.

5. Apr 16, 2007

### Staff: Mentor

Some alternatives are
• Use Euler's formula, $e^{i\theta} = \cos \theta + i\sin \theta$. You need to find the value $\theta$ such that $\cos \theta = 0$ and $\sin\theta = -1$.
• Use the cotangent instead of the tangent, paying attention to the sign of the argument.
• Use the two argument form of the inverse tangent, http://en.wikipedia.org/wiki/Arctangent#Two_argument_variant_of_arctangent

6. Apr 16, 2007

### rsnd

its 1 e^ {-I * Pi/2} ?

7. Apr 16, 2007

### cristo

Staff Emeritus

8. Apr 16, 2007

### jernobyl

Ah, thanks for that. I'm confused but I kinda understand the answer...

9. Apr 17, 2007

### HallsofIvy

Staff Emeritus
Since you say some, at least of the answers are in degrees, between 0 and 360 degrees, start at the positive real axis and measure the angle counterclockwise to the negative y-axis. You should see that the angle is a multiple of 90 degrees.

10. Apr 17, 2007

### jernobyl

Yeah, thanks. I'm just not sure, like...one of the answers to one of the questions was -7.15 degrees...and how can that be, if you're meant to measure it counterclockwise from the positive real axis? Sigh.