# Complex numbers question.

by jernobyl
Tags: complex, numbers
 P: 31 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 - $$\theta$$...
 Mentor P: 8,262 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)
 P: 31 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...
Mentor
P: 8,262

## Complex numbers question.

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

 Okay, err, I've plotted -i on the argand diagram...-i is at "-1"...I'm really not getting it here...
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.
 Mentor P: 13,636 Some alternatives areUse 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/Arctang..._of_arctangent
 P: 26 its 1 e^ {-I * Pi/2} ?
Mentor
P: 8,262
 Quote by rsnd its 1 e^ {-I * Pi/2} ?