# Arguement - complex numbers

1. Dec 9, 2009

### ChrisBaker8

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

Find the modulus and the principal value of the argument for the complex number $$\sqrt{3}$$ - i

3. The attempt at a solution

I know the modulus is just 'square both, add, and square root of sum', so r = $$\sqrt{2}$$, but I don't know how to find the second part. I know vaguely that the argument = $$\theta$$, but I don't know where to go from here.

Do I need to convert the complex number into polar or euler form?

2. Dec 9, 2009

### Dick

What two numbers did you 'square both, add, and square root of sum' to get sqrt(2)? The complex number a+bi can be drawn as the hypotenuse of a right triangle in the complex plane with a horizontal leg of length a and a vertical leg of length b. Haven't you seen this picture? The argument is the angle the hypotenuse makes with the x-axis. So you have tan(argument)=b/a. Remember trig? Principal value is a convention for choosing which of several possible angles might satisfy the tangent equation. Look it up, hopefully it will come with a nice picture.

3. Dec 9, 2009

### ChrisBaker8

okay, I think I get the argument now

for the modulus, I added $$\sqrt{3}$$ $$^{2}$$ and i$$^{2}$$ to get 3 - 1, then square rooted to get $$\sqrt{2}$$

is that wrong?

4. Dec 9, 2009

### Staff: Mentor

Yes. You square the real part, sqrt(3), and the imaginary part, -1, add them, then take the square root. The imaginary part is the coefficient of i.

5. Dec 9, 2009

okay, thanks