Question about Modulus and Arguments?

Question

how do you find the argument of

-1+i

without a calculator??? please explain your working. Thanks

Answer 1

Draw yourself an Argand diagram. (http://en.wikipedia.org/wiki/Argand_diag...

The angle between -1+i and the vertical is clearly 45 degrees, so its angle from the positive real axis is clearly 135 degrees = pi/2 + pi/4 = 3pi/4.

So the argument is 3pi/4.

And its modulus is Sqrt(1^2+1^2) = sqrt2.

Answer 2

Well, for -1 + i, the real part a = -1 and the imaginary part b = 1.

The first step to find the argument is then to compute

arctan(b/a) = arctan(1 / -1) = -π/4.

If the real part were positive, this would be the answer.  However, since the real part is negative, we add π to the number we just found to get:

arg(-1+i) = 3π/4.

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On a side-note, b/a will be undefined when a=0.  However, this is solved really easily, since only two cases are possible.