argument of a complex number on an argand diagram?

Question

On wikipedia, it says that the argument of a complex number is worked as follows, depending on what quadrant the complex number lies within on an Argand diagram:

1. arg(z) if in 1st and 4th quadrants: arctan(y/x)
2. arg(z) if in 2nd quadrant: arctan(y/x)+pi
3. arg(z) if in 3rd quadrant: arctan(y/x) - pi

What I need to understand is why we need to add pi in the second quadrant and why we need to take away pi in the 3rd quadrant. It makes absolutely no sense to me! 

Wouldn't it make sense to do pi - arctan(y/x) if the complex number is in the 2nd quadrant?

Someone please please please explain it to meeee!!!!

Any help would be much appreciated!

Thanks.

Deepak · Accepted Answer

Here's the reason: The range of the arctan function is only (-pi/2, pi/2). When you are in the second quadrant, the actual value of arg(z) has to be between (pi/2, pi]. However, this is outside the range of the arctan function. Consequently, the arctan function, when applied to y/x, will automatically subtract the answer by pi, to obtain a value in the range (-pi/2, 0]. This is perfectly fine as far as the arctan function goes because tan (a) = tan(a - pi) for every a. Because of this automatic subtraction, you must add pi to arctan(y/x) to bring it into the second quadrant. 

The explanation for the third quadrant is similar. There, the value of arg(z) is in the range [-pi, -pi/2) and the arctan function automatically adds pi to bring it to the range [0, pi/2). Hence, pi must be subtracted from the result of the arctan function.

In summary, the arctan function produces answers only in quadrants 1 and 4 by mapping quadrant 2 to quadrant 4 and quadrant 3 to quadrant 1. If you want answers in quadrants 3 and 4, you must adjust the answer by adding or subtracting pi.

az_lender · Answer

In the second quadrant, x is a negative number, so (y/x) is a negative number.
If you "ask your calculator" to find arctan of a negative number, 
your calculator will give you the smallest negative angle that works,
so it's giving you a 4th quadrant value, when you need a 2nd quadrant value.
That's why you have to add pi.

In the third quadrant, (y/x) is positive, 
so if you "ask your calculator" to find arctan of (y/x),
it's going to give you the smallest positive angle, which is a 1st-quadrant value.
You need the 3rd quadrant value; if you want the value between -pi and +pi,
you would subtract pi to get the smallest negative value
(whose magnitude is less than that of the smallest positive value).

Anonymous · Answer

you probably are not realizing what the principal values of the arctan function are
just google arctan principal value
to find sites to explain it (they'll likely show graphs too, which I can't show here, which is why I direct you to use google)

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argument of a complex number on an argand diagram?

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