I Conceptual Question About Polar Coordinate System

[CheeseSandwich]

I am learning about the polar coordinate system, and I have a few conceptual questions.

I understand that in Cartesian coordinates there is exactly one set of coordinates for any given point. However, in polar coordinates there is an infinite number of coordinates for a given point. I see how they are derived visually in the diagram below, and I see how the coordinates are derived by the expressions below.

Here are my questions:

1. I see how infinite coordinates can be derived, but why can there be infinite coordinates for one point?
2. How can a point in the first quadrant be equivalent to a point in the third quadrant?
3. Why can't we represent the point (5, π/3) in the second or fourth quadrants? Why does n have to be an integer and not a rational number?

Thanks!
CheeseSandwich

 

[mfb]

I see how infinite coordinates can be derived, but why can there be infinite coordinates for one point?

Why not? Some coordinate systems have unique coordinates, some do not. You can fix this for polar coordinates by restricting the range of parameters, e.g. requiring that the radius has to be positive and that the phase has to be $0 \leq \theta < 2 \pi$. This still leaves the center with an undefined phase, but you can say that the center has a phase of 0 - this is completely arbitrary.

How can a point in the first quadrant be equivalent to a point in the third quadrant?

It is not, why do you think so?

Why can't we represent the point (5, π/3) in the second or fourth quadrants? Why does n have to be an integer and not a rational number?

I don't understand that question. (5, π/3) is in the first quadrant. Instead of 5, you can also have any other real number. This example uses 5.

 

[Mark44]

How can a point in the first quadrant be equivalent to a point in the third quadrant?

As already explained, it can't. However, if your question really is, "How can an angle in the third quadrant be associated with a point in the first quadrant?"
This can happen if you allow r to be negative. For example, (5, 4π/3) is a point in the third quadrant. (-5, 4π/3) is a point in the first quadrant, despite the fact that 4π/3 is an angle in the third quadrant.

 

[CheeseSandwich]

mfb, thank you for replying.

Why not? Some coordinate systems have unique coordinates, some do not. You can fix this for polar coordinates by restricting the range of parameters, e.g. requiring that the radius has to be positive and that the phase has to be 0≤θ<2π0≤θ<2π0 \leq \theta < 2 \pi. This still leaves the center with an undefined phase, but you can say that the center has a phase of 0 - this is completely arbitrary.

I understand how to work with the numbers, I am only curious about why the polar coordinate system itself allows for infinite ways to represent a point.

It is not, why do you think so?

Whoops, thanks to Mark44 I see my error in this statement. I meant "How can an angle in the third quadrant be associated with a point in the first quadrant?" Sorry for the confusion. Again, here I understand how to get the numbers, but I'm wondering what it is about the polar coordinate system that allows for this in the first place.

I don't understand that question. (5, π/3) is in the first quadrant. Instead of 5, you can also have any other real number. This example uses 5.

Isn't 5 the radius in this example, meaning that whatever the angular coordinate is, the point is always 5 units away from the pole? Am I wrong that by representing this point in the third quadrant, the radius 5 becomes -5? But, theoretically, if I could represent this point in the second quadrant, wouldn't the radius 5 still be 5?
I'm really wondering why I can't take any point from the first quadrant and represent it with a coordinate in the second or fourth quadrants. Instead of rotating a point by π, why can't it rotate it by π/2 and end up in the second quadrant? I've been told that this is just the way polar coordinates work, but I'm wondering what the reason is.

I hope this makes more sense.

 

[CheeseSandwich]

Mark44, thank you for replying.

As already explained, it can't. However, if your question really is, "How can an angle in the third quadrant be associated with a point in the first quadrant?"
This can happen if you allow r to be negative. For example, (5, 4π/3) is a point in the third quadrant. (-5, 4π/3) is a point in the first quadrant, despite the fact that 4π/3 is an angle in the third quadrant.

Thank you for correcting me. I suppose this is where I got the wrong idea:

Are the equals signs in this example saying that these points are associated with each other rather than equivalent with each other?
How can (5, 4π/3) be in the third quadrant unless the radius is negative?
I still don't understand how (-5, 4π/3) is in the first quadrant.

I must have a huge gap in my intuition about this.

 

[Mark44]

Mark44, thank you for replying.
Thank you for correcting me. I suppose this is where I got the wrong idea:

Is the equals sign in this example saying that these points are associated with each other rather than equivalent with each other?

Edit: Mistake on my part. These aren't the same points. I didn't check them all, but the first point is in the first quadrant, and the second point is in the third quadrant.
All four sets of coordinates represent the same point. In terms of the reference point, '=' is an appropriate connector.

How can (5, 4π/3) be in the third quadrant unless the radius is negative?

(5, 4π/3) is in the third quadrant, period. You might not be understanding how the radius is measured. Start with a ray that extends from the pole (origin) out along the pos. x axis. Let the ray sweep counterclockwise through an angle of 4π/3. If the radius is positive, go out that ray 5 units. This gives you a point in the third quadrant.

I still don't understand how (-5, 4π/3) is in the first quadrant.

I must have a huge gap in my intuition about this.

Yes.
Again, start with a ray going out from the pole to the right. Sweep the ray counterclockwise (for positive angles) 4π/3. The ray is now in the third quadrant. Since r < 0, go out 5 units in the opposite direction from where the ray is pointing. That will be a point in the first quadrant.

 

[CheeseSandwich]

Mark44, thank you for your help.