What are polar coordinates of the origin and why is $\arg(0) =$ undefined, an option?

Question

I have come here from reading this and but it didn't really answered my question.

Wikipedia says the cartesian co-ordinates are converted using $$x=r \cos \theta$$ $$y = r \sin \theta$$ where $r≥0$ but that doesn't make sense because those conversions are found using the definitions of $\sin$ and $\cos$ which are $$\sin \theta = \frac{y}{r},$$ $$\cos \theta =\frac{x}{r}$$ where $r>0$ and thus $r≠0$.

So for $(0,0)$ those conversions are not valid.

Also from this answer, we may let $\theta = \arg (0,0)$ be undefined or $\mathbb{R}$.

My question is why is "$\arg (0) =$ undefined" even an option when all real numbers are perfect candidates for it? Why is the "convention" of leaving it undefined even a thing? What are we afraid of? Multiple values? Because surely, $\arg(z)$ has multiple values even when $z≠0$. So why "undefined" for $z=0$?

Answer 1

The origin has polar coordinates $ (r,\theta)$ defined as $$ ( 0, t )$$ for any polar angle $t$.

Let me say with some caution... in rectangular coordinates since the origin is $$ (0,0)$$ we tend to continue the habit or custom and assume it as some sort of convention.

In cases where our curve is going through the origin it presents a minor difficulty. For example the circle with polar equation $r= \sin (\theta-\dfrac{\pi}{4} )$ can still be thought of as having polar coordinates $(0,0)$ at the origin. However we like to take the point representation as

$$(0, \dfrac{\pi}{4} )$$

rather than $(0,0) $ in order to avoid discontinuity in the polar angle at $\theta=\dfrac{\pi}{4}$, derivatives takes taken would be smooth continuous.

Answer 2

You have two questions here which are only slightly related to each other.

TL;DR: The polar coordinates of the origin are $(0,\theta)$ for any real $\theta.$

TL;DR: Setting "$\arg(0) = \text{undefined}$" is an "option" because we write definitions to suit our needs. But the "option" is just that, an option and not a necessity.

Polar coordinates

Polar coordinates are a sometimes-useful way of mapping pairs of real numbers to points on a Euclidean plane. In order to be able to make full use of polar coordinates, for example to plot the function $r = 2 \cos\theta,$ we want every pair of real numbers $(r,\theta)$ to map to a point.

We can get that point conceptually by rotating the point with Cartesian coordinates $(r,0)$ through an angle $\theta$ around the origin. To give a formula for the Cartesian coordinates of that point, we use the functions $\sin$ and $\cos$. The function $\sin$ can be defined in many ways; some frequently-cited definitions are:

1. The ratio of the opposite side to the hypotenuse in a right triangle.

2. The $y$ coordinate reached by traveling a certain distance around the unit circle.

3. The solution of the ODE $f''(t) = -f(t)$ with conditions $f(0)=0$ and $f'(0)=1.$

4. $\sin t = t + \dfrac{t^3}{3!} + \dfrac{t^5}{5!} + \dfrac{t^7}{7!} + \cdots + \dfrac{t^{2k+1}}{(2k+1)!} + \cdots.$

Only the first definition comes from the ratio of two numbers, and it needs to be supplemented with a lot of seemingly arbitrary extra rules to work for angles outside the interval $\left(0,\frac\pi2\right).$ So we might prefer the second definition to use for polar coordinates, since we can use it in all four quadrants without any further fuss.

In any case, $\sin$ is simply a function from real numbers to real numbers. You can put any real number in as its input parameter, and out comes a value. Similarly with $\cos.$

In any event, we find that after rotating the point with Cartesian coordinates $(r,0)$ through an angle $\theta$ around the origin, its image has Cartesian coordinates $(r\cos\theta, r\sin\theta).$

This is perfectly well-defined even if $r = 0,$ because we didn't tell the $\sin$ or $\cos$ functions what $r$ is, only what $\theta$ is. Even if you somehow found a book that used the symbol $r$ in its definition of $\sin,$ the $r$ in that definition is meaningful only within that definition and is a completely different variable from the $r$ in the polar coordinates $(r,\theta).$ Borrowing a bit of the language of computer programming, the variables we see in the definitions of functions are only "local" variables, not "global."

So when choosing polar coordinates $(r,\theta)$ for a point, let $r = 0$ and pick any real $\theta$ you want; the $\cos$ function is defined on all real numbers, so $\cos\theta$ is defined and is a real number and therefore $r\cos\theta = 0\cos\theta = 0.$ Similarly $\sin\theta$ also is a real number and $r\sin\theta = 0\sin\theta = 0.$ It's that simple. The Cartesian coordinates of this point are $(0,0).$

Conversely, if you're looking for "the" polar coordinates of the origin, any polar coordinates of the form $(0,\theta)$ for real $\theta$ are possible.

Defining or not defining $\arg(0)$

The complex argument function $\arg$ is used in complex analysis. It can be defined as a single-valued or multi-valued function; see the Wolfram MathWorld article for details.

There is some relationship between the complex argument and the angle in polar coordinates, but they are not the same concept. For example, while the polar coordinates $\left(-\sqrt2,\frac\pi4\right)$ name a point with Cartesian coordinates $(-1,-1),$ we are not going to accept $\frac\pi4$ as the argument of $-1 - i,$ or even as one of the many values of the argument of $-1 - i.$

We might or might not define $\arg(0).$ I refer you to the comments by Daniel Fischer under the question you cited, what is the argument of 0?, one of which is:

Yes, either undefined, or any real number is an argument of $0$. Whichever choice is more convenient.

What is "convenient," then? It might be convenient to define $\arg(0)$ because we have a particular use for it.

On the other hand, it might not be convenient. We might have defined $\arg$ as a single-valued function with a finite range such as $(-\pi,\pi].$ To add $0$ to the domain, we'd have to pick a single value for $\arg(0).$ If we defined $\arg$ as a multi-valued function, on the other hand, the values of $\arg(z)$ for any non-zero $z$ are the equivalence class of a single value in $(-\pi,\pi]$ modulo $2\pi.$ Defining $\arg(0)$ as real numbers breaks that pattern.

Consider the identity on complex numbers, $$ \arg(wz) = \arg(w) + \arg(z). \tag1 $$ As described in the MathWorld article, if $\arg$ is single-valued, this identity is only true modulo $2\pi.$ But if we define $\arg(0),$ then $\arg(0) = \arg(0\cdot i) = \arg(0) + \arg(i),$ which implies that $\arg(i) = 0$ modulo $2\pi,$ which is false.

On the other hand, if $\arg$ is multi-valued, we might want to manipulate $(1)$ algebraically to obtain $$ \arg(z) = \arg(wz) - \arg(w). \tag2 $$ Then if $w = 0$ we get $\arg(z) = \arg(0) - \arg(0).$ I think it would be challenging to find an interpretation for this in which both $\arg(1) = \arg(0) - \arg(0)$ and $\arg(i) = \arg(0) - \arg(0)$ were correct equations.

Not defining $\arg(0)$ means that we can write something like $(2)$ as an identity by using the convention that identities apply only when all their terms are defined, without having to explicitly list the exceptions. That might be considered "convenient."

So nobody decides how to define $\arg$ because they are "afraid of" something. They define $\arg$ in a way that suits their purposes. Not defining $\arg(0)$ is only a "convention" insofar as there are a lot of mathematical works in which that decision suited the authors. It's not a universal convention, because sometimes it suits someone to define $\arg(0).$