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I saw a post a couple of posts about "proving that $\pi$ = $4$", and I thought of using a naive way to say that it is not impossible.

We know that $\pi$ is commonly defined like this:

$$\pi = \frac{\text{Circumference}}{\text{Diameter}}.$$

How will you measure the circumference? That is equivalent to saying that "What metric I will choose to measure the length of this curved line?" The important point is that the metric we choose will affect what value of $\pi$ we will get.For example, we will get $\pi = 4$ if we choose the taxicab metric $(|\Delta x|+|\Delta y|)$ and we will get $\pi = 3.1415...$ if we choose the Euclidean metric $(\sqrt{\Delta x^2 + \Delta y^2})$.

In short, we will get different answers based on the underlying assumptions we are taking beforehand. Is my approach wrong?

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I'd say that in the end it simply depends on how you define $\pi$. Most people (especially non-mathematicians) know the definition you mention, i.e. $\pi$ is the ratio between the circumference and the diameter of a unit circle. Implicitly we use the euclidean metric to define what a unit circle is (because who wouldn't think of the euclidean metric in everyday life first, right?).

Now I think what you are doing is exactly the 'easiest' way to justify an assessment like $\pi = 4$. But we'll have to deviate from the 'everyday life definition' above in the following way. Let's take a different metric, e.g. $d(x,y) = \max(|x|,|y|)$. What is the unit circle $S$ in this metric? It's

$$S = \{(x,y)\in \mathbb{R}^2 | d(x,y) = \max(|x|,|y|) = 1\} $$

You can easily draw this circle, and you will see that it turns out to be the square of length 2 centered around the origin, which has of course circumference 8. If we now apply our definition of $\pi$ being equal to the circumference divided by the diameter, we get $\pi = \frac{8}{2} = 4$. (Small remark: you get a different result when using the taxicab metric as you proposed. You can also draw the unit circle with respect to the taxicab metric to figure it out!).

So as often: It depends on your definition.

I hope this helps!

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  • $\begingroup$ I guess the definition of pi we choose beforehand is also a assumption. $\endgroup$ – Prithu biswas Feb 2 at 8:17
  • $\begingroup$ Sure, I mean in principle $\pi$ is just a greek letter. A lot of people think of $\pi$ as the mentioned ratio just because it has been defined that way a long time ago, and has been used consistently enough that it stuck. So if you don't say anything, it's ok to assume that by $\pi$ we mean this ratio. But $\pi$ (the letter) can also be used for plenty of other definitions. $\endgroup$ – noam.szyfer Feb 2 at 8:55

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