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?


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!

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.