Can we ever know the exact value of π

Question

Edit as @copper.hat said, this topic is meant to be"less about mathematics and more about semantics"

Hi so i was discussing the definition of π and the subject of finding an exact value for π with a friend of mine, and here are some thoughts that came up. Edit: In reference to the Hindu–Arabic numeral system (0-10)

• If one could create or at least imagine a perfect circle, and know the radius and, for example, area of said circle, shouldn't one be able to find an exact value for π?
• π has been proven to be an irrational number and therefore cannot be expressed as a ratio of integers and, when written as decimal numbers, do not terminate, nor do they repeat. But couldn't that be the case because our numeral system might be flaved?
• So could one ever come up with an exact value for π by for example inventing a new numeral system or is it impossible for a human to ever find it?
• And if someone if of the opinion that you can never find an exact number for π, wouldn't that mean that we could never never know the exact area of a circle? The exact area should exist as far as i know but is it just that we wouldn't be able to know it?

I'll update more thoughts if the discussion gets going, or simply rest my case if i have been thinking about this from the wrong angle.

Be nice :)!

Answer 1

The concept of a circle, in the context of $\pi$, is a purely mathematical one, as far as it's defined and used regularly by scientists and mathematicians. You could say that in your life, you've encountered "circles", but $\pi$ relates to the mathematical object "circle", which can be defined, for example, by $$C = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$$ as a purely mathematical object. So asking whether you can "actually find the exact area of a circle" assumes that you can actually find circles in your own reality (by your own conception, I suppose), which assumes that the universe you live in respects the mathematical system you have prescribed it: in your case, you're saying that you can only find the exact area of a circle if it can be described with the Hindu-Arabic numbering system precisely: but this assumes that your universe follows the laws of the Hindu-Arabic numbering system. Furthermore, you're positing that the "exact area" makes sense in your reality, which is another assumption.

Basically I'm saying you should think more on your first statement: "If one could create or at least imagine a perfect circle".

Also, one could say that the Hindu-Arabic numbering system is not "flawed" in a mathematical sense, but applying it to the real world without serious thought may lead to contradictions and complications like the one you've described.

Hope this helps!

Answer 2

The exact value of $\pi$ is $\pi$. It is a perfectly well-defined and specific real number.

Answer 3

In base-$\pi$, $\pi = 10$. Can't get more exact than that.

Answer 4

You have stated that an exact definition number is a number whose definition does not involve an infinite series.

Then $\frac{\pi}{4}$ is the probability that, if you pick a random spot uniformly in a square of side length 2, you will pick a point within the unique unit circle centered within the square.

$\pi$ is the unique ratio of a circles circumference to its diameter.

$\pi$ is the unique ratio between the square of the radius of a circle and its area.

$\pi$ is the unique real number such that $e^{\theta i \pi}$ as $\theta$ varies from 0 to 2 describes a uniform speed path once around the unit circle in a counter-clockwise direction.

These are a definitions of $\pi$ that is "exact" and never uses an infinite series. (The definition of "area", "circumference", "random spot", and "$e$" may use limits.)

Converting this definition to a decimal representation, or determining if it is greater or less than some other value, will require an equation involving a limit or other infinite process.