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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 :)!

Jonatan
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  • $\pi$ is defined as ratio of diameter of circle and the circumference, and just because we can't express it in terms of fraction, it doesn't mean we don't have exact value of $\pi$. – Jack Yoon Sep 28 '16 at 17:41
  • You can approximate the value of to any degree of accuracy,though. –  Sep 28 '16 at 17:50
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    $\pi$ is not only irrational but also transcendental, we even can't construct $\pi$ on the number line in simple ways. – Ng Chung Tak Sep 28 '16 at 17:50
  • Some thoughts: for 1), it depends a lot on what you mean by "find". For the remaining points, it's hard to say - if you're looking for a method that is useful for real-life calculations, then decimal expansions do just fine, and if you're looking for a method that facilitates your algebra and pure maths, then just writing "$\pi$" is perfectly suited to doing that as well. – πr8 Sep 28 '16 at 17:51
  • What do you mean by "know"? $\pi$ does have an exact value that is well defined and we can compute it to as many decimal places as we'd like. In what sense do you think we do not "know" the value? – Alexis Olson Sep 28 '16 at 17:51
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    This question makes crucial use of the concept of knowing a real number. That concept does not (as far as I am aware) have an accepted mathematical definition, so there can be no mathematical answer to this question. If I were to try to define this concept, I might say that we know a number when we have an algorithm for computing its decimal digits, one after another. In that sense, $\pi$ is certainly known. If you have some other mathematical sense of "know" in mind, you should say what it is. – Andreas Blass Sep 28 '16 at 17:52
  • So what i mean is that if we have a perfect circle. We now have a way to calculate an infinitely close value of π. But shouldn't we be able calculate an exact value for pi in the the Hindu–Arabic numeral system, since the area of an perfect circle is fixed, or is the Hindu–Arabic numeral system flawed? – Jonatan Sep 28 '16 at 17:55
  • @Johnny Might also be worth clarifying what you mean by "flawed". Finite decimal expansions are "flawed" in the sense that they contain gaps, but infinite decimal expansions (such as $1/3=0.333\cdots, \pi=3.14\dots)$ manage to "plug" these gaps, in that any real number can be represented in this form. – πr8 Sep 28 '16 at 17:58
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    What is your definition of an "exact number"? Would you claim that "we could never never know" the exact length of the diagonal of a square with side length $1$? – Christian Blatter Sep 28 '16 at 17:59
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    While this is an interesting discussion, it is less about mathematics and more about semantics. Loath as I am to do this, I am going to add a close vote. – copper.hat Sep 28 '16 at 18:00
  • @Christian Blatter To my understanding, and my definition. An exact number in the Hindu–Arabic numeral system would be any number or fraction that wouldn't have to be defined through an infinite series. – Jonatan Sep 28 '16 at 18:01
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    One has to distinguish between our intuitive grasp of noninteger numbers and their bureaucratic representation. It's clear that $\sqrt{2}$ or $\pi$ are "difficult" numbers, but with due respect to the "Hindu-Arabic" culture we should accept ${1\over3}$ as a true number even if it not possible to express it in the "Hindu-Arabic" number system in finite terms. – Christian Blatter Sep 28 '16 at 18:54

4 Answers4

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

Robert Israel
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  • Yes indeed, but π is not represented in the The Hindu–Arabic numeral system, but i'll edit my question. – Jonatan Sep 28 '16 at 17:44
  • @Johnny Neither $e$, $\phi$, $G$, $\hbar$, $\gamma$.... – Enrico M. Sep 28 '16 at 17:46
  • Yes, exactly but for example e is defined as an infinite series of (1+1/n)^n when n -> infinity. And e isn't derived from as much of a real life application as Pi, since pi is defined from an area of a circle which should be exact. – Jonatan Sep 28 '16 at 17:49
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    @Johnny You can define $\pi$ as a series or a limit as well. – Alexis Olson Sep 28 '16 at 17:53
  • @Johnny Say you had a "perfect circle" in real life. How would you measure the area of circle to obtain the exact value of $\pi$? It would almost certainly be a precription where you add up (smaller and smaller) parts to cover the whole area. That is basically the same as what a decimal expansion is: $3+0.1+0.04 +\ldots$. – Winther Sep 28 '16 at 17:54
  • @Winther yes exactly! But shouldn't the value of π be able to represented exact? Not only infinitely close, since the area of the circle is fixed? – Jonatan Sep 28 '16 at 17:58
  • @Johnny when you say "exact", do you mean "as a finite decimal number"? – πr8 Sep 28 '16 at 18:01
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    @Johnny The problem here is that you seem to have your own definition of what exact means. – Winther Sep 28 '16 at 18:01
  • @Winther Yes, could be i'm not really experienced mathematician at all and haven't studied that much math. I'm simply trying to explain my thoughts and questions, and I'm sorry that I'm not being clear enough! – Jonatan Sep 28 '16 at 18:04
  • For example, $\pi = 2 \arcsin(1)$ or $\pi =\displaystyle 4 \sum_{n=0}^\infty \dfrac{(-1)^n}{2n+1}$. – Robert Israel Sep 28 '16 at 18:05
  • @Robert Israel isn't that just an infinitely close approximation? – Jonatan Sep 28 '16 at 18:09
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    @Johnny No, this is one of (infinitely) ways of having $\pi$ written as convergent sum. – Jack Yoon Sep 28 '16 at 18:11
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    No. Partial sums of the series (where you replace the $\infty$ by a finite number) are approximations, but the equation itself is exact. – Robert Israel Sep 28 '16 at 18:12
  • @Robert Israel Yes but apart of my question was about the flaws of the HA numerical system and isn't ∞ "invented" in order to make sense of the HA numerical system? – Jonatan Sep 28 '16 at 18:21
  • The fact that $\pi$ can't be represented as a finite decimal is a feature, not a bug. No, $\infty$ was not "invented" in order to make sense of the HA numerical system. It has very little to do with that system. – Robert Israel Sep 28 '16 at 19:42
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In base-$\pi$, $\pi = 10$. Can't get more exact than that.

B. Goddard
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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!

Ashwin Iyengar
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  • Which isn't to say that there are no circles in the real world: I mean, drop something in a still pool of water and you'll see a wave propagate that looks like a circle: but you have to understand them as something experienced by you before you understand them as a sort of mathematical entity. – Ashwin Iyengar Sep 28 '16 at 17:54
  • I think you've interpreted my question the best so far. But correct me if I'm wrong. Doesn't modern mathematics all derive from the Hindu-Arabic numbering system, and in a sense π was invented after the HA numbering system in order to explain the circle. What I mean by this is that the mathematical number π can't be written, for example, in to a computer in order for one to get an exact number in either 2-base or 10-base numerical system. But shouldn't we be able to calculate one since the area of a circle is fixed? – Jonatan Sep 28 '16 at 18:18
  • Again, put the statement "the area of a circle is fixed" into question. Also mathematics derives, in some sense, from logical axioms, which have somehow given us the "real numbers", which can be defined as limits of sequences of rational numbers. Keep in mind that although computers can do some math, they can't do all of it: for example, in the most intuitive number systems, you can imagine infinity: but a computer has finite storage. Analogously, using the most intuitive number system, you can define the real numbers, but a computer can't concretely pin them down, by construction. – Ashwin Iyengar Sep 28 '16 at 18:22
  • @ Ashwin Iyengar That actually answers my question, and agrees with what i intended to mean. It's obvious that π is the exact number for π. But you can't use π in as you said computer calculation. But maybe i can elaborate with computers in mind; is it impossible to write an exact representation of pii or is the numerical system we use in that sense how we build our computers flawed? – Jonatan Sep 28 '16 at 18:30
  • Well it's "flawed" if the existence of irrational numbers is a flaw, but I would disagree. – Ashwin Iyengar Sep 28 '16 at 19:26
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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.

Yakk
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