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I have seen this question asked countless times online, but almost every time it is misunderstood. What property of a circle causes the ratio between the circumference and 2 times the radius (pi) to equal 3.14159265358979...? What it is it about a unit circle that makes this number somewhere between 3 and 4, and not 4 and 5, or 2 and 3. In other words, I am looking for an informal mathematical proof of the numerical value of pi.

I am not asking: a) For the definition of pi b) For a tutorial in how to measure the ratio of pi with physical instruments c) For an explanation on why we approximate it 3.14, and not 3.14159, or 22/7 d) For someone to remind me that "pi is irrational and 3.14 is not the real value" or "the length of the circumference is different in every circle. Only the ratio is the same."

I understand the explanation is probably not simple. Explanations of the properties irrational numbers usually include infinite fractions or calculus. I just want an explanation of why it is such as seemingly arbitrary value. Thank you in advance!

Note: thanks to your search function I realized that a similar question had already been answered on this site. Why is $\pi $ equal to $3.14159...$? This answer referred to Archimedes's approximation of pi. The details of the explanation were a little confusing though, at least for me. If you want, a description or explanation of the algebraic formula he used to approach the value of pi will answer my question. Thank you!

Electro-blob
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    I'm not sure if there is a very satisfactory answer to what you're looking for. Why should any fundamental constant of nature be what it is, instead of any other? Or, on the other hand, who are we to deem them arbitrary? They're only arbitrary by human reckoning and probably only because they don't look nice - thus why we don't ask similar questions of, say, integers more often than not. – PrincessEev Nov 16 '19 at 03:40
  • That said, one reason behind the value of $\pi$ is tied to how we measure distance (the usual, Euclidean distance), assuming we consider a "circle" to be all of the points equidistant from a given center, and consider a circumference to be the perimeter/circumference of the shape those points make. Changing our definition of distance to other metrics changes what one would consider to be the value $\pi$, relative to that metric: it can be any value in the interval $[3,4]$, depending on which metric you use. – PrincessEev Nov 16 '19 at 03:40
  • More details, examples, and explanation can be found in this PBS Infinite Series video: https://www.youtube.com/watch?v=ineO1tIyPfM – PrincessEev Nov 16 '19 at 03:40
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    Well, it clearly should be less than $4$, since $(2r)^2=4r^2$ is the area of the square which contains a circle. Likewise, it should be greater than $2$ since $(\sqrt2r)^2=2r^2$ is the area of the inscribed square. You can get closer with polygons with more sides. – Rushabh Mehta Nov 16 '19 at 03:42
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    And as to why we approximate it as $3.14$ rather than $3.14159$, that's simply because $2$ decimal places is often enough. If we need more, we use more. – Rushabh Mehta Nov 16 '19 at 03:45
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    The area of the unit circle is $\pi = 2\int_{-1}^1 \sqrt{1-x^2} dx$, which you can approximate as closely as you like. Not the most efficient way to compute pi, but it demonstrates that it is a particular number which you can bound above and below as closely as you like (assuming you're familiar with integrals). – Dustan Levenstein Nov 16 '19 at 03:53
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    I will point out that the comment by DonThousand is precisely the intuition behind the archimedean approximation... So you draw an inscribed and a circumscribed square and see their perimeters. Then move up to an inscribed and circumscribed hexagon and see their perimeters. The inscribed polygon is always a bit too small of a perimeter and the circumscribed polygon is always a bit too large. Keep increasing the number of sides and the perimeters of the inscribed vs circumscribed keep getting closer together, and this gets closer and closer to the perimeter of the actual circle. – JMoravitz Nov 16 '19 at 03:53
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    Eventually, with enough sides, you will find that half of the perimeter is somewhere between $3.14158$ and $3.14160$ and you can repeat this process ad nausem until you get as good of an approximation as you like. Archimedes himself stopped at $96$ sides. – JMoravitz Nov 16 '19 at 03:55
  • Usually, $\pi$ is defined as twice the smallest positive root of the function $f(x)=\cos(x)$. To calculate $\cos(x)$ , you can use the taylor-series , for example. But "why" $\pi$ has the given and not another decimal expansion, is rather a philosophical than a mathematical question. – Peter Dec 01 '19 at 14:18
  • Might be off-topic , but two fun-facts about $\pi$. $1)$ Somewhere in the USA, before $1900$ (I do not remember the year) , $\pi$ was defined by law to be $4$ $2)$ The first $360$ digits of $\pi$ after the comma in the decimal expansion end with $360$. Some people do not consider this to be a coincidence. – Peter Dec 01 '19 at 14:24
  • Because the number of letters in the words "Yes! I knew I could calculate pi easily!" are 3,1,4,1,5,9,2,6. Hey $\pi$ had to conform with something. – Oscar Lanzi Jun 08 '20 at 00:43

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