How do we specifically get to 3.14 from axioms through logic to that particular transcendental number being the case?  I'm guessing it's the sum of an infinite series. But how do we go from "there's a circle" to "this particular infinite series describes the ratio of the diameter to the circumference", other than getting out a tape measure, and noticing that the series seems to produce the same number (sans any proof). I'm looking for the proof of why the circumference/diameter = 3.14... specifically.
71187
When I was a child I was taught $\pi$ (Circumference/Diameter) is an irrational number  and can be approximated to $22/7$ but $= 3.(142857)(\ldots)$.      But where does this value comes from?  In short How do I derive $\pi$?
42119
Consider any integer, suppose a number 3. We can break any such integer down into a sequence of 1's. So, we can write: $$ 3\equiv1+1+1 $$ 3 represents what we get when we add 1 to 1 to another 1. Conceptually, what I am getting confused with, is the fact that notionally, at least according to this definition, there is one unit at the beginning, to which we add another, and yet a third. There seems to be a sequence here, which would imply there being an order. Am I missing something obvious?
910
I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.
42671
Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$ When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi = 1.61803398875...$. What if I were to plug in $x=2$? $$2+\cfrac{1}{4+\cfrac{1}{8+\cfrac{1}{16+\ddots}}}$$ Using a caluclator, it seems to converge upon the irrational number: $2.24248109286...$ Is there any way I can represent this irrational number algebraically?
57877
How can I prove that the number formed by concatenating the primes in order i.e. $0.235711131719...$ is irrational.  I know I have to demonstrate that it has no period, but I'll be so thankful if someone can explain very clear, including all cases.
24151
Most math texts claim that $\pi$ is an irrational number. However, I'm having a little bit of trouble understanding that.  Since nobody has calculated all of the digits of $\pi$, how can we know that either:   one of the digits repeats (as in $\frac{10}{3}$) the number eventually terminates     Note: Please be very descriptive in your answers... I don't have anything beyond high school math.
2753
I've been told that the rational numbers from zero to one form a countable infinity, while the irrational ones form an uncountable infinity, which is in some sense "larger". But how could that be? There is always a rational between two irrationals, and always an irrational between two rationals, so it seems like it should be split pretty evenly.
46761
Inspired by a paper (from 2001) entitled :  Why is $\pi$ = 3.14... instead of 6.28... ?  Setting $\pi$ = 6.28 would seem to simplify many equations and constants in math and physics.  Is there an intuitive reason we relate the circumference of a circle to its diameter instead of its radius, or was it an arbitrary choice that's left us with multiplicative baggage?
110231
I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we can verify for every $\pi$ digit discovered till now with a "standard formula", how do we know that the formula is correct after those digits?  Also, what's the implication of these kind of BBP formulas for other "divine"/fundamental constants? can we learn anything else about the value of such constants and our universe through these? Any insight into the "supreme fascist" mind?
46696
I am a high school math teacher who likes to understand the development and rationale behind formulas, albeit I am not a math expert by any means.  Could I get some help in trying to explain how the formaula for the sum of a finite geometric series actually can be derived?  I need a a rather simple explanation initially, so that I can grasp the essential nature of the formula.  Thanks for allowing me some space here.
90857
Earlier today I thought I proved that the following series diverged:  $$\sum_{n=2}^\infty\frac{\phi(n)}{n^2}$$  as a result of a misapplication of the prime number theorem. I mistook $\phi(n)$ for $\pi(n)$ in the statement. Is this salvageable? I've been trying to find lower bounds for $\phi(n)$ that I may pass to, but the literature on this is a bit dense for me.  Thanks
78442
Just as circumference of circle will remain $\pi$ for unit diameter, no matter what standard unit we take, are the speeds of light and sound irrational or rational in nature ?   I'm talking about theoretical speeds and not empirical, which of course are rational numbers.
67247
Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal  assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for non-tautological reasons, $\aleph_\alpha$ is provably larger than the cardinality of the continuum? I would hope not since it would seem pretty silly if something like $\alpha = \omega_0$ worked and we could say "well gee we can't prove that $c = \aleph_1$, but it's definitely one of $\aleph_1, \aleph_2, \ldots , \aleph_{73}, \ldots$". I (obviously) don't know jack squat about set theory, so this is really just idle curiosity. If a more precise question is desired I guess I would have to make it     For any countable ordinal $\alpha$ is the statement: $c &lt; \aleph_\alpha$ independent of ZFC in the same sense as the continuum hypothesis?   assuming that even makes sense. Thanks!
8142
I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much
48579
Is it possible to generate transcendental numbers of arbitrary irrationality measure ? I am thinking of something like an infinite series which depends on a parameter fixing the irrationality measure.   All I have found so far is bounds on the irrationality measure of famous numbers, I have not been able to find something going the other way around.
106378
The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle.  How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same for every circle?  How do we know that $\pi = {C\over d}$ for any circle? Is there a proof that states the following: Given any circle we know that $\pi = {C\over d}$. Doesn't such a statement require a proof considering $\pi$ is used so widely on problems involved with circles, spheres, etc. How do we truly know that the value $\pi$ is correct for all circles?
106849
(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.