We know, for example, the constant $\pi$ is the perimeter of a circle with diameter $1$ unit. In the similar manner how would we explain the constant $e$. I have searched a lot for it. But I couldn't comprehend it in a practical way. Can anybody help me to understand the constant $e$?
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Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.    I know there are many ways of calculating (or should I say "ending up at") the constant e.  How would you explain e concisely?  It's a rather beautiful number, but when friends have asked me "what is e?" I'm usually at a loss for words -- I always figured the math explains it, but I would really like to know how others conceptualize it, especially in common-language (say, "English").     related but not the same:
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We all know this magical mathematical constant.   My question being , how and why pi just shows up in every other physics derivation or formula or even statistics for that matter .    Is it just coincidence or some actual reason?
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Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk. If you divide the circumference by the diameter, you will get exactly $\pi$. Now you start rotating the disk. A book I have claims that now the ratio will be different (lets call it $\pi_\circ$ to avoid ambiguity.) This has caused to reevaluate my understanding of special relativity (this is a set up to general relativity, but the problem itself only requires the special theory.) My problem is that, sure, the measuring rods (or tape) would contract on the circumference, but wouldn't the circumference contract a similar amount, therefore giving a measurement of $\pi$ in every reference frame according to your own measuring rods. Of course, this reasoning goes on to make me question how I understood all of special relativity up this point.
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Let $C$ denote the unit circle  centered at the origin in Complex Plane  What is the value of   $$ \frac{1}{2\pi i}\int_C |1+z+z^2 |dz,$$ where the integral is taken anti-clockwise along $C$?   0 1 2 3   What I have answered is 0 because it seems like $f(z)$ is analytic at 0 hence by Cauchy's Theorem.
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When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.  Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi r^2$.  Is this just a coincidence, or is there some deep explanation for why we should expect this?
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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.
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In this wikipedia, article  its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all, I would like to see different proofs, for this result. (If there are any elementary ones!)  One, interesting observation, which one can think while seeing this problem, is: How does one propose such type of problem? Does, anyone take all closed curves, and calculate their area to come this conclusion? I don't think that's the right intuition.
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What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
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Is there a fundamental way to look at the universal constants ? can their orders of magnitude be explained from a general points of view like stability, causality, information theory, uncertainty?   for example, what sets the relative magnitudes of Planck's constant compared to say charge of electron or is it just a matter of choice of units.   Does the physics become more or less cumbersome, insightful if we set all fundamental constants to 1 in appropriate units.
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I am fairly sure that I understand the how Monte Carlo integration works but I am not understanding the formulation of how it is used to estimate Pi.  I am going by the procedure outlined in the 5th slide of this presentation   I understand the preliminary steps.  Pi is equal to 4 times the area of a quarter of the unit circle.  And the area of the top-right quarter of the unit circle centered at (0,0) is equivalent to the integral of the curve that is the top-right quarter of the unit circle in $0&lt;x&lt;1$ and $0&lt;y&lt;1$.    What I don't understand is how this integral is  $\iint I((x^2+y^2)&lt;1)P(x,y)dxdy$  where $P(x,y)$ is uniformly distributed in the unit square around the quarter circle (i.e. it is always equal to 1 if $0&lt;x&lt;1$ and $0&lt;y&lt;1$ and 0 otherwise).  So this would mean that  $I((x^2+y^2)&lt;1)P(x,y)$ is the function that is the top-right quadrant of the unit circle at $0&lt;x&lt;1$ and $0&lt;y&lt;1$ but I do not understand how this is true since the indicator function can only be 1 or 0.  I understand that it is probably written in this way to make Monte Carlo sampling easy (i.e. it is an expectation so just sample from $P(x,y)$ and get the average of the samples applied to $I((x^2+y^2)&lt;1)$) but it just does not make intuitive sense to me why that integral represents the area under that curve.  Could someone provide an intuitive explanation of this.  Maybe show how that integral was derived in a step-by-step way?  EDIT:  I was able to gain a better understanding by relating the expectation to an area.  I will explain it here in case it helps anyone.  First start with relating Pi to the area of the top-right quadrant of the unit circle  $\pi=4\times A_{tr}$  Then we place the top-right quadrant into the unit square.  And under a uniform distribution over the unit square, the area of the circle quadrant is proportional to the probability of obtaining a sample from it.  It follows that the following equality holds  $P(x^2+y^2&lt;1)=\frac{A_{tr}}{A_{square}}$  and $A_{square}=1$ so  $P(x^2+y^2&lt;1)=A_{tr}$  And substituting into the original equation  $\pi=4\times P(x^2+y^2&lt;1)$  and it is also true that $P(x^2+y^2&lt;1)=E[I(x^2+y^2&lt;1)]$ which is equal to the original double integral.  So I understood it by relating the area to a probability then relating that probability to an expectation that is equivalent to the integral.  Let me know if I have made any mistakes.
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Can someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle.
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The first question that comes into my mind here is whether any cylinder that touches(at 4 pts) the circumference of the sphere and does not go out of it, has equal volume?   Second, how do i mathematically limit the volume of the cylinder to be less than that of a sphere? Squeeze theorem?   Please help, thanks!
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I often see $I(t)\propto |E(t)|^2$. What is the exact form ? Which constants are missing to make this an equality and why are they so often omitted ?
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Foreword  This question was inspired by initial mistakes in . I wanted to explore the strange circle with $A&gt;\pi r^2$ and got lost into geometrical jungle.  A  is usually described by it's height $h$, radius of the base $a$ and radius of the sphere $r$. Besides we have a relation $a^2+(r-h)^2=r^2$ so the cap actually only has two degrees of freedom.  When considering a case of circle on a sphere (like the Earth), it's the surface of a spherical cap, but we can't see the height or other classic parameters of this spherical cap. Instead we encounter parameters like area $A$, perimeter $C$ and diameter $d$. I will try to write formulae for these: $$A=2\pi rh$$ $$C=2\pi a$$ $$d=2r\theta$$  Where $\theta$ is introduced with $\sin\theta=\frac{a}{r}$ or $\cos\theta=(1-\frac{h}{r})$.  As the cap only has two degrees of freedom, it should be a good problem with unique solution if only two values are known.  A-C Problem  The simple example: $A$ and $C$ are given. I show this only as example before the other tasks.  We can easily find $a=\frac{C}{2\pi}$ and then solve:  $$A=2\pi rh = 2\pi r (r-\sqrt{r^2-a^2})$$  to obtain  $$r=\frac{A}{2\sqrt{\pi(A-\pi a^2)}}$$  Parameters $a$ and $r$ are now known and it is straightforward to find $h$, $d$, $\theta$ and anything else.  C-d Problem  $C$ and $d$ are given.  First step once again is easy: $a=\frac{C}{2\pi}$. The knowing of $a$ suggests that we might prefer the $\theta$ definition with the sine. We obtain: $$d=2r\theta=2r\arcsin\frac{a}{r}$$  That's where I got stuck. Is there an easy way to solve this? Is there a hard way to solve this?  I tried solving numerically with Wolfram Mathematica and it went into complex numbers. When solving the geometrically equivalent $\sin\frac{d}{2r}=\frac{a}{r}$ I got some roots, but the search seemed very unstable and not finding the root closest to initial value. Besides it was finding a lot of different roots while I am pretty sure that there should only be one $r$ for each pair of $d$ and $a$. What would be the right formulation to solve this problem numerically?  A-d problem  $A$ and $d$ are given. This seems the hardest one. We obtain a system of equations:  $$A=2\pi r h$$ $$d=2r\arccos\left(1-\frac{h}{r}\right)$$  (The first equation suggested solving the problem in $h$ and $r$).  Is this system solvable analytically? Or numerically? My experiments with Wolfram Mathematica never converged.   Of course, I could express $h$ or $r$ from the first equation and leave only the second in one of these forms: $$d = 2r\arccos\left(1-\frac{A}{2\pi r^2}\right)$$ $$d = \frac{A}{\pi h}\arccos\left(1-\frac{2\pi h^2}{A}\right)$$  But I have no idea how to solve these either.
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Background:  I recently began taking calculus and it has come to alter the way I look at circles, and curves.  The equation of a circle is $\pi r^2$, traditionally in school we have always left the answer in terms of pi (i.e.) if the radius $r=2$ then the area $A = 4\pi$.    Question:  If one were to attempt to write the area of a circle in decimal form (i.e.) if the radius=2 then the area $A = 4\pi$, but $\pi$ doesn't have an end, it has (per what I have learned in school) an infinite number of decimal places so it is $3.14159\ldots$ therefore if one multiplied $4\cdot 3.14159\ldots$ one would have to approximate ones answer.    Does that mean that it is impossible to calculate the exact (without approximation) area of a circle?  Thanks for any responses,  Joel
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The author is trying to prove it is impossible to square a circle of radius $1$ under the assumption that $\pi$ is transcendental. Even with this assumption, the author explains that if the circle can be squared, then we could draw a line with distance $\pi^{1/2}$. Everything good at the moment. Then he claims that, if such construction is possible, we could also draw a line of distance $\pi$, and that the proof would, under the stated assumption, be complete.     The thing I don't get is how, from a line with distance $\pi^{1/2}$, we could create a line with distance $\pi$.
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Just a curiosity. I just saw a video where pi was being calculated by the series  $$1-(1/3)+(1/5)-(1/7)+...$$  My question is how can $\pi$ be irrational if we can calculate it using rational numbers?