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2

What you can do is divide the angle at the center of the circle by $n$, which gives angles of $\frac {2 \pi}n$ radians at the center. Now draw radii from the center to the circle at this spacing and make the points where the radii hit the circle. As all the central angles are the same, the lengths of the sides between the radii are all the same. The angles ...


7

As of today, we don't know if $e+\pi$, $e\cdot\pi$ are irrational numbers. This is an open problem. From the expansion $$(x-\pi)(x-e) = x^2 - (e + \pi)x + e\pi$$ one sees that the above polynomial can't have rational coefficients. Thus at least one of $e + \pi$ and $e\pi$ is irrational. We are inclined to think both are transcendental numbers. A ...


1

I'd say it's unbounded but it'll be hard to prove. Here is a picture of the trajectory of the first 100000 points and a graph of the distance to the origin for this trajectory. The largest distance is $\sqrt{45520} \approx 213$.


4

It is extremely unlikely that you're going to be able to get any answer for any question of this sort. Questions about the decimal expansions of irrational numbers are totally unconnected to anything we'd normally consider a good mathematical property of the numbers. To illustrate how complete our lack of understanding is here, we cannot currently rule out ...


2

You can define a circle knowing the centre and the radius (distance $r$).   A circle is the set of all points, on a 2D-plane, at distance $r$ from the centre. That's a concise and elegant definition; try doing so using the diameter (distance $d$). Then, having defined circles using the radius, it becomes convenient to also define the radian measure of ...


0

That image contains a number of factual errors, but the most important one is the misleading assertion that irrationality implies disjunctiveness. One can easily construct an non-disjunctive, irrational number. Let $ r = \sum_{n = 0}^\infty 2^{-n} \begin{cases} 1 & \text{if } 2 | n \\ s_n & \text{else} \end{cases} $ for any non-periodic sequence $ ...


0

Many of the answers above have covered this in a rigorous way, so I'll try to put some intuition behind it. Let $ C_n $ be your parameterization's circumference. You assume that $ \pi = \lim_{n \to \infty} C_n $. The problem lies in that $ \pi $ isn't defined as such limit. In fact, the argument is fundamentally flawed by assuming $ [\forall n \in \mathbb N,...


1

Yes. For each rational number $a$, let $f(a)$ be any rational number $b$ such that $\vert b-\pi\vert< \frac{\vert a-\pi\vert}{2}$. And of course this works for any irrational in place of $\pi$. (The "$1\over 2$" is needed to make sure that $f^n(x)\rightarrow \pi$, rather than just $\vert f^{n+1}(x)\vert<\vert f^n(x)\vert$.) We can even get rid of the ...


0

The roots of $\frac{\sin{x}}{x}$ are the non-zero roots of its numerator $\sin{x}$: $\pm\pi$, $\pm 2\pi$, $\pm 3\pi$... and so on. Using these roots, Euler factored the series for $\frac{\sin{x}}{x}$ as the infinite product: $$\frac{\sin{x}}{x}=\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$$ If you use Euler's formula and set $x=\frac{\pi}{2}$, ...


0

$\pi/2$ is the least positive solution of $0=f(x)=(\sin x) -1.$ Since $f'(x)>0 >f(x) $ for $x\in (0,\pi/2)$ we can use Newton's method to approximate $\pi/2:$ Take any $x_1\in (0,\pi/2)$ and let $$x_{n+1}=x_n-f(x_n)/f'(x_n). $$ We can use the angle-sum formulas together with the power series for $\sin$ and $\cos, $ to more easily compute $\sin x_{n+...


0

I wrote a little program to explore this question by considering all possible numerators and denominators and ranking them by their percent deviation from the "true value" of Pi. #!/usr/bin/env python3 import math #Let's explore all three digits numerators and divisors #Obviously, numerator must be > denominator candidates = [] for numerator in range(...


1

If a number has a finite expansion, in a rational base, using rational digits, then the number is rational. This is because the sum and product of rational numbers is rational. Note: Even some rational numbers have non-terminating expansions in base 10. For example, $1/3$.


2

The function $x^{\frac{1}{x}}$ is strictly decreasing for $x>e$, the maximum is $e^{\frac{1}{e}}$. Therefore: $e\le a<b$ => $a^{\frac{1}{a}}>b^{\frac{1}{b}}$ => $a^b>b^a$ => $a^b-1>b^a-1$ => $(a^b-1)\ln b>(b^a-1)\ln a$ => $b^{a^b-1}>a^{b^a-1}$ => $b^{a^b-1}\frac{\ln a}{a}>a^{b^a-1}\frac{\ln b}{b}$ => $a^{b^{a^b}}>b^{a^{b^a}}$...


4

Starting from $\pi^e\lt e^{\pi}$, we have, by taking the logarithm twice and doing a trivial bit of algebra, $$\pi^e\lt e^{\pi}\implies e\ln\pi\lt\pi\implies1+\ln\ln\pi\lt\ln\pi\implies\ln\ln\pi\lt\ln\pi-1$$ We'll use the two ends of the above in the following, which begins by taking a logarithm, then does some trivial algebra, and ends by exponentiating ...


8

We use the following fact in the proof: Let $c > 0$. Then $\ln(x + c) < \ln(x) + c$ for $x \geq 1$. For notational convenience, we use the notation $f(x) \rightarrow g(x)$ to denote that $g(x) = \ln f(x)$. We have $$ e^{\pi^{e^\pi}} \to \pi^{e^\pi} \rightarrow e^\pi\ln \pi \to \pi + \ln\ln \pi $$ and $$ \pi^{e^{\color{red}{\pi^e}}} < \pi^{e^{\...


-9

Do you know why $3^{2} \ge 2^3$ (2 less that 3) but $3^{4} \ge 4^3$ (3 less than 4)? When that is change? The limit is $e$ If you know $e^\pi \ge \pi^e$ and know this reason, then you know that question which is bigger $e^{\pi^{e^{\pi}}}$ or $\pi^{e^{\pi^{e}}}$ is the same with question which is bigger $3^{4^{3^{4}}}$ or $4^{3^{4^{3}}}$ Second you can ...


-6

By WA, your first expression equals approximately $$4.3928231369985664320825774950093661823846807534793830... × 10^{30}$$ and your second expression $$1.871461075707814429342563096957936841416990167360729... × 10^{31} $$ which means that your second expression (the one where $\pi$ is the base) is slightly bigger.


0

The fundamental concept here is discontinuity. The arclength of a curve is a discontinuous function of its path, in the sense that two paths can be arbitrarily close (in the visual or point-by-point sense) but have dramatically different arclengths. You can take any discontinuous function and build a dumb apparant-paradox in the same style. The sign of a ...


0

Pi is the ratio between the circumference and the diameter. If you multiply a length (diameter) by pi you get the circle that has that length has a diameter. If you multiply by 3, you don't get the whole circumference, because you will be missing a piece. In order to get a circle again, you need to "curve" the space. Depending on whether you curve the space ...


0

Here is a short implementation in Python: from decimal import Decimal, getcontext def pi(precision): getcontext().prec=precision return sum(1/Decimal(16)**k * (Decimal(4)/(8*k+1) - Decimal(2)/(8*k+4) - Decimal(1)/(8*k+5) - Decimal(1)/(8*k+6)) for k in xrange(precision)) print pi(1000)


3

Your argument also proves that $e^{\ln2}$ is not rational.


4

This proof is not correct. The fact that $e$ is irrational means that you can't write $e=\frac{p}{q}$ where $p$ and $q$ are both integers. Your $p$ and $q$ are not integers (at least not obviously so), so you don't get a contradiction. Every number $x$ can be written as a fraction $\frac{p}{q}$ for some $p$ and $q$ (for instance, $x=\frac{x}{1}$); this ...


2

The irrationality measure of $\pi$ is less than $8$. This means that $$ \Bigl|\pi-\frac{n}{m}\Bigr|>\frac{1}{m^8} $$ for all $n$ and all $m$ suficiente large. Thus, for $m$ large enough, $$ \epsilon(m)\ge\frac{1}{m^8}. $$


0

pi can be experimentaly proven to differ significantly from 4, eg: use a piece of string and a cylindrical can. considering this demonstration in the cotext of the faulty proof should quickly lead to an understanding of flaws of the rectilinear approximation of the circumference.


1

What the sine function really is, for example, is simply the ratio between two of the sides in a right triangle. If we know one angle is 90 degrees, and we know the angle we are working with (the one we are taking the sine of), then we know the third angle since they have to add to 180. Then, we know the ratio of the lengths of any two sides of the ...


3

I think there are really two questions here: How did Archimedes find the length of the side of a right triangle opposite to a specified angle? How does a calculator evaluate trig functions? For #1: Honestly, I think he just drew it and measured it. This then leads to the question, "How did they measure length in general?" and maybe that's actually what ...


0

The calculator uses Taylor series for trigonometric functions. To find the true value, you will need to add an infinite number of terms, but for calculator's precision, usually a reasonably small number is enough. These are the ones the calculator is likely to use: $$ \begin{split} \sin x &= \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - ...


1

You're almost there in your reasoning. However, you're treating the terms diameter and circumference as UNITS in your explanation. This is odd because the diameter is not a unit; nor is the circumference. They are scalar measurements of some other unit, such as centimeters, for example. If you have a diameter of 100cm, then the circumference would be about ...


1

This is a rounded representation with all exact digits, so you can infer $$3.141592653585<\pi<3.141592653595.$$ (Equality is not possible as $\pi$ has an infinite decimal expansion.)



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