1
vote
0answers
39 views

What is the limit of $k^2|\pi-n(k)/k |$, where $k$ minimizes $|k\pi -n|$?

Let $k\in \mathbb N$ and for any such n, let $k=k(n)$ minimizes the distance $|k\pi-n|\leq 2 \pi$. It is clear that, by fixing the value of $n$, it is possible to choose $k$ (and vice versa). ...
0
votes
2answers
70 views

Lambert's Original Proof that $\pi$ is irrational.

I am trying to find Lambert's original proof that $\pi$ is irrational. Wikipedia has a little description but it is quite lacking. Can someone direct me to Lambert's original proof or post his proof ...
8
votes
0answers
115 views

The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
6
votes
1answer
177 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
3
votes
3answers
128 views

Is it possible that $\pi$ is finite in other numerical bases?

In base $\pi$, the number $\pi$ is $1\cdot \pi^1 + 0\cdot \pi ^ 0 $, which is equal to $10$. So, is $\pi$ an irrational number in all bases or not?
0
votes
1answer
29 views

Is there any combination of numbers which upon division gives the exact number of P?

In other words, there are (probably) infinite combination of numbers/operations which leads to irrational numbers. So I wonder, if there is one which gives exact number representation of P(π)? Do we ...
3
votes
2answers
97 views

Understanding proof that $\pi$ is irrational

Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part: Since $n!f(x)$ has integral coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and ...
0
votes
3answers
117 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
2
votes
1answer
106 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
0
votes
4answers
95 views

why the occurrence of 4,5,6 and 9 in pi differs?

i´m playing around with pi, i have this document with the first 5million decimal numbers after comma. http://www.aip.de/~wasi/PI/Pibel/pibel_5mio.pdf and i build a script that i put in for example ...
0
votes
1answer
80 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
6
votes
1answer
177 views

Predicting digits in $\pi$

Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational? Basic assumption is that the person doesn't know a ...
-3
votes
1answer
142 views

how $\pi$ is irrational if it is a ratio [duplicate]

How can $\pi$ be an irrational number if it is a ratio of the circumference over the diameter? Thanks!
3
votes
2answers
546 views

Pi might contain all finite sets, can it also contain infinite sets?

In a previous, and quite popular, question it was discussed about whether or not $\pi$ contains all finite number combinations. Let us assume for a moment that $\pi$ does in fact contain all finite ...
-1
votes
2answers
2k views

Are “perfect” circles mathematically impossible (and do irrational numbers exist)? [closed]

It occurred to me that while $\pi$ is an irrational number, it is nevertheless the ratio between the circumference and diameter of all circles. This seems like a contradiction. Thinking about it ...
5
votes
3answers
661 views

Is $e^{i+\pi}$ irrational or not?

Since we know that the value of $e$, $i$, and $\pi$ are irrational reals, how about $$e^{i+\pi}\;?$$ Is it still irrational (that is, not a Gaussian rational)? The problem make me curious until now.
1
vote
1answer
132 views

Area of a circle is $A = \pi r^2$. Is it possible that both $A$ and $r$ are perfect integers.

Can you produce an example where both the area of a circle and it's radius are integers?
20
votes
1answer
541 views

Any proof to $\pi^{e}$'s irrationality?

I've searched for this for a while but get nothing... There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
2
votes
3answers
415 views

Can an irrational number have a finite number of a certain digit?

This question came up because I was wondering the following: If the digits of PI are placed in ascending order, what is the <insert-large-finite-number-here>th digit? I believe that the answer ...
13
votes
2answers
1k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
2
votes
0answers
122 views

Irrationality of $\pi$ from the spigot algorithm?

The spigot algorithm for BPP formula gives hexadecimal digits of $\pi$ one at a time. Is it possible to prove directly that this algorithm cannot be computed with bounded-memory? (From R J Lipton It ...
5
votes
2answers
409 views

Irrational Numbers Containing Other Irrational Numbers

Does $ \sqrt{2} $ contain all the digits of $ \pi $ in order? Does it contain all the digits of $ \pi $ in order an infinite number of times? Does $ \pi $ contain all the digits of $ \sqrt{2} $ in ...
3
votes
2answers
326 views

Is there any sans-calculus proof of irrationality of $\pi$?

Is there a proof that will convince someone who doesn't understand calculus, of $\pi$'s irrationality .
5
votes
1answer
117 views

A rope and Pi's irrationality

Here is a question which has been puzzling me for some time. You have a thin rope of an integer length $L$. You can bend it to create a rectangle of perimeter $L$. Fine so far. Next, through some ...
17
votes
2answers
933 views

Why is $\pi$ irrational if it is represented as $c/d$?

$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction. Then why is $\pi$ an irrational number?
23
votes
5answers
3k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
16
votes
2answers
429 views

Which results depend on the irrationality of $\pi$?

Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it ...
5
votes
1answer
396 views

Is it ever really Pi Time?

Walking with my son at 3:14pm the other day, I mentioned to him, "Hey, it's Pi Time". My son knows 35 digits of $\pi$ (don't ask), and knows that it's transcendental. He replied, "is it exactly ...
20
votes
1answer
2k views

Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...