Exploring the Irrational: Pi and E in Math and Beyond

[Jedi_Sawyer]

Does anyone know if it has ever been proved that pi divided e, added to e, or any other mathematical operation combining these two irrational numbers is rational. Another thing does anybody know of any other irrational numbers that is not some derivative of pi and e.

 

[D H]

$$e^{i\pi} = -1$$

 

[pwsnafu]

another thing does anybody know of any other irrational numbers that is not some derivative of pi and e.

0.123456789101112...

 

[Jedi_Sawyer]

Yeah I understood that Euler's idenity combined e, i, and pi and you can find a good write up of why that is in Penrose's book "The Road to Reality". I guess I should have specified I was not interested in polar coordinate solutions where pi is integral to the coordinate system.

 

[Jedi_Sawyer]

well the fraction 0.1234... seems to be a well have a predictable pattern for what any arbirary next digit should be, so induction would solve what the next digit should be. I guess it could be an irrational number if we give up on randomness.

 

[genericusrnme]

well the fraction 0.1234... seems to be a well have a predictable pattern for what any arbirary next digit should be, so induction would solve what the next digit should be. I guess it could be an irrational number if we give up on randomness.

There is an algorithm for working out, in order, the digits of sqrt(2) and sqrt(2) is certainly not rational.
I'm not sure what you mean by randomness though..
Give me an example of a number with 'randomness' please.

 

[SteveL27]

Does anyone know if it has ever been proved that pi divided e, added to e, or any other mathematical operation combining these two irrational numbers is rational. Another thing does anybody know of any other irrational numbers that is not some derivative of pi and e.

It's an open question. Nobody knows whether or not some rational linear combination of e and pi is rational or not.

 

[DrewD]

I can't imagine that there is any way using only multiplication and addition of $\pi$ and $e$ that would give a rational number unless it simplified to something trivial like
$$\frac{e}{\pi}-\frac{e}{\pi}$$
I bet this can be proven for all transcendental numbers if it is true, but as it is, there is not a well defined question. Since euler's equation wasn't what you were looking for, you should be more precise about what you mean by "mathematical operation"

 

[Jedi_Sawyer]

There are a lot of numbers that are irrational. Your'e all right about that. I'm actually looking into the origin of randomness, how could a truly random number ever be generated, and I have to do more thinking on it before I can phrase the question rignt.

Thank you over and out.

 

[HallsofIvy]

There is also the famous 0.10010001000010000010000001... and variations on that.

 

[SteveL27]

There are a lot of numbers that are irrational. Your'e all right about that. I'm actually looking into the origin of randomness, how could a truly random number ever be generated, and I have to do more thinking on it before I can phrase the question rignt.

Thank you over and out.

There are lots of numbers that can not be described by any finite-length algorithm or formula. These are the non-computable numbers. Any number, even an irrational, that can be computed to any desired precision by an algorithm, can't reasonably be called random.

http://en.wikipedia.org/wiki/Computable_number

 

[acabus]

http://en.wikipedia.org/wiki/Transcendental_number#Numbers_which_may_or_may_not_be_transcendental

 

[Mickey1]

You might also want to look at the discussion on Wu's riddle forum

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgiPost name: Combinations of Pi and Sqrt(2)

The problem:

"Say I am given a number X = A*[sqrt]2 + B*[pi], where A and B are integers.
Given X, how can you find A and B, without using brute force?"

 

[acabus]

http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Analysis

See the fifth one down.

 

[soothsayer]

There are a lot of numbers that are irrational. Your'e all right about that. I'm actually looking into the origin of randomness, how could a truly random number ever be generated, and I have to do more thinking on it before I can phrase the question rignt.

Thank you over and out.

It is deceptively difficult to actually come up with a completely random set of numbers. The best way to do it is via physics systems. For example, nuclear decay is totally random, and by listing of numbers you get out of a geiger counter, you'd have a truly random list of integers, however, you'd be limiting your integers to a preferred range. Certain numbers would be more likely than others.

 

[micromass]

It is deceptively difficult to actually come up with a completely random set of numbers.

This might be true, but you are first going to have to actually define what you mean by random. Without a working definition, a discussion about it is meaningless.