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1. Prove that is irrational.
Prove that is irrational.
Prove that e is irrational.
2. (Original post by Ano9901whichone)
Prove that is irrational.
Prove that is irrational.
Prove that e is irrational.
Do you need help with this or are you posting it as a challenge?

If you need help, start with the first one and tell us if you know how to prove that is irrational. If you don't then you should start with that.

is similar but slightly harder.
3. (Original post by notnek)
Do you need help with this or are you posting it as a challenge?

If you need help, start with the first one and tell us if you know how to prove that is irrational. If you don't then you should start with that.

is similar but slightly harder.
Proving is easy.
4. (Original post by Ano9901whichone)
Proving is easy.
Okay. Do you need help with proving is irrational?
5. (Original post by notnek)
Okay. Do you need help with proving is irrational?
Well I think I have it. I would say that assume is rational such that it can be written as .
Squaring gives .
Since is clearly divisible by 17 (prime) it follows that p is divisible by 17 where gcd =1.
, by the same logic we deduce that . This is a contradiction as both p and q cannot be divisible by 17 as they have gcd of 1. So this forces us to conclude that is irrational.
How's that?
6. (Original post by Ano9901whichone)
How's that?
Fine.
7. (Original post by Ano9901whichone)
Well I think I have it. I would say that assume is rational such that it can be written as .
Squaring gives .
Since is clearly divisible by 17 (prime) it follows that p is divisible by 17 where gcd =1.
, by the same logic we deduce that . This is a contradiction as both p and q cannot be divisible by 17 as they have gcd of 1. So this forces us to conclude that is irrational.
How's that?
Good.
Alternative
Note from p^2=17q^2 consider p,q in prime factors the LHS has an even number of exponents the RHS has odd number of exponents, hence a contradiction.
For the log one you do the same and simplify some equations and get a parity argument for p,q>0.

Note sometimes the first argument does not work as nicely.
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8. Oh, and for e is irrational. Look at its series expansion and then multiply both sides by , then prove that the remaining terms with factorials in the denominator is bounded in and you have a contradiction.
9. (Original post by Zacken)
Oh, and for e is irrational. Look at its series expansion and then multiply both sides by , then prove that the remaining terms with factorials in the denominator is bounded in and you have a contradiction.
Alternatively a much easier way is just to note that e's continued fraction expansion is infinite - this was the first proof of e's irrationality, and was published by Euler.
10. (Original post by HapaxOromenon3)
Alternatively a much easier way is just to note that e's continued fraction expansion is infinite - this was the first proof of e's irrationality, and was published by Euler.
Sounds too much like cheating unless you prove that infinite continued fractions => irrationality and that e's continued fraction expansion is infinite.
11. (Original post by Zacken)
Sounds too much like cheating unless you prove that infinite continued fractions => irrationality and that e's continued fraction expansion is infinite.
It is not very difficult to compute e's continued fraction expansion, which shows that it is infinite. This paper does it in just over a page: see Proposition 1 and Theorem 1.

The result about infinite continued fractions is trivial to prove by contrapositive - clearly every rational number's continued fraction must terminate as rational numbers are themselves just fractions.
12. (Original post by HapaxOromenon3)
It is not very difficult to compute e's continued fraction expansion, which shows that it is infinite. This paper does it in just over a page: see Proposition 1 and Theorem 1.

The result about infinite continued fractions is trivial to prove by contrapositive - clearly every rational number's continued fraction must terminate as rational numbers are themselves just fractions.
Nice paper, thanks for the link. The proof via series expansion seems shorter than that, though. I can comfortably do it in 5-8 lines.

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13. (Original post by Zacken)
Nice paper, thanks for the link. The proof via series expansion seems shorter than that, though. I can comfortably do it in 5-8 lines.

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Not with full rigour you can't. It takes about a page of working - see
http://www.mathshelper.co.uk/Proof%2...Irrational.pdf
14. (Original post by HapaxOromenon3)
Not with full rigour you can't. It takes about a page of working - see
http://www.mathshelper.co.uk/Proof%2...Irrational.pdf
Lol
15. (Original post by physicsmaths)
Lol
At what do you laugh, fool?
16. (Original post by HapaxOromenon3)
At what do you laugh, fool?
The document and now you.
All of that can be shortened to 8 lines as Zacken said.

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17. (Original post by physicsmaths)
The document and now you.
All of that can be shortened to 8 lines as Zacken said.

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You have no evidence of that.

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