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#1 2007-04-12 09:24:02

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

A sudden thought

Some Irrational numbers can be changed into rational by a function , like 

Then what about

, If they can't be changed prove it .

I first it will be interesting to put in the exercise section , but since I dont know how to prove it , so , help me?

Last edited by Stanley_Marsh (2007-04-12 09:24:16)


Numbers are the essence of the Universe

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#2 2007-04-12 10:08:20

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: A sudden thought

Well there are "trivial" examples like

where

I mean trivial in the sense that pi appears in the equation, which is kind of cheating. But a similar function to that isn't trivial:

where

See my thread http://www.mathsisfun.com/forum/viewtopic.php?id=3923 for how to calculate Riemann's zeta function at integer values.

Edit: The concept of an algebraic number wouldn't be useful. An algebraic number is one who is the root of a polynomial with integer coefficients. The transcendence of pi and e don't matter to us here though since I assume our functions aren't limited to just polynomials with integer coefficients.

edit 2: I forgot about e. Here's one that works:

where

If you put restrictions on what kind of functions we can have the problem would be more difficult. I've used some "uncommon" functions. Actually the trigonometric functions work for pi and ln x works for e, I don't know why I didn't think of that before, haha.

Last edited by Zhylliolom (2007-04-12 10:25:15)

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#3 2007-04-12 11:06:26

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: A sudden thought

Why get so advanced?  sin(pi) = 0.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2007-04-12 11:07:10

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: A sudden thought

Or if you want to cheat...

f(x) = 0.  f(pi) = 0. f(e) = 0


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2007-04-12 12:28:26

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: A sudden thought

Yeah, I noted that at the end of my post. I was thinking too hard at first smile. I'd consider a constant function to be a trivial answer though.

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#6 2007-04-12 13:00:45

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: A sudden thought

yeah..... , Sin(pi)=0 .. Pi is quite interesting , cant be changed into rational by powering


Numbers are the essence of the Universe

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#7 2007-04-12 16:24:08

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: A sudden thought

A friend of mine once marvelled at the fact that

how two irrational (in fact transcendental) numbers and a complex number could combine to produce a rational number (in fact, an integer). smile

Last edited by JaneFairfax (2007-04-12 17:37:05)

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