Can We Prove that Non-Rational Decimals Go On Forever?

[1MileCrash]

"Infinitely repeating decimals may not exit, they may terminate somewhere down the line but we can never tell."

I've heard it claimed that numbers that pi or e may terminate eventually and that there is no way of knowing.

Is there no mathematical way to show that a non-rational non-terminating decimal will have infinitely many digits and that it cannot possibly ever terminate?

Do we only proclaim pi to be non-terminating because we've "never reached the final decimal place?" That can't be, can it?

 

[Office_Shredder]

If a number has, say, only 157 digits past the decimal place, then you can always write it as a fraction by just taking the number past the decimal and dividing it by 10¹⁵⁷.

For example, .12359694929120132 = 12359694929120132/10¹⁸.

So if a number is irrational it can't have a terminating decimal

 

[1MileCrash]

Yes, I understand that. A rational number is a number that can be expressed as a fraction and all terminating or repeating decimals can be expressed as a fraction therefore your logic follows. But this isn't really what I'm asking.

Allow me to ask the question in a different way.

We accept that e is an irrational number, cannot be expressed as a fraction and is a non-terminating decimal. But why? Is there an underlying mathematical proof?

Is it possible that e terminates after one-billion decimal places and actually is a rational number? Or can it be proven that e without a doubt never terminates?

Ergo, what leads us to declare a number irrational? If it's a really, really, really high number of decimal places, do we eventually say "screw it, it's irrational." or is there a mathematical way to conclusively show without a shadow of a doubt that e (example) will never terminate?


Thanks

 

[JG89]

A number is rational if it can be expressed as a fraction of integers. That's the definition. A number is irrational if it CANNOT be expressed as a fraction of integers.

There are proofs that show pi, the number e, the square root of 2, and the square root of any number that isn't a perfect square are all irrational.

Proof of square root 2 being irrational http://en.wikipedia.org/wiki/Irrational_number#Square_roots

 

[1MileCrash]

A number is rational if it can be expressed as a fraction of integers. That's the definition. A number is irrational if it CANNOT be expressed as a fraction of integers.

There are proofs that show pi, the number e, the square root of 2, and the square root of any number that isn't a perfect square are all irrational.

Proof of square root 2 being irrational http://en.wikipedia.org/wiki/Irrational_number#Square_roots

Okay, thanks, that's what I was looking for.

So to say "pi might be rational, we just haven't gotten to the last decimal yet" is utter nonsense since we can mathematically prove that it is irrational.

Some people seem to be under the impression that if a decimal just keeps going for a "long time" than it is declared to be irrational on the spot. I figured that couldn't be the case.

 

[CRGreathouse]

Some people seem to be under the impression that if a decimal just keeps going for a "long time" than it is declared to be irrational on the spot. I figured that couldn't be the case.

Right. When we can't tell, like with Euler's constant gamma, we say "unknown" or "believed to be irrational".

 

[arildno]

Furthermore, a number need not be irrational in order to have a provably non-terminating decimal representation. The number 1/3, for example, is a rational number with a non-terminating decimal representation.