Irrational Numbers: Expressible as Infinite Summations?

[jeffceth]

I apologise if this belongs in another place, but:

Can all irrational numbers be expressed as infinite summations, ie like Pi and e?

I'm looking for: provable, disprovable, or neither. This is essential to something else I am working on.

sincerely,
jeffceth

 

[Hurkyl]

It depends on what you mean by "expressed". There is certainly an infinite series that converges to any irrational number you like. For example:

$$\pi = \pi + 0 + 0 + \ldots$$

Ok that was a little too trivial. It turns out that by rearranging the terms of this sequence in the right way, you can make the left hand side of this equation any real number you want:

$$\ln 2=1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \ldots$$


However...

In the language in which we write mathematics, we can only write countably many equations, proofs, theorems, articles, et cetera. However, there are an uncountably many number of irrational numbers. Thus, there exist irrational numbers for which we cannot explicitly write sequence that converges to that number.

The good news, however, is that for any irrational number we can express uniquely, we can write down a sequence that converges to it (though not necessarily in a way that is computationally useful).

 

[HallsofIvy]

Certainly every real number, rational or irrational, can be written in decimal form- therefore an "infinite summation".

 

[jeffceth]

Originally posted by Hurkyl
It depends on what you mean by "expressed". There is certainly an infinite series that converges to any irrational number you like. For example:

$$\pi = \pi + 0 + 0 + \ldots$$

Ok that was a little too trivial. It turns out that by rearranging the terms of this sequence in the right way, you can make the left hand side of this equation any real number you want:

$$\ln 2=1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \ldots$$


However...

In the language in which we write mathematics, we can only write countably many equations, proofs, theorems, articles, et cetera. However, there are an uncountably many number of irrational numbers. Thus, there exist irrational numbers for which we cannot explicitly write sequence that converges to that number.

The good news, however, is that for any irrational number we can express uniquely, we can write down a sequence that converges to it (though not necessarily in a way that is computationally useful).

I am especially interested in the fact that we can only write countably many mathematical expressions, and there are uncountably many irrationals. Logically, this makes sense, but can it be mathematically proven that at least one such irrational number exists that cannot be expressed uniquely by form of any expression we currently have?

sincerely,
jeffceth

 

[Hurkyl]

Yes. It's easy enough to prove there are an uncountable number of irrational numbers.

Mathematical statements are defined to be strings of finite length of characters from a finite alphabet, and it's easy enough to show that the set of all of these strings is countable.

Thus, by the pigeonhole principle, there exists an (uncountable number of) irrational numbers that cannot be expressed uniquely with a mathematical statement.


We can appeal to physics to show that any other form of expression must have a countable number of states. Pick a fundamental unit for every observable quantity (e.g. length, momentum, charge) that is far smaller than anything we can measure. Then, in principle, we can write the state of the universe as a finite string of characters from a finite alphabet by "digitizing" every elementary particle in the universe by measuring all of the obvservable quantities of every particle and writing them down. Each of our measurements can only yield a finite number of results, there are only a finite number of observable quantities, and there are only a finite number of elementary particles in the observable universe, so there are actually only a finite number of distinguishable states of the universe.