# Irrational numbers

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

Staff Emeritus
Gold Member
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).

Homework Helper
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

Staff Emeritus