system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: How can someone be absolutely sure that a number is irrational, not a fraction with obscenely massive numerator & denominator?

Like pi or e. Could they actually be rational numbers with an extremely large denominators (something as large as Graham's number)?
gpt
The definition of an  irational number is that it CAN NOT be written as, p/q, where p, q are integers and q != 0.

It is possible to prove this for numbers. For example we can use a proof by contradiction to proove that sqrt(2) is irrational.

Proof that sqrt(2) is irrational:

Assume that sqrt(2) is rational. By definition of rational numbers we can right a/b = sqrt(2). Further assume that a/b is reduced, so that a and b do not share any common factors. (If they do then simplify the fraction and you have your new a and b). Then:

a/b = sqrt(2)

(a/b)^2 = sqrt(2)^2

a^(2)/b^(2) = 2

a^(2) = 2b^(2)

This means a^2 is an even integer, which is only possible if a itself is even, and a = 2k, for some integer k.

Then:

a^(2) = 2b^(2)

(2k)^(2) = 2b^(2)

4k^(2) = 2b^(2)

2k^2 = b^2

This means b^2 is even, which is only possible if b itself is even.

But if both a and b are even then they share 2 as a common factor. This is a contradiction of our original assumption.

Therefore sqrt(2) can't be rational. 


As the other person says the proof for the irrationality of pi and e are not as simple.