Question

how to prove irrationality of integers or equations
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Answer 1

There are many proofs of irrationality, and some of them are quite different from each other. The simplest that I know is a proof that log23 is irrational. Here it is: remember that to say that a number is rational is to say that it is a/b, where a and b are integers (e.g. 5/7, etc.). So suppose log23=a/b. Since this is a positive number, we can take a and b to be positive. Then

2a/b=3.

2a=3b.

But that says an even number equals an odd number. That is impossible. Hence log23 cannot be rational.

The most well known and oldest proof of irrationality is a proof that 2–√ is irrational. I see that that's already posted here. Here's another proof of that same result:

Suppose it is rational, i.e. 2–√=n/m. We can take n and m to be positive and the fraction to be in lowest terms. Then a bit of algebra shows that (2m−n)/(n−m) is also equal to 2–√ but is in even lower terms. That is a contradiction. Hence it is impossible for 2–√ to be rational.

It is also not very hard to show that e, the base of natural logarithms, is irrational.

To show that π is irrational is much harder—in fact so hard that it was not done until the 18th century.

Another proof of irrationality begins by proving that when you divide an integer by another integer, if the decimal expansion does not terminate, then it must repeat. I posted an explanation of that here. Once you've done that, you can construct a non-repeating decimal. For example:

0.10110111011110111110…

(a 1, then a 0, then two 1s, then a 0, then three 1s, then a 0, then four 1s, then a 0, and so on