How do we know that irrational numbers never settle into a pattern? Do we just assume it or is there any kind of proof?

“Settle into a pattern” is a vague term. If you refer to the decimal fractions (e.g. 3.14…), then if the fraction is finite or repeating the number is rational (basically follows from the definitions of rational numbers and positional fractions). We know that numbers like √2, π and e are irrational, and hence, their decimal fraction representations are infinite and non-repeating.

However, some irrational numbers can be expressed as periodic continued fractions. For example √2=1+1/(2+1/(2+1/(…

So what you’re asking about is probably more about the properties of decimal fractions, and not irrational numbers.

By definition. If they did, they’d be rational.

If a number has some repeating pattern, then we call that rational. Rational numbers can all be expressed by a fraction of two integers.

You can prove that a number is irrational by showing that it's not possible for a certain number to be expressed as such a fraction. A common proof technique for achieving this is a proof by contradiction, where you assume that a number can be expressed by M/N and those two integers M and N are the smallest integers with this property, then showing that you can find an even smaller pair of integers with the same property, hence a contradiction.

https://en.wikipedia.org/wiki/Proof_that_pi_is_irrational

https://en.wikipedia.org/wiki/Proof_that_e_is_irrational

You can easily find the proofs for the other irrational numbers with a quick web search.

There's a proof. The main point is that any number that settles into a repeating pattern is rational, so irrational numbers can't possibly have a repeating pattern.

Notice that the pattern has to repeat, so 1.010010001000010000010000001000000010000000010000000001 etc is a pattern, but because it doesn't repeat it's irrational, whereas 0.123456789123456789123456789123456789123456789123456789 and so on repeats, and it's rational.

I'll do an example, which hopefully makes it clear how it works.

If x = 0.123123123123123123123...... then
1000x = 123.123123123123123123123123123....
but then if we subtract the original number we get that
1000x - x = 123
so 999x=123
and so x = 123/999

If you had something that takes a while to start repeating, you could just multiply it by 10000000 or something so that it starts repeating straight after the decimal point and use the same argument, and surely 10000000y can be written as a fraction exactly when y can. But you can do it directly, a bit like this one:

If y = 44.332211007438743874387...... then
100000000y = 4433221100.7438743874387438....
and then
1000000000000y = 44332211007438.743874387438....
so that 1000000000000y - 100000000y
= 44332211007438 - 4433221100 which means that
999900000000y = 44327777786338
therefore y = 44327777786338/999900000000 so it's rational.

Now personally, I think that's less informative than the first one with the middle argument, so whilst it's more direct, I think doing the simple case and then arguing why it covers the complicated case is actually better.

Interestingly, proving the other truth that all rational numbers eventually repeat is much harder!

Suppose the decimal repeats after n places. For example, .252525... repeats after 2 digits. You can find the fraction that represents this number by dividing the n repeating digits by n 9s. .252525... = 25/99.

Since this number can be represented by a ratio of integers, it must be rational.

You can use similar tricks for numbers that repeat after an initial nonrepeating string.