Assuming that pi is normal, which is probably true, but hasn’t been proven.
Hence Hari_Seldon’s qualifier of “virtually certain”.
Deja vu!
And if a thread goes on long enough, it’s virtually certain that posts will repeat the same points over and over.
It’s normal for that to happen.
We can however say when a number cannot exist in a certain number system. If we posit that the number 2 exists, and define multiplication, then we can say concretely that if x^2 = 2, then x cannot be a rational number. Perhaps from there we conclude that \sqrt 2 does not exist at all. But alternatively, we can invent the real numbers. The same goes for x^2=-1 and imaginary numbers, or even weirder things like \epsilon^2=0, \epsilon \neq 0. The invention of new number systems that contain numbers that provably cannot exist in previous systems is one of the most common things in math.
The OP may find this video helpful (less than five minutes long):
(As an aside: The video does not let us know what happened to Hippasus who discovered irrational numbers but the legend is he was thrown out of a boat and drowned by his fellow mathematicians who were really upset by the notion of irrational numbers which they found inelegant in a universe they thought was mathematically elegant.)
Nitpick: real algebraic numbers.
That depends on how you construct \sqrt 2. If you do it by saying it’s the number between the set of all (positive) rationals whose squares are less than 2 and the set of those whose squares are greater than 2, then the obvious thing to do gives you all the reals, including the transcendentals.
Please don’t rationalize the irrational.
Well, technically we could just invent the quadratic field \mathbb{Q}(\sqrt 2) if we want the “smallest” number system that contains \sqrt 2. But as Topologist noted, it depends on how we’re constructing things.
What do you mean Pi is non-repeating? It is exactly equal to 1 (in base pi). 
The condition that @CalMeacham describes is actually somewhat weaker than normalcy, so it’s possible for a number to meet that condition without being normal. I don’t think that weaker condition has been proven for pi, either, though.
And not only did the Greeks of Pythagoras’ day believe that all numbers were rational, that was the basis of their entire notion of proportionality. It wasn’t until Euclid that a new formulation of proportionality was formed, that could account for irrational numbers (most of Euclid is believed to have just been a compilation of things already known, but his techniques of proportion (based on similar triangles) were all his).
Easy to see in constructed cases. Take the number mentioned earlier, 0.012345678910111213141516… Then insert zeroes between the numbers based on how many digits each appended number has: 0.00010203…0010001100120013…000997000998…
The number is >50% zeroes, so it’s not normal. But it still contains every possible finite sequence of digits.
Interesting. I hadn’t seen that before.
Why does the number of zeros matter in that way? What would a number with a single appended zero be?
If you had a normal number, and prepended a single zero, it would still be normal–the single extra would be “diluted” by the infinite other numbers back to an even mix. That is, normal numbers only need an even distribution of strings in the limit as the length goes to infinity. Some finite number of digits that don’t hold to the patter won’t matter.
ETA: The example I cited also wouldn’t have worked if I inserted a single zero between the substrings. Although it would seem to have a preponderance of zeroes at first, once you get to very long substrings, the extra zero doesn’t make much difference. That’s why I had to add zeroes equal to the number of substring digits–it’s easy to see then that the mix will be >50% zeroes.
In a normal number, all strings of any given length are equally common. So each digit would appear 1/10 of the time, each two-digit string 1/100 of the time, and so on. When over half the digits are 0, it’s clear that 0 does not appear 1/10 of the time.
But no finite number of added digits matter, because a number is only normal when you consider all of its infinite digits.
Well, it depends what you mean by repeat.
If it its “2nd ,. third, fourth, and all following versus, same as the first” , then that proves its rational.
Consider 0. 123456 915 915 915 915 915 915 … etc. It ends in “915” repeating forever.
Just multiply it by , in this case , a million .(6 0’s matching the number of the non-repeated leadup. )
That gives its 123456 . 915 915 915 915 “915” repeating.
Repeating in that way means rational, just put the repeated digits over the same number of (B-1) digits (B being the base…for base 10, its 9’s. For base 7, its 6. )
So if its base 7 0.2145 2415 2145 “2145” repeating, thats just 2145/6666 , all calculated in base 7 there… right ?
So we can write the 915 example as ( 123456 + 915/999 ) / 1000000 , which is a rational number obviously. So repeating infinitely is always rational.
If you mean, there are repeating sections , there must be repeating sections, but they don’t occur at a predictable place, nor more than random chance suggests.
As to the density of rationals vs irrationals. There is at least one rational between any two irrationals, and there is at least one irrational between any two rationals.So its always possible to find a rational to count as the partner for any irrational… they are in the same class of count-ability. no density difference ?
?? A lengthy mathematical treaty is just a long form of algebraic calculation.
modern algebra is really just a short hand form
Their way is more about lines. “line 20 : the result of line 15 plus 5 , all divided by pi.”
Modern way, l20=(l15+5)/pi , but if write y instead of l20, and x instead of l15,
y=(x+5)/pi
Modern way does that the advantage that it doesn’t shutter your brain to the idea that the value of x can come from later lines in the paper…
Its rather that they only had low order problems to solve… so clumsy line algebra was working, and they didnt deduce some things like ability to solve n simultaneous equations with m unknowns.
True. I wasn’t really talking about that issue (although on reflection it wasn’t all that clear.) My point about existence is about the sequence of numbers in the decimal representation. We can find an infinite number of ways of creating a representation, simply by choosing different integer bases. Irrationals still have non-periodic representations no matter what base you choose. But in what sense does this infinitude of representations have anything to do with the number being represented?
Decimal representation ^{\rm \dagger} is just shorthand for something like x = a_010^0 + a_110^{-1} + a_210^{-2} + \cdots where a_i \in \mathbb{N}, a_i < 10
So why are we so hung up on these sequences? They are not of themselves really part of the real numbers, but rather an ancillary artefact.
Existence is a thorny subject in the philosophy of mathematics. You are only a step away from invented versus discovered arguments. Madness lies in wait.
{\rm \dagger} - numbers less than 10.