Well given that it goes on for an infinite number of digits you are bound to find random sets of seemingly repetitive strings of digits but in the sense that I suspect that you mean it is purely random. If it ever started to repeat one string of digits forever then it would be a rational number, but it is not ... it is irrational.
My favorite is the Carl Sagan speculation in his sci-fi novel Contact, that PI has embedded within its infinite digits the image of a circle how ever you choose the interpret the digits. There's a lot of creative PI art that can be found via Google Image search "PI art" and here's a couple of links: http://www.google.com/url?sa=i&rct=...fZRp8zgfqIRGqQtj8f_Jk-lQ&ust=1400951799993306 http://www.google.com/url?sa=i&rct=...fZRp8zgfqIRGqQtj8f_Jk-lQ&ust=1400951799993306 but no circle's been found yet.
You are essentially asking whether pi is a normal number. Whether it is or isn't is unknown. It's very hard to prove whether a number is normal. Consider the following base 10 number: 0.101001000100001000001... This number is irrational, but it is not normal.
http://en.wikipedia.org/wiki/Pi#Properties This section has a short discussion of what you're asking.
Some amazing properties of Pi that you may be interested in: https://scontent-b-sin.xx.fbcdn.net...247_833595243328155_8157491092640025910_n.jpg
That site says ##\pi## is an infinite non-repeating decimal "meaning that every possible number combination exists somewhere" in the decimal expansion of ##\pi##. No, that means that ##\pi## is irrational. Whether it satisfies that phrase in quotes is an open question.
Yes. The link provided by adjacent contains a lot of misinformation. I'm only leaving the post up because the link is very popular on the internet, so it would be good to debunk it here. So all the readers should be aware that it is very problematic.
Pi is irrational, is it not? Would you mind explaining why it's problematic and contains misinformation?
A sequence of decimal digits can be non-repeating without having to contain all possible subsequences. For instance, 0.101001000100001... is irrational. It's decimal expansion never repeats. But its decimal expansion also does not contain any 2's.
Stating some number is irrational means that it is not rational ie you can't define it as the ratio p/q of two integers p and q where q not equal to zero. Taking that as a definition you can't leap to the conclusion that all possible combinations of digits will appear. As an example, you could get a non-repeating sequence of digits without the digit 9 appearing anywhere in the sequence and still have an irrational number but not with every combination of digits.
@adjacent - take 1/90 and add 0.2 to it, does that number have any arbitrary length of the string of 2's in it? Using your logic, it must - despite all indications to the contrary. And I could do the same thing with 1/900 and 0.23 or 1/9000 and 0.234...should I draw a map? Yes, I know that this continuing fraction isn't transcendental. Extending this simple argument to 0.101001etc. is not exactly rocket science. -=-=- On a more interesting note: @acesuv: take a look here for some patterns in π: https://en.wikipedia.org/wiki/Generalized_continued_fraction#.CF.80 -=-= the whole continued fraction thing is just fascinating...
The number 0.123456789010010001000010000010... is also irrational and contains numbers from 0 to 9. It still doesn't satisfy the property you want.
I think that the author of the [STRIKE]post[/STRIKE] image meant that a digit can be used as many times you want. He must have thought without considering the reality.
It is proved that you can have all the combinations of digits.( I have just realised that this is wrong) Sorry,not the post. The image.