Maths - There Should Be More Of Them
It is not lost on this chap that …
… in the US, one studies Math, whilst in the UK, one studies Maths. Now, not to be pernicity, but I do think the English have it since it is clear there are many Maths that can be studied. Take this exchange for example - which emerged on an email list I belong to. Quite why this chap is on that list to begin with, is a totally different story, which we may - or may not - elect to cover in a future dialogue. What is clear is that this is a math that is not part of some other maths that he does know about. IT ALL STARTED QUITE SIMPLY …
"I’ve been wondering, since irrational numbers are infinitely long, and non-repeating, must any finite sequence of digits exist somewhere in this infinite sequence? For example, looking for “10” in π, we find, 3.141592653589793238462643383279502884197169 … but what if I wanted a sequence of zeros and ones that would spell out all of Shakespeare a million times, back to back?"
FIVE MINUTES LATER, IN CAME THE REPLY
"No, they need not. 1.1010010001000010000010000001… For any particular irrational or transcendental number, it can be very difficult to determine what if any regularities there are in the sequence of digits that may make it impossible to find any particular desired sequence in there."
AND THEN SIX MINUTES LATER
"Irrational numbers don’t need to possess the quality of including any finite digit sequence in a fixed-base expansion. For example, the following (in decimal) 0.11010010001000010000010000001… is of course irrational, but doesn’t contain even a single digit 2. Normal numbers have all possible finite sequences uniformly represented and therefore have the property of containing them all. There are many examples of normal numbers, such as a zero followed by a decimal point followed by concatenation of all integers, and pi is conjectured to be normal, but this is not yet proven, to my knowledge."
_"As far as I remember, it is an open question whether any finite sequence __appears in the expansion of π. In_ principle _however, sure, why not? I__nfinity is very long indeed – but it does not follow from the fact that __the expansion is not repeating."_
THEN ‘EXTERNAL SOURCES’ WEIGHED IN
"Technically, ’toan’ only required a “disjunctive sequence”: [ https://en.wikipedia.org/wiki/Disjunctive_sequence ] … though of course normal numbers’ decimal expansions are disjunctive sequences."
MORE SUCCINCTLY …
“Actually, there are un-countable many irrational numbers whose sequences contain every finite sequence of numbers except for those that contain Shakespeare.”
FINALLY CONCLUDING THAT …
(perhaps even with a ‘drop-o-the-mic’) "… of course, but for any such (rational or irrational number) you have (countably) infinitely many that do."
This chap is very glad that there are people out there that will be able to understand this exchange.
Graham comments … … that he’s happy for the other Chap that his friends are trending along similar lines of geekdom. Not quite the same at this end … as one of this Chap’s own wrote,
“So what is the point about Möbius strips anyway?”
and for good measure added …
At least at your end somebody understands.
Ahem … are you sure?