Wikipedia:Reference desk/Archives/Mathematics/2016 November 1

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November 1

Squares plus 1

Is the product of 2 consecutive members of the sequence of squares plus 1 (1, 2, 5, 10, 17, 26, etc.) also a member? GeoffreyT2000 (talk, contribs) 04:05, 1 November 2016 (UTC)

Yes, it is. The product is n^2+n+1 squared plus 1, where n^2 is the smaller of the two original squares.Bh12 (talk) 05:33, 1 November 2016 (UTC)

(n²+1)((n+1)²+1) = (n²+n+1)²+1. Bo Jacoby (talk) 08:40, 1 November 2016 (UTC).

This is a special case of the Brahmagupta–Fibonacci identity. --RDBury (talk) 00:01, 2 November 2016 (UTC)

Half Signs

How do I write the half (1/2) in the WP editing page? What's the coding? 103.230.104.30 (talk) 18:29, 1 November 2016 (UTC)

Assuming you mean ${\frac {1}{2}}$, look at the code for this page to see how I did it. Another option is ${\tfrac {1}{2}}$. -- Meni Rosenfeld (talk) 19:10, 1 November 2016 (UTC)

The WP style guide for fractions is discussed at Wikipedia:Manual_of_Style/Mathematics#Fractions. Basically, the styles (1/2) or a horizontal bar like ${\tfrac {1}{2}}$ are preferred. --Mark viking (talk) 19:15, 1 November 2016 (UTC)

You do not need the fraction bar. Use exponentiation. Write 2^-1 or $2^{-1}$. Bo Jacoby (talk) 16:40, 2 November 2016 (UTC).

As a single character, ½. —Tamfang (talk) 02:37, 3 November 2016 (UTC)

Uncountably infinite irrationals

The set of all irrationals is uncountably infinite, as can be proven by cantor's diagonal. How do you prove that there are infinite irrationals between 1 and 2? Is there some sort of way of establishing a one to one correspondence? I'm thinking of a proof somewhat like the one to prove that the set of all even numbers is equivalent to the set of all whole numbers, through matching one from each set to one from another. Margalob (talk) 22:42, 1 November 2016 (UTC)

There are no infinite irrationals between 1 and 2. In fact, every number between 1 and 2 is finite. --Trovatore (talk) 22:46, 1 November 2016 (UTC)

OK, that was a little aggressive, sorry. It's a peeve of mine. Never say "infinite foos" when you mean "infinitely many foos". Please please please. Never ever ever. Thanks. --Trovatore (talk) 22:47, 1 November 2016 (UTC)

It seems that the OP wanted to know how one proves that the set of irrationals is uncountably inifinite, and that's why they wrote "infinite" (rather than "infinitely"), but they forgot to add the "uncountably". HOTmag (talk) 23:16, 1 November 2016 (UTC)

It is also incorrect to say that there are uncountably infinite irrationals between 1 and 2. --Trovatore (talk) 23:19, 1 November 2016 (UTC)

As I've already pointed out, they meant an uncountably infinite [set of] irrationals. HOTmag (talk) 23:25, 1 November 2016 (UTC)

Yes, but without "set of", this is still utterly wrong. I want to be very clear on this. This is not a matter of style or preference. As a matter of English, this is absolutely incorrect. --Trovatore (talk) 23:40, 1 November 2016 (UTC)

I guess the OP knows that. They probably forgot to add "set of", which they didn't forget to add in their first sentence. HOTmag (talk) 08:01, 2 November 2016 (UTC)

As to the actual question. Yes, you could come up with an explicit 1-1 correspondence between the irrationals between 1 and 2, and all the irrationals. It might be a fun exercise to come up with a clever one. But I doubt that's the approach that will best build your understanding. The main point is that there are uncountably many reals between 1 and 2, and only countably many rationals. Since the union of two countable sets is countable, the set of irrationals between 1 and 2 must be uncountable. The Cantor diagonal argument easily proves the claim that there are uncountably many reals between 1 and 2. --Trovatore (talk) 22:59, 1 November 2016 (UTC)

It seems that by "irrational" the OP meant "real", as one can realize by reading the OP's first sentence. HOTmag (talk) 23:35, 1 November 2016 (UTC)

Every real number in the uncountably infinite set obtained by Cantor's argumemnt, is larger than 1. So, you can replace every x in that set by 1/x +1, and then you receive an uncountably infinite set of reals between 1 and 2. Now, by Trovatore's argument above, you can understand that also the set of irrationals between 1 and 2 is uncountably infinite. HOTmag (talk) 23:16, 1 November 2016 (UTC)

Most versions of Cantor's diagonal proof that I've seen, including the one in our Wikipedia article, show that the set of reals between 0 and 1 is uncountable. Then there is a second step to show a bijection between (0,1) and all the reals. So if you omit that second step and just map (0,1) to (1,2) in the obvious way you have what the OP wants. CodeTalker (talk) 23:57, 1 November 2016 (UTC)

It all looks nice, but there are problems, that have been glossed over by the mathematical community. Count Iblis (talk) 18:46, 2 November 2016 (UTC)

No, there really aren't. --Trovatore (talk) 19:14, 2 November 2016 (UTC)

If you don't find it a problem that math is too much based on meaningless philosophy. The problem is not that it doesn't work, formally it works ok., but there is a price to pay in the form of a lot of baggage that comes with working with uncountably infinite sets. Count Iblis (talk) 20:00, 2 November 2016 (UTC)

The philosophy is not in fact meaningless. --Trovatore (talk) 20:03, 2 November 2016 (UTC)