Wikipedia:Reference desk/Archives/Mathematics/2014 October 8
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October 8
[edit]number theory
[edit]Let x and y be real numbers, then prove that:
- a) [x + n] = [x] + n, for any integer n
- b) [x] + [-x] = 0 or -1 according as x is an integer or not.
- c) [x + y] ≤ [x] + [y] — Preceding unsigned comment added by 117.208.156.211 (talk) 12:48, 8 October 2014 (UTC)
- Is this homework? It looks like homework. These are all basic facts that follow from the definition of the floor function. The point of the exercise is to teach you how to prove basic facts with rigor, so that you can eventually prove difficult facts with rigor. Some teachers will allow proofs of the form LHS = ... == ... == RHS: QED, while some other instructors find that that format is a little sloppy. Follow the examples in your textbook, and be very careful to justify each step, based on some definition of e.g. integers or real numbers or floor functions. If you can post an attempt, someone here can probably critique your work. SemanticMantis (talk) 15:04, 8 October 2014 (UTC)
- Express x as m+a, where m=[x], and see where that gets you. —Tamfang (talk) 06:57, 9 October 2014 (UTC)
- By the way, I think one of those demonstranda is wrong. —Tamfang (talk) 06:58, 9 October 2014 (UTC)
- Good point. I don't think it's helping the OP too much to mention here that it's part c) that is incorrect. Probably a transcription error. All the correct forms are stated in the article linked above. SemanticMantis (talk) 15:13, 9 October 2014 (UTC)
Also, side note — this is not number theory, as the term is usually understood. Number theory studies the properties of the natural numbers, not the real numbers (albeit the proofs often use real analysis). --Trovatore (talk) 07:01, 9 October 2014 (UTC)