Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20061225 14:17:08
InequalitiesI was shopping for some Christmas presents in Borders (a book store) and I of course stumbled my way into the math section. Found a really cheap and interesting book on inequalities. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #2 20061225 16:56:20
Re: Inequalities"In Between" meaning that: "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #3 20061226 03:05:20
Re: InequalitiesI had exactly the same reaction. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #4 20061226 04:54:17
Re: InequalitiesMy phraseology may not be the best here, but you get the idea. The proof is easy indeed, I could do it in my head (which is my excuse for not having it as polished as possible here). or is true, since √2 is irrational and therefore cannot be equal to a rational number, and thus the result follows from trichotomy. Now consider the inequality where the ? denotes the unknown direction of the inequality. Since m and n are natural numbers, their sum is nonzero and positive, and thus we may multiply each side my m + n without reversing the inequality: Once again, n is a natural number and thus is nonzero and positive, so we may divide each side by n and maintain the direction of the inequality: Now subtract m/n and √2 from each side to obtain which gives (after dividing through by 1  √2, flipping the inequality, rationalizing, then flipping again so that we now have the original direction (these steps are left to the reader as an exercise)) which gives the direction of the inequality as the opposite of the "initial condition" m/n ¿ √2, where ¿ denotes the original (and opposite) inequality direction, so that if m/n < √2 then (m + 2n)/(m + n) > √2 or if m/n > √2 then (m + 2n)/(m + n) < √2. Thus, either or is true for natural numbers m and n. #5 20061226 09:45:59
Re: InequalitiesNice Proof. "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #6 20061227 02:09:19
Re: InequalitiesI tried following through Zhylliolom's method with k instead of 2, and I'm pretty sure you're right. Why did the vector cross the road? It wanted to be normal. #7 20061227 10:29:25
Re: InequalitiesAh yes, if √k is rational, then there is an m/n that is equal. But otherwise the "between" works. "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #8 20061227 15:56:33
Re: Inequalities
The book is called "Analytic Inequalities", by Nicholas D. Kazarinoff. It's aim is to fill a gap left between algebraic (high school) math and analysis, specifically that of inequalities. Analysis is just full of them, and many students don't have a lot of experience with them. I was one of those. I did just find in my introduction to analysis course, however did struggle a lot with inequalities. There is a very crucial hint given, but I will wait a day or two to post it. Note that no computer nor calculator may be used. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #9 20061227 20:53:58
Re: InequalitiesAnalytic Inequalities at Amazon.com. This looks like a very interresting book. Might buy it #10 20061229 13:05:54
Re: InequalitiesHere is your hint: As with most proofs on the natural numbers, the recommended way to proceed is by induction. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #11 20070125 16:02:00
Re: InequalitiesI can't prove it, but it seems right, since x^2 + x + (x+1) = (x +1)^2 igloo myrtilles fourmis #12 20070126 04:37:16
Re: InequalitiesWe know that: So: and so: Thus: And so: which means that: And so: Whew! The other side is exactly the same, just start off with the fact that: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." 