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

You are not logged in. #1 20070306 15:39:49
Jane’s exercisesTwo integers are said to be relatively prime iff their highest common factor is 1. Thus, 2 and 5 are relatively prime, as are 4 and 9. #2 20070306 15:45:34
Re: Jane’s exercisesThe one easiest to answer is #2, since any two consecutive even numbers are such that one of them is certainly divisible by 2. Hence, the product would be divisible by 8. Character is who you are when no one is looking. #3 20070306 16:09:02
Re: Jane’s exercisesWell, you’re on the right track with your answer to #2, but you need to elaborate just a little bit. #4 20070306 16:37:10
Re: Jane’s exercisesAn even number can be expressed as 2x. The next even number would be 2x + 2. The product would be 4x^2 + 4x = 4x( x+ 1). Either the x or the (x+1) has to be even and therefore divisible by 2. Combine that with the 4, and you have your 8. #5 20070307 02:58:29#6 20070307 03:23:52
Re: Jane’s exercisesn² 1 can be factorised into (n+1)(n1). We are given that n is odd, and so n+1 and n1 are both even. Hence, by the answer to 2), n² 1 is divisible by 8. Why did the vector cross the road? It wanted to be normal. #7 20070307 07:39:16
Re: Jane’s exercisesExcellent job. #8 20070311 17:13:00
Re: Jane’s exercisesMore exercises from me. 4. If x is real, what is 0^{x}? 5. If x is a real number, [x] denotes the greatest integer less than or equal x. Prove that for any real numbers x and y, In other words, the greatestinteger function is a superadditive function. (Hint: If n is any integer such that n ≤ x, then n ≤ [x].) 6. A cycloid is the curve traced by a fixed point on a circle rolling along a straight line. If the radius of the generating circle is r, the parametric equations of the cycloid are Find the area bounded by one arch of the generated curve (0 ≤ θ ≤ 2π) and the xaxis. Last edited by JaneFairfax (20070311 17:18:22) #9 20070320 04:11:53
Re: Jane’s exercises1. every number that isnt a prime can be factorized in its prime factors, for example 36 can be factorized in 2*2*3*3. #10 20070320 11:12:54
Re: Jane’s exercisesYou’re more or less on the correct line of thought. However the problem is to show that c is divisible by ab. It’s true that ab is less than or equal to c, but this doesn’t show that c is divisible by ab. #11 20070320 12:45:04
Re: Jane’s exercisesLast edited by JaneFairfax (20070320 18:19:07) #12 20070320 12:56:22
Re: Jane’s exercisesI thought that 0^{0} was indeterminate? There's a problem because 0^n = 0, but n^0 = 1, so there we have a contradiction. Why did the vector cross the road? It wanted to be normal. #13 20070320 18:18:33
Re: Jane’s exercises0^{0} is defined as 1. Try it on a calculator. so the graph is not rightcontinuous at x = 0. Last edited by JaneFairfax (20070320 18:22:26) #14 20070320 21:09:12
Re: Jane’s exercisesWhat calculator are you using? Any that I try just tell me Ma Error or something similar. Why did the vector cross the road? It wanted to be normal. #15 20070320 22:18:26
Re: Jane’s exercises
well since ab is created by the prime factors in c, c must be divisible by ab? #16 20070321 12:49:50
Re: Jane’s exercises
A decent one.
I’ve read your proof again and I can see what you’re trying to get at – so, yes. #17 20070321 14:26:23
Re: Jane’s exercisesLetting 0^{0} be 1 is a standard convention, which helps eliminate special cases where some theorems break down. There are some combinatorial and settheoretic justifications for this convention. However, in analysis 0^{0} is treated as an indeterminate. Moreover, consider x^{x} = e^{x ln x}. There is no real number corresponding to ln 0, so x^{x} is not defined at x = 0; in other words 0^{0} is undefined. The calculator says it is equal to 1 because of the common convention mentioned above; but a real decent calculator would notify you something like "Warning: 0^0 replaced by 1" . #18 20070404 01:12:37
Re: Jane’s exercisesThanks for the tip Jane. Last edited by Kurre (20070404 01:12:56) #19 20070404 02:22:38
Re: Jane’s exercisesWell, I’ve been trying to create some more exercises for this thread. Here is something I just this moment made up. 8. Use the above result to show that NB: For #8 you must use the result in #7. Last edited by JaneFairfax (20070417 08:27:46) #20 20070405 09:12:50
Re: Jane’s exercises
right?
Numbers are the essence of the Universe #22 20070405 18:13:19
Re: Jane’s exercises
No. You have the right idea, though.
True, but as I’ve stated, I want you to use the result in #7 to do #8. #24 20070409 22:17:12#25 20070412 15:08:01
Re: Jane’s exercises9. Prove that a monotoneincreasing sequence of real numbers that is bounded above converges to its least upper bound. 