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You are not logged in. #1 20130928 16:07:22
Explanation?Hi. I'm not well educated in math, just a high school student interested in math. I was playing around with numbers and noticed a pattern. If I lined up consecutive perfect squares like {0,1,4,9,16,25,36...} and then took the difference between each pair of consecutive numbers, I got {1,3,5,7,9,11,13...}. Then I took the difference between each consecutive pair of numbers and got {2,2,2,2,2...}. Taking the difference between those gives {0,0,0,0,0...}. Interesting. #2 20130928 17:43:57
Re: Explanation?
These are no coincidence, just inherent property of numbers. You could easily try and prove them. 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #3 20130928 17:47:02
Re: Explanation?hi Ma123, will generate a similar set of differences. Each time the highest power goes up by 1, it takes one more set of differences before you get {0,0,0....} Is it useful? Well yes. If you have investigated a pattern of numbers and you find the differences, it will tell you what the highest power of n is and thus can help you to find the formula for the pattern. There's a bit about it here: http://www.mathsisfun.com/algebra/seque … rule.html It's great that you found this out for yourself. It's a sign of a true mathematician Welcome to the forum. Bob ps. One way to prove why it works involves using this: http://www.mathsisfun.com/algebra/binomialtheorem.html You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #4 20130928 19:45:13
Re: Explanation?Hi Ma123; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20130928 22:05:09
Re: Explanation?
Please explain 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #6 20130929 01:17:07
Re: Explanation?Which part? The multiplication? In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20130929 05:37:09
Re: Explanation?Could you explain both the approximation of derivatives and the multiplication? That sounds cool #8 20130929 05:59:54
Re: Explanation?Hi; Form a table of differences, starting with the first difference. 4th difference is 384 so divide by 4! 384 / 4! = 16 Coefficient of x^4 is 16, store that. Subtract that from the original expression: Form a table of differences of that: 3rd difference is 576 so divide it by 3! Coefficient of x^3 is 576/6 = 96, store that. Subtract that. Form a table of differences of that: 2nd difference is 432 so divide it by 2! Coefficient of x^2 is 432/2 = 216, store that. Subtract. Form a table of differences of that: First difference is 216 so divide it by 1! Coefficient of x is 216, store that. Subtract. Form an x y table of that: Zeroth difference is 81 so that is the constant term. Put it all together: Multiplication of polynomials by differences. Wunderbar! Did you know it was because of work like this that Charles Babbage designed his difference engine and later his analytical engine. For that reason he is called the father of computers. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #9 20130929 17:51:32
Re: Explanation?hi bobbym, Bob You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #10 20130929 19:57:01
Re: Explanation?Hi Bob; Now you just make a difference table from that. 625  81  544, 2401  625 = 1776 6561  2401 = 4160 etc. repeat for each line until you come to constant differences. The first line are called the first differences. The second line is differences of the differences and is called second differences. I left out the operator notation so as to not confuse the OP. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #11 20130929 20:14:10
Re: Explanation?OK, got it now. Thanks. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #12 20131001 10:04:04
Re: Explanation?That's really cool! So how can you approximate derivatives with that? #13 20131001 18:19:31
Re: Explanation?Hi; This last line is the zeroth difference, the first line of the difference table. Now just take that line and plug in to a formula: This agrees with the exact answer of 0.3535533... very well There is an even more direct way if you want to see that to. A thought: It is important to remember that numerical mathematics and classical school taught math are very different. Things like integration are hard in math but easy in numerical math. In the same way differentiation is comparatively easy in math but difficult in numerical math. Difficult in the sense that numerical differentiation using differences is highly unstable and subject to roundoff error. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #14 20131003 11:06:29
Re: Explanation?Thank you! That is really cool! #15 20131003 12:38:14
Re: Explanation?I converted it into sigma notation, but I'm not sure how to write that on the computer. Last edited by Ma123 (20131003 15:01:33) #16 20131003 17:09:47
Re: Explanation?Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #17 20131003 17:12:29
Re: Explanation?Somewhat. Is that summation useful at all? I thought it was pretty cool how it followed coefficients from Pascal's triangle. #18 20131003 17:18:16
Re: Explanation?Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #19 20131003 17:33:44
Re: Explanation?Sure, I will find 4! using the summation: #20 20131003 17:41:23
Re: Explanation?Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #21 20131003 17:42:37
Re: Explanation?Yeah! That's what I was looking for. Is this not a known formula? #22 20131003 17:48:19
Re: Explanation?Hi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #24 20131003 17:52:33
Re: Explanation?When we speak of practical uses then we get into efficiency. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #25 20131003 17:55:24
Re: Explanation?I couldn't imagine it would be. Oh well, it was still fun to derive! 