October 23rd, 2015, 12:24 PM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 622 Thanks: 85  Did You Make Any Math Discoveries On Your Own?
I'm not referring to things you were the first person to discover (that's hard). I'm referring to anything discovered on your own before being taught it. I started Product Of The Least Common Multiple and Greatest Common Factor Of Whole Numbers about one thing I discovered. Another thing discovered working with small numbers is that 4*6 is 1 less than 5*5. I did it for different numbers and thought that it was more than a coincidence. Although I couldn't explain it at the time, I had discovered that (x + 1)(x  1) = $\displaystyle x^2  1$ by myself.

October 23rd, 2015, 01:55 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,957 Thanks: 1844 
What was the "it" that you did with different numbers?

October 24th, 2015, 01:23 PM  #3  
Member Joined: Oct 2015 From: Arizona Posts: 57 Thanks: 0 Math Focus: Analysis  Quote:
When I was thirteen years old, I invented the Law of Logs. It was obvious. I was taking trigonometry simultaneously and had just learned about the Law of Sines, so I was like, "why aren't we doing exponential decay the same way?" Now, 36 years later, I have given My Math Forum the (dubious) honor of being the site where I would unleash my original idea on the world. In the meantime (1999) I invented Axiomatic Economics, so I was able to anticipate what sort of reaction I could expect. There would be reactionaries who would insist that everything was hunkydory until I stuck my foot in the pot and stirred things up: The reactionaries would boast of their decades of experience: Quote:
They would need a reminder of what "proof" means: The reactionaries would boast of how well read they are in the current method, which is not surprising since they have spent a lifetime doing things exactly the way they have always been done, and have passed through many editions of essentially identical textbooks: Quote:
The reactionaries would quote verbatim from the Education classes they took in college: Quote:
And, when all else fails, the reactionaries would misquote me: It is amazing how much the meaning of a sentence can be changed by removing the letter "f" in order to change "If" into "I."  
October 24th, 2015, 03:32 PM  #4 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 622 Thanks: 85  
October 28th, 2015, 05:48 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,957 Thanks: 1844 
So the "it" was the comparison of (x  1)*(x + 1) and x²  1, where "x" can vary, but "1" stays the same. If you had compared (x  n)*(x + n) and x²  n, you might have gone on to find that the two match if n is 0 or 1, but not otherwise.

October 28th, 2015, 08:17 PM  #6 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 2,000 Thanks: 132 Math Focus: Trigonometry  Me, too. And when I tried to write it in algebraic form, I was disappointed to know that it has been taught as early as middle school. Neither I or my schoolmates realized this because our teachers have never taught to count, for example, 24 x 26, using this method.

October 29th, 2015, 09:15 AM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
I remember proving, with great labor, the formula 1 + 2 + ... + n = (n+1)/2 * n. It seemed like it had to be true, but it was very hard for me to justify why it must be true for all n. I wondered if anyone else had discovered this.... I think I was in 2nd grade when I found the formula/trick but I didn't prove it until a few years later.

October 29th, 2015, 09:24 AM  #8 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,914 Thanks: 774 Math Focus: Wibbly wobbly timeywimey stuff. 
Just a couple of little things: Pascal's triangle and a way to find the area of an ellipse without using Calculus. Dan 
October 30th, 2015, 04:25 AM  #9 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
A bunch of us did a lot of work here on My Math Forum about an observation made by mathbalarka and originally explored by the mathematician Liousville. Diophantine Equation with Multiple Variables I'm pretty sure we all discovered something 'new' , at least on a personal level and maybe beyond because there didn't seem to be much literature on the subject at the time. It was a great collaboration and exploration of ideas from many My Math Forum members , I miss those days a little bit 
October 30th, 2015, 04:36 AM  #10 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,132 Thanks: 717 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
I came up with a formula for the area of any regular polygon with side length a and number of sides n when I was 16 or so and was quite excited to find that this formula wasn't in any of the school textbooks. Eventually, a couple of years later, I found a random maths textbook in a University library that was written in the 1960s that had the formula in it and, when internet finally got really big, it was trivial to find the formula online. Nevertheless, it was a nice feeling anyways I sometimes wonder if the joy of coming up with a formula like that spurred me on to do physics research! 

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