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October 23rd, 2015, 11:24 AM   #1
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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.
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October 23rd, 2015, 12:55 PM   #2
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What was the "it" that you did with different numbers?
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October 24th, 2015, 12:23 PM   #3
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I'm not referring to things you were the first person to discover (that's hard).
Not so! It's as easy as falling off a log[arithm].

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 hunky-dory until I stuck my foot in the pot and stirred things up:

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Originally Posted by Benit13 View Post
Current textbooks are fine
The reactionaries would boast of their decades of experience:

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Originally Posted by skeeter View Post
I taught all levels of high school math and physics for 23 years
Since 100% of the reactionaries' experience is doing things the way they have always been done, it is easy for them to do a little data dredging to find the rare occasions when it actually works and then claim that these examples represent an unbroken record of unmitigated success.

They would need a reminder of what "proof" means:

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Originally Posted by Grozny View Post
That's not proof, that's anecdotal evidence.
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:

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Originally Posted by Benit13 View Post
There are virtually no references to existing work. Can you demonstrate that there is indeed a problem? Can you demonstrate that your proposed method hasn't been considered already by others? Can you demonstrate that a new method is required rather than other strategies for improving teaching?

This is the key flaw in your work... in order for it to be taken seriously, you need to conduct a thorough literature review. This means a lot of reading!
But the one thing they will NOT do is find a counter-example to my claim that the Law of Logs is original. The burden of proof is actually on them to find a counter-example to my claim of originality, not on me to read a lot of books looking for someone else who already thought of my idea. One counter-example would disprove my claim of originality, but a thousand examples of extant authors writing crap would just get me accused of cherry-picking.

The reactionaries would quote verbatim from the Education classes they took in college:

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We invented games, activities, competitions and invented all sorts of methods for presenting ideas.
For many of them, those endless and repetitive classes in the Education building were the ONLY classes they took in college - they never once darkened the door of the Mathematics building. Why bother? Their actual teaching method is to read the students' textbook one chapter ahead of the kids and then morph into a game show host when it comes time to explain the material.

And, when all else fails, the reactionaries would misquote me:

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"I hate for math ..."
Your personal opinion does not belong in the abstract.
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."
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October 24th, 2015, 02:32 PM   #4
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What was the "it" that you did with different numbers?
I compared 4*6 and 5*5. I did it with different numbers such as 3*5 compared to 4*4 and other sets of numbers that could be expressed as x-1, x, and x+1.
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October 28th, 2015, 04:48 PM   #5
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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.
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October 28th, 2015, 07:17 PM   #6
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Quote:
Originally Posted by EvanJ View Post
Although I couldn't explain it at the time, I had discovered that (x + 1)(x - 1) = $\displaystyle x^2 - 1$ by myself.
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.
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October 29th, 2015, 08:15 AM   #7
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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.
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October 29th, 2015, 08:24 AM   #8
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Just a couple of little things: Pascal's triangle and a way to find the area of an ellipse without using Calculus.

-Dan
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October 30th, 2015, 03:25 AM   #9
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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
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October 30th, 2015, 03:36 AM   #10
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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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