My Math Theory

In My Humble Opinion
1

On a lighter note (couldn’t we all use something lighter now?), I thought I would share with you something I discovered in math. At the present time, I am sure there is nothing new to be discovered in math. And therefore, I probably am not the first to notice what I am about to share. It is still interesting though. In case you don’t know, I am theorizing there may be an easier way to find irrational numbers. Engineers used to replace complex math with simple addition, with slide rules. My theory is much like that.

I have shared this with other message boards. And invariably, people say it must be a fluke, that just is the product of a calculator. But if you read it all the way thru, you will see it is not (the ‘pattern’ is already there).

Please read:

I think I may have come upon a new discovery in mathematics (do we still make “discoveries” in math?). But first some background.

Anyways, I play around a lot with calculators. And I’m not ashamed to admit it either. You know you can learn a lot that way, believe it or not. And I have even shared some of these things with others on the internet in the past.

This next one is equally as bizarre as the rest. When you take the square root of .111111… you get .3333333… naturally, since the square root of one-ninth is one-third. But one time, just as a lark, I thought I’d square root .11 alone. Then .111 (again, only three digits), etc… Long story short, you get the following pattern: 0.33333333331666666666624999999998. As you can see, the .33333… pattern is followed by an intrusive 1666666… pattern, and a 2499999… pattern (leading ultimately to 25, presumably).

It happens with other numbers too. Take .44444… The square root of this repeating decimal is .66666…, two-thirds, naturally. But when you do the same thing, you get 0.66666666663333333333249999999996. A “333…” pattern emerges, and then again that “25” pattern.

It doesn’t just happen with these. Consider .9999… That equals one, of course. But when you do the same, you get 0.99999999994999999999874999999994. Now, you get “5” and “75” as your hidden pattern.

Also odd, is that these patterns are “put off” until infinity. Which I guess is permissible, even if they are never part of the actual number.

What is the explanation for these strange patterns? Because personally, I think I have hit upon something big and (possibly) undiscovered. I think I may have also hit upon a simpler way of finding irrational numbers. (That is, if they show unique patterns too–just think of how the slide rule uses simple addition and subtraction to find multiplication. Think about it.)

And BTW, I don’t think it is just a phenomenon found in calculators alone. the square root of .1 is 0.3162277… (note the “16” already there). In short, the pattern is already there, for all to see. It’s clearly not a fluke.

So what do you all think? As I said, I offer it mainly as a diversion. But it is interesting. And what about Pi? Could we find solutions to Pi using simple addition, as it were?

:slight_smile:

2

You seem to have a lot of interesting theories. Do you ever follow them through? If yes: which one was it?
And the answer to your last question is: NO

3

let me check this before I reply

4

As with many things, calculus can help.

For small cases where d is small, the square root of (x-d) is going to be close to the
square root of (x) -1/2 (1/square root of (x)). For 0.11, d is 0.001111111 (1/900), so the square root of (0.11) is very close to square root of (0.111…) - 3/2 * 1/900 = 1/3 - 1/600 = 0.33166666666 - very close to what you got. What we’ve done is estimate the square root of 0.11 by using the first derivative of the square root function. We can get a better answer by using the second derivative too, which will get an answer even closer to what you got (but I won’t do that work now). It is fun though.

5

You can compute sqrt(1/9 - 1/(9 10k)) using the Binomial theorem

6

I think a lot of my math faculty colleagues would be surprised to hear there isn’t much left to be discovered in math, either theoretical or applied.

7

There’s a number between 112,566,743,320 and 112,566,743,322 that hasn’t been discovered yet.

8

I don’t understand what you mean by “a simpler way of finding irrational numbers.” Most numbers are irrational. Most square roots are irrational. And most importantly for this claim, all square roots of numbers that are not a perfect square are irrational. Since you’re starting with non-integers you’re bound to get an infinite decimal root, although you can never prove that using a calculator, which returns only finite results.

9

You find that- tenure granted!

10

Great. That’s an appropriate end to your OP, to ask the experienced mathematicians to explain what’s going on, because what you wrote about was indeed somewhat interesting.

But you make yourself a laughing stock by continuing with this…

Really, how on earth could you think any of this is remotely plausible?

11

A slightly simpler (IMO) version of the same proof:
Consider the prime factors for any integer p. If all have even powers, p is a perfect square, and we can stop. Otherwise, there is at least one prime factor with an odd power. Consider now the two numbers pb2 and a2 for any integers a and b. pb2 has, like p, at least one odd-powered (prime) factor. a2 has all even-powered factors. Thus they are different numbers and we can write:
|pb2 - a2| >= 1
And:
|p - a2/b2| >= 1/b2

Thus, for any rational a/b, its square must differ from p by at least 1/b2.

Another one, even simpler:
Take an integer p with square root r. Assume r=a/b for integers a and b in lowest terms. Then a2/b2 is in lowest terms as well since squaring a number cannot create a new prime factor, only change its power. But we know that a2/b2 = p/1 in lowest terms, so b2=1 and b=1. Hence, a square root is only rational for perfect squares.

12

While I agree, I suspect there’s not much at the level the OP is looking at.

That’s not a sleight, either: the same is true for the level of math I tend to look at, too. I always wind up just proving something that was already proven, but it’s often fun. I did well in high school math (top of my class in Calculus, even) but that’s nowhere near the expertise I see in those actually attacking unsolved problems in math. I need the math communicators to heavily simplify anything they do for me to understand it.

13

Why do you say that?

E.g., inspired by Ramanujan, but in 2016, not earlier, we got the identity

           ∞                     ∞
         _____                 _____
         ╲                     ╲
          ╲                     ╲
           ╲       7             ╲        7
            ╲     n               ╲      n
17 =  32⋅   ╱  ──────── - 8192⋅   ╱  ──────────   
           ╱    π⋅n              ╱    4⋅π⋅n
          ╱    ℯ    - 1         ╱    ℯ      - 1
         ╱                     ╱
         ‾‾‾‾‾                 ‾‾‾‾‾
         n = 1                 n = 1
14

Huh? That was not proven with the math either of us know. I doubt the OP even understands the concept of summation given what he said. I do know what summation is, to some extent, but I have no idea how the person found and proved the forms they give would in general produce primes. That paper seems to me not to actually show their work–though I’m sure it makes more sense to those who know what he did.

I actually seem to remember watching a video on this. But it sure looks like it requires number theory to me, which I do not know. And, since it’s about large primes, I suspect it also takes decent computer power, like most prime discoveries.

15

Whereas I am even more certain that there is something new to be discovered in math, even in low-level math like arithmetic. Gödel proved 90 years ago that there is always something new to be discovered in math, no matter how much you’ve already discovered.

Though, that said, this wasn’t one of them.

16

He did not start with the proof, nor was there a proof indicated on the linked couple of pages. But he was able to discover some interesting-looking identities by numerically playing around with those sums.

17

Some quick emails will save them from having to wake up tomorrow morning.

18

That’s not what he said though. He didn’t qualify “at his level”. He made a declarative statement, which he then contrasted with the fact that he discovered something possibly “big”.

If he’d like to explain for himself what he meant, that’s fine. But I don’t think you speaking for him is better than my going by his precise words. Especially for someone who wants to be considered a researcher.

19

On the subject of maths being finished, it occurs to me that in decimal expansions alone we still have the question of how to prove whether an expansion is normal or not, including whether pi is normal.

Although ISTM, that just looking at a finite string of numbers is unlikely to yield such a proof, so the OP is unlikely to find it using his existing methods.

20

Up to a point - and only a point - I get where the OP is coming from. (Treat the next paragraph as TLDR as there is nothing of substance in it.)

I also like to play around with numbers to find simple patterns and quirks - it’s fun. I used to do it all the time in big lecture halls during uninteresting classes in grad school, but only with a pencil and paper, no calculator. One I remember is discovering patterns in each column of digits for 2^n. For example (ignoring 2^1 and starting with 2^2 or 4), the tens column repeats, probably through infinity, in this pattern: 00136251249986374875. Each column has its own pattern and if you knew them all you could use them to create the next number in the sequence, without actually multiplying the previous number by 2. There are also multi-digit patterns, if you want to get a little fancier. (The pattern written out here could be written with the last two digits, ie 04, 08, 16, 32, 64, 28, 56, and so on.)

But … so what? I’m sure the sequences are easily enough proved, should someone care to write out a proof (though why they’d waste their time I can’t imagine). However, there is no reason to think said proof would lead to any exciting mathematical insight that would help solve any existing problems in mathematics. It’s just arithmetic in action. If we used base 12 instead of 10, the patterns would be different, through equally predictable.

So where I part company with the OP is when he says:

I think I have hit upon something big and (possibly) undiscovered.

I have no such illusions. Even though some of the interesting math problems out there are fairly understandable (getting the gist of the Millennium Prize Problems, or at least some of them, isn’t impossible for an interested lay person), SOLVING them is another matter entirely, as is understanding why they’re important.

So, how likely is it that an ordinary person goofing around with a calculator is going to stumble on a genuinely significant mathematical property? Pretty darn slim. My advice? Jim, just have fun with it and don’t veer into grandiosity.