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.

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