logo Bad Maths -- Started 2020/11/17 -- Updated 2021/9/6
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I never learned math in school, so as an adult I have been trying to teach it to myself. The things I have found on this page are probably wrong, and the things I am not wrong about were likely discovered long ago, but this page is just for fun so hakuna matata. Infinite Series & Randomness: If you have an infinite series of whole numbers from one to positive infinity, and you randomly select a number from that series, then the length (number of digits) of that selected number will also be infinite. This is because the average number of digits for numbers in a series of integers from one to infinity is itself infinite. Conversely, there is no actual randomness in infinity; if you are to randomly pick a number from one to ten, each number has a one-out-of-ten chance of being picked, but if you are to randomly pick a number from one to infinity, each number has infinitely low odds of being picked. I posit that since these odds asymptotically approach zero, the odds of randomly picking any number out of an infinite series is actually zero, meaning you simply could not pick a number. Goldbach's Conjecture: If not already familiar, read this. All even numbers are predicated off of 2, and 2 is a prime, so all reduction of numbers predicated by 2 can be reduced to primes of the predication value, which in this case is 2 (so 2 primes). The framework then is such that any member of a factor-tree based on some number X will inherit all properties of X and will be reducible to an X-unit-count that share one of those given properties.
The way this looks for the existing conjecture: Every number factorable by 2, past 2, can be expressed by 2 units of some property of 2 (namely primehood).
The way this looks for new conjectures: 1 - Every number factorable by 1 (all whole numbers), past 1, can be expressed by 1 unit of some property of 1 (namely empty product or unity/identity). 3 - Every number factorable by 3, past 3, can be expressed by 3 units of some property of 3 (namely merseene primes or fermat primes). π - Every number factorable by π, past π, can be expressed by π units of some property of π (namely irrational numbers or transcendental numbers). 2 - Every even integer greater than 2 can be expressed as the DIFFERENCE of 2 primes. > Goldbach's Conjecture modified to show that unit relations are arbitrary so long as the unit relations are of non-arbitrary units themselves.
If the form of this new framework is valid, then its reduction to symbolic logic should return its formal proof. The Percent-Unit Circle: This is a bad idea for many reasons but it's funny so I included it. If we remade the unit circle at base 100 instead of base 360, the numbers work out a lot cleaner and base 100 means you can use percent values to ascertain positions on a grid. This is much easier to visualize mentally for most people and it's objectively much faster to parse through. This also makes it possible to give %-%-% formated coordinates for points in 3D space. Again much easier to cognize than using 360 or π. As an important irony, this turns the unit circle into an actual single unit since 100% equals 1 whole, instead of "2 units of π," which by it's very description is not a unit circle but a two-unit circle. By happenstance this is also better for relativistic frameworks used in mapping galaxies. I assumed someone had already made this but I couldn't find it online so I did it myself and it gets used sparsely in some of the games we've dev'd on Snerx. If you want to look at other dumb shit we've done with percent-based relativistic frameworks you can check some of those out here. Non-Zero Decimals of Whole Numbers: 1/3 = 0.33, and 2/3 = 0.66, but what numbers out of a whole give us the other repeating decimals? If we wanted 0.66 out of 1 whole instead of 2, we get the following. ___________
1 / x = 0.66
1 = 0.66 • x
x = 1 / 0.66
x = 1.5
‾‾‾‾‾‾‾‾‾‾‾ But I contend that this number is actually 1.50015. Why and how? Dividing 1 by 0.666, we get 1.5015, by 0.6666 we get 1.50015, by 0.6666666666 we get 1.50000000015, and so on. By 0.66 what we get is an infinite series of infinites, namely the infinite bar between 15's, that is 1.50015001500, repeating. This is the same as saying 1.5 with an infinite series of zeros following it, and then after infinite zeros there is a 15 followed by another infinite series of zeros, etcetera. Another way of saying this is that as the antecedent (divisor) grows in decimal length, so too do the number of zeros between the numbers of the decimal of the consequent (quotient). Therefore with an infinitely-repeating-decimal divisor you get infinitely repeating zeros followed by a finite series of numbers, the set of which itself then infinitely repeats, in the quotient.
I've had people argue with me that, "This is not how fractions work," and if we were using whole-number fractions, they would be right, as one divided by two-third's becomes three over two and then cleanly resolves as one-and-a-half. But we aren't concerned with whole-number fractions here; the property I describe shows that the numbers in decimal format are not 'cleanly' divided. 1 divided by 3 gets you 0.33, but as now described, 1 divided by 0.33 does not seem to get you 3.
I'm sure this already has a formal name but I can't find anything about it online so I've independantly developed it here as Snax's Bar Numbers lol. I have written out some of the bar numbers below so you can see their weird properties.
0.11 is 1/3 of 0.33 so 3 by 3 should mean 9, and in fact we see that 1/9 does equal 0.11; this then should be the grounding for the others as the other bars are multiples of this first one. ________________________
1 / x = 0.11
1 = 0.11 • x
x = 1 / 0.11
x = 9.00900900 repeating
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ This means 900 is the bar number attained from 0.11. To re-iterate why this happens we can follow the non-bar series, which results in: _________________________
1 / 0.1 = 10
1 / 0.11 = 9.0909
1 / 0.111 = 9.009009
1 / 0.1111 = 9.00090009
1 / 0.11111 = 0.0000900009
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ You keep adding zeros per n decimals of 1 from here, ultimately giving us the bar number
we attained, 900, for the series of 0.11. A list of these follows. For 0.11 we get 9.009, or 900 as the bar number.
For 0.22 we get 4.50045, or 4500 as the bar number.
For 0.33 we get 3.003, or 300 as the bar number.
For 0.44 we get 2.2500225, or 22500 as the bar number.
For 0.55 we get 1.80018, or 1800 as the bar number.
For 0.66 we get 1.50015, or 1500 as the bar number.
For 0.77 we get 128571428571428571428700, as the bar number (see below).
For 0.88 we get 1.125001125, or 112500 as the bar number.
For 0.99 we get 1.001, or 100 as the bar number. N.b., 0.11 and 0.99 are inverses of each other but there are no other inverses. Notice also the strangness of 0.77's bar number and how no other bar creates the same level of noise (so far). 0.77 is more dynamic and there appears at first to be no upper bounds on the series length or mutations, however it does resolve, but I don't know how many digits out it takes to resolve since I only tried up to 12 and then skipped to 30. ______________________________________________________________________________________
1 / 0.7 = 1.42857142857
1 / 0.77 = 1.29870129870
1 / 0.777 = 1.28700128700
1 / 0.7777 = 1.28584287000128584287000
1 / 0.77777 = 1.28572714298571557144142870000128572714298571557144142870000
1 / 0.777777 = 1.28571557142985714414285842857271428700000128571557142985714414285842857271428700000
1 / 0.7777777 = 1.28571441428572714285842857155714287000000128571441428572714285842857155714287000000
1 / 0.77777777 = 1.28571429857142870000000128571429857142870000000
1 / 0.777777777 = 1.28571428700000000128571428700000000
1 / 0.7777777777 = 1.28571428584285714287000000000128571428584285714287000000000
1 / 0.77777777777 = 1.28571428572714285714298571428571557142857144142857142870000000000
1 / 0.777777777777 = 1.28571428571557142857142985714285714414285714285842857142857271428571428700000000000
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ This is strange since it resolves to a non-zero infinite sequence in the bar number and further there are two infinite sequences within the general infinite sequence (demarcated by the double-bar). I believe this also serves as proof that the numbers past the infinite repetitions are non-trivial since there is not an infinite sequence of '00' at the tail of 142857, but rather '14287' instead. The bar number for 0.77 does not have a finite period length either, even though all the others do.
You might say, "Okay wahterever, but what practical application could this possibly have?" And to this I say that fractional divisors in equations for physical systems that result in very fuzzy statistical outcomes (like in quant) could probably be cleaned up by acknowledging that the quotients are not so 'clean' and this 'infinite-zero-finite-sequence-repeating' property (name pending) should not be ignored since infinite values appear often in some quant systems. From a meta-logical perspective, when this is factored in, it does result in 'clean' outcomes.
"But what about 0.12, or 0.69, or 4.20?" Most of the numbers I've looked at don't
result in much of anything interesting, e.g. if we look at 1.11 we get 0.9009 (or
900), totally in line with what we've already looked at. However, some numbers truly
have unique properties, like 1.22, which resolves as follows: ______________________________________________________________________________________
1 / 1.2 = 0.833
1 / 1.22 = 0.819672131147540983606557377049180327868852459016393442622950
1 / 1.222 = 0.818330605564648117839607201309328968903436988543371522094926350245499181669394435351882160392798690671031096563011456628477905073649754500
1 / 1.2222 = 0.818196694485354279168712158402880052364588447062673866797578137784323351333660612011127475045000
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‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ So what is the bar number for 1.22? Part of why this one is so weird to me is that the bar number is not constant. The number we get changes depending on if the infinite series of decimals is even in length or odd in length. If the infinite series of
decimals for 1.22 is even in length, then the bar number resolves as 819669421... and if the ifninite series of decimals for 1.22 is odd in length, then the bar number resolves as 818330578..., which I have written in my notes simply as '0.81bar 2‽'. In both cases, WolframAlpha suggests there is a repeating period length for the
sequence that follows 81 but the complete sequence is infinite and so WolframAlpha does not say what that period length is.
The series' period that follows 81 grows rapidly as you include more decimals and the same number that follows 81 resolves granularly to a definite series, yet is assumedly infinite in length at its absolute resolution. The fact that the number it resolves to also alternates depending on the even-ness or odd-ness of the infinite decimal series for 1.22 is strange in itself but what makes this more challenging for me is that given the meta-epistemics of maths (or what I know of this subject in limited fashion) is that a 'true' calculation of 1 divided by 1.22 could not actually resolve to any number since not only can the series not be determined in finite time but the series alternates its determination dependant on how the bar 'feels' (as a function of even-ness or odd-ness). This is quite perplexing and I need someone much smarter than me to explain this.
We have of course only looked at repeating decimals divided out of 1, and could go through the same infinite list of decimals and divide them out of 2, or 94, or π, and get new infinite lists of bar numbers, most of which would probably never be touched or be useful to anyone or anything. But I think it's neat. Help a 'cian Out: If any of you know a real mathematician, I need help understanding things that I can't find papers or videos for online. Things like why isn't there a constant for primes or coprimes despite so many theorems showing regular sets wherein primes occur (10 mod 1, 10 mod 3, 10 mod 7, and 10 mod 9)? Why are primes a function of division only, why is there not an analogue set of prime-like entities for multiplicatives? Since there isn't an analogue, is this just asymmetric logical operation in the system? Wouldn't this just mean the mathematical framework is broken and no longer consistently useful? Why did mathematicians stay autistically devoted to broken frameworks? I know these have answers, none of these are new questions, but I don't know where to look to find the answers.