logo Bad Maths -- Started 2020/11/17 -- Updated 2021/11/30
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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. Since by definition it is the average, half the numbers in the infinite series must be longer, but half of infinity is still infinity, and half of that is infinite still, thus all the numbers available to pick must be 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 is valid, then its reduction to symbolic logic will return its formal proof, but I'll let someone else do that work. Serious Problems for Cantor's Diagonalization: The main thrust of my arguments here are that Z and R are not of separate cardinalities. There are three issues I've found with diagonalization while trying to do the metalogic for it. The first occurs when trying to reverse the list; if you put the natural numbers in a list denominated by the real numbers instead of the other way around (the way Cantor does it), you attain the same outcome of new infinites, meaning the conclusion would be that the set of natural numbers is larger (and uncountable) than the set of real numbers (which are then the countable set - a contradiction since this set contains the natural set), and this is of course the opposite of what Cantor concluded. To demonstrate this, look at the following images modified from Veritasium's video Math Has a Fatal Flaw:
Above, you see all I have done is swapped the natural index numbers with the list of real numbers from 0.0 to 1.0. The randomized list of natural numbers is enumerated just the same in the right list as the real numbers are in the left list. All we do now is apply the diagonalization technique Cantor uses on the new list the same as the old list, shown below.
What you see here is that the new natural number we generate from the diagonalization method similarly 'does not appear in the list', much the same as the new real number Cantor generates from the diagonalization method. ~ THIS IS A PROBLEM ~
What this means is that had Cantor started with indexing the Reals instead of the Naturals, he would have concluded that the infinite set of natural numbers was qualitatively larger than the set of real numbers. I believe he purposefully avoided doing it this way because the set of natural numbers is contained inside the set of real numbers and having the set of natural numbers be larger than the set of reals means the set of real numbers would be both a countable infinity while being large enough to contain an uncountable infinity - an apparent contradition. Either I am just overtly wrong in a way I do not understand, or I am right and Cantor's method is flawed, or as a third option, both I and Cantor are right and the contradiction is solved by simply saying our initial intuitions were wrong and in fact uncountable infinities are actually qualitatively smaller than countable infinities, which then explains how the countable set of reals can contain the uncountable set of naturals.
However the above is resolved, it is only one of three issues I have discovered with diagonalization. For the second issue, I believe his derivation of different cardinalities is due to his mixing of potential infinity with actual infinity, something already known to be improper in metaphysics as far back as their discovery by Aristotle. Whether you use my reversed list or Cantor's original list, both ways require the use of potential infinity for the index numbers and actual infinites for the numbers enumerated to the right of the index. So of course this would appear to be different kinds of infinity, because you have begged the question and baked a pre-supposed conclusion into the formulation of its proof. This seems like an obvious problem to me, but if it is not clear to you the reader as to why this is wrong or how this works, then I point to the third issue with diagonalization below.
Cantor's diagonalization requires the ordering of his real numbers to be random, as we shall see that reordering the list from smallest to largest real numbers demonstrates the diagonalized new number in fact already appears on the list. If Cantor's list was in order from smaller to larger infinite decimals, we would see the list ordered as 0.111, then 0.112, 0.113, and so on. If we then applied diagonalization we would get 0.2 from the first real, 0.02 from the second, 0.002 from the third, and so on, resulting in 0.22 as our newly generated real number. But of course, when ordered from smallest to largest decimal, 0.22 would obviously appear later in the list.
I contend that if you don't believe I applied Cantor's diagonalization properly, then you have not been careful to note the kind of infinites I used. In my example, I do not mix potential with actual infinity, and thereby the list of 0.11n's we enumerated earlier results in an infinite number of reals that lead with 0.11, meaning we generate an infinite series of 2's. If you think this problem is resolved by baking the idea of uncountability back into real numbers, then simply swap the naturals with the reals as I did in the pictures earlier and you'll see the problem I just described reoccurs.
So even if one or two of these issues can be resolved, the third will still re-assert itself, making the notion of distinct cardinality between sets of infinites in math an untennable position. If I am horribly wrong about this, and I hope I am, I would greatly appreciate someone who knows better taking the time to explain to me why that is the case. 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. Multiple-Infinite Decimals: 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 = 9.0000900009
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 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 systems. From a meta-logical perspective, when this is factored in, it does result in 'clean' outcomes.
As proof for this, look to the standard that the number 0.001 is equal to 0. Since there are different kinds of infinities in math, not including my objections to the counter in a previous section on this page, the countable infinity in the number 0.001 will be overcome when divided by a number that is an uncountable infinity. The infinite part of 0.001 can be skipped over by an uncountable infinity, leaving the 1 at the end as a non-arbitrary part of the divisor. This makes what I've been calling the 'bar' numbers above worth considering for application in cleaning up infinites.
"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 seen. 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
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 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.... 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. I Need Help Learning: 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.