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#rational numbers
escuerzoresucitado · 6 months
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Integers like their personal space. Real numbers are clingy huggers. Rational numbers look like they're super touchy-feely but they're actually just standing really close together.
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By random chance, the decimal expansion of pi briefly begins repeating itself after 6954 digits before resuming its randomness
3.1415926535...987571415957811...
That's the longest repeated string in the first million digits, so a really good rational approximation would have the first 6954 digits repeat infinitely. You can create any rational number with n repeating digits by taking a string of length n and dividing it by an equally long string of nines.
0.123123123... = 123/999
0.141591415914159... = 14159/99999
Of course, these may not be in simplest form
0.123123123... = 123/999 = 41/333
0.363636... = 36/99 = 12/33 = 4/11
0.102911029110291... = 10291/99999 = 251/2439
So the first 6954 digits of pi over 6954 nines could very likely be simplified. At that magnitude, the odds that they are co-prime (share no factors) is very slim
Scratch that. I plugged the first 6954 digits into a prime factorization calculator, and the first three factors it was able to find were 99551 × 18298401827 × 104229615391 (100k, 18.3 billion, 104.1 billion). It took 30 minutes to find those three, so it could take HOURS or DAYS to calculate the remaining factors because it still has over 6,900 digits to get through, but this makes it seem more likely that the string is co-prime with 9999999...9999999 just because there are multiple orders of magnitude between factors, so the odds of any two random 12+ digit numbers being the same are slim. I don't know, I'm not a statistician.
3 + (1.4159...98757 x 10^6953) / (10^6954 - 1) ≈ pi
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falseandrealultravival · 10 months
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Are there more irrational numbers than rational numbers? ~ Set Theory: Mathematics Note -11 (Essay)
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Cantor (1845-1918) conceived of "set theory" and used this concept to cut into "infinity". This is a field that other mathematicians have not dabbled in because of its monstrous appearance.
The weapon is "one-on-one correspondence". For example, natural numbers 1, 2, 3, 4... and even numbers 2, 4, 6...
Which one is more? Suppose you asked me. It is normal to think that "the even numbers are the natural numbers minus the odd numbers, so there are more natural numbers", but Cantor's answer is different. An even number 2 is the first, an even number 4 is the second, and so on. Since all natural numbers and even numbers can be in one-to-one correspondence, it means that there are the same number of natural numbers and even numbers. In this way, we can conclude that there are as many fractions (= rational numbers) as there are natural numbers.
"Then what about irrational numbers?" --Here is a very interesting proof. The goal is to reach a conclusion based on the law of contradiction that "there are more irrational numbers than natural numbers."
If it is possible to have a one-to-one correspondence with the natural numbers, the irrational numbers will be expressed as decimals (rational numbers will be converted to recurring decimals) and all of them will be arranged to create a list. Now, take a number that differs from the first number in the list in the first decimal place, and from the number in the second list in the second digit, and so on. The number obtained by the procedure is not in the list ! ・・・ I was able to derive a contradiction due to contradiction. Irrational numbers are infinitely more than natural numbers. In an elegant proof, this is called the diagonal argument.
Kronecker (1823-1891) raises an objection. This person was quite eccentric and professed that the natural numbers were created by God, and that all other numbers were created by humans. Kronecker seems to have had the ambition to rebuild all mathematics on the basis of natural numbers. That's why he couldn't help but hate Cantor, the guy who came to the conclusion that there are more irrational numbers than natural numbers. Kronecker persecutes Cantor at every opportunity. As a result, he suffered from mental illness and spent the last half of his life in and out of mental hospitals.
(2023.04.22)
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mekesh · 1 year
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What is a Rational Number?
A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero. learn more about rational number from the CK-12 FlexBooks.
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egophiliac · 6 months
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GRIMS COMING!!! You gonna pull for him??👀👀
I'm gonna try, but...
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I have mere hours to decide if I want to make one last attempt at Malleus or save a few to try for Grim...and this is all before the new event reveal on the 16th. truly the most difficult choice of our modern times. the gacha is getting its revenge for all of my Lilias.
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evercelle · 1 year
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time to raise the bpm ♪
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pink-sheep-sorcerer · 11 months
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he loves MC so much lol literally beaming i can't
(no my mind can't be changed please start a Solomon love chain i have so many more examples)
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aro-ortega · 8 days
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me when i think about or am reminded of chens canon apartment
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minhosimthings · 2 months
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Jake and Changbin would definetly put a rose quartz in their s/o's room even if they are together
Someone please come and scream at me to study JAKE IS DISTRACTING ME
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dabidagoose · 4 months
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Well it's been a good ride soldiers see you all in purgatory glassheart fans🫡
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haggishlyhagging · 11 months
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Thus, for the Western world, a description was abstracted from substance and deified into an "objective system." Thus Pythagoras introduced "logical reasoning" (rather than organic reasoning) into the domain of religion and ritual, laying the foundations for the later "intellectual theology" of Christian scholars and the Western world in general. This combination of mathematics and theology characterizes religious philosophy in Classic Greece, in medieval Europe, in Reformation and modern times. Consequently, "God" was no longer intuition or ecstatic vision, not profound ontological experience, but a kind of rational machine "explained" by a few scholarly men. Plato, St. Augustine, Thomas Aquinas, Descartes—all inherited this numerical philosophy, and shared a logical admiration of the "timeless" and static, utterly transcendent and aloof "pure spirit" of the Father God. Utterly abstracted from flesh and earthly cyclicity, this Father God was as unappealable as pure number. Pure number is always elevated above the messy world of mere life. During three hundred years of witch-hunts in Europe, this rationalistic theology was used as a legal and clerical weapon against what still remained of the ancient womencultures. Medieval scholiasts not only debated endlessly, and in dead earnest, about how many angels could dance on the head of a pin; during obscene torture, priests and civil judges also hounded their female victim with obsessive questions about the precise number of imps she employed, the total number of times she had intercourse with the Devil, the exact size and length of his "member," and so forth. The dogmatic insistence on number was supposed to cast an aura of "rationality" and "objectivity" around their otherwise bizarre and sick proceedings. In modern forms—e.g., psychological measurements, behavioral norms, notions of quantifiable "sanity" and "adjustment"—deified number is still used as a power-tool against "social deviants."
-Monica Sjöö and Barbara Mor. The Great Cosmic Mother: Rediscovering The Religion of the Earth.
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skitskatdacat63 · 1 year
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2005 French Grand Prix - Fernando Alonso (ft. Kimi Räikkönen & Michael Schumacher)(my personal post-race highlights)
+ bonus uh...whatever Michael's doing here
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mylittleredgirl · 8 months
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decided to more or less restart the strange new worlds experience from the beginning and just watched “ask not.” i was going to post about how wild it is that starfleet just does this stuff to undergrads and then i remembered that a psychological trauma test was part of wesley’s SATs when he was literally fifteen. no wonder no one is normal in starfleet.
also i haven’t watched the new season except in gifs, but “we need to know that you’ll honor your commitment to starfleet even when those you hold dear are on the line, even when old wounds are triggered and loopholes appear” feels like idk. a theme. that might become important later.
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steeb-stn · 2 years
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Obi-wan and anakin actually shutting up and just standing there dumbly for like 5 seconds after padme collapses, as if her fall breaks the spell, then they move as one and and get her back on the ship and to medical care. 20 hrs later, the three of them in the isolated little medical center w no one but the medi-droids and two wailing babies, avoiding each other’s eyes and wondering what the hell theyre gonna do next
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moxymaxing · 8 months
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stuck between the two competing mindsets of “scary is just a kid who’s being manipulated” and “I’m gonna kill anyone who does anything against normal which negatively affects him”
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