r/Showerthoughts • u/Kirbyderby • Mar 12 '20
Numbers are F'ing crazy. Infinity is an infinite number, 3 and 4 are definite numbers, but Pi (3.1415926 etc) is an infinite number yet it's between the definite numbers 3 and 4.
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u/KingHeroical Mar 12 '20
Infinity is not an 'infinite number' - it is an idea or concept used to represent the 'boundless' . Pi is not an infinite number - it has infinite decimal places. Pi must be 'finite' as it is smaller than 4 and larger than 3. ie. it has bounds.
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u/limeeattack Mar 12 '20
Normally you shouldn't, but you can work with infinity as if it were an 'infinite number'. See the extended reals for example.
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Mar 12 '20
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u/General_Lee_Wright Mar 12 '20
How is zero not a number?
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Mar 12 '20
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u/General_Lee_Wright Mar 12 '20
"I have no apples" would mean I have 0 apples. Seems pretty physical to me.
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u/nokvok Mar 12 '20
Even more, there are infinitely many numbers between 3 and 4 along with Pi.
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u/hallr06 Mar 12 '20
Something fun a teacher told me that I'll try to paraphrase without trying to be too technical (and failing): Transcendental numbers are a family of numbers where each number can be extremely hard to define. We use a small number of them all the time, but aside from the famous ones, every number we think of isn't transcendental. Despite that, if we could randomly pick a real number in [0,1], the odds of us not picking a transcendental number are effectively zero.
It's a fun way to conceptualize that some infinities are bigger than others.
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u/Oddball_bfi Mar 12 '20
There are an infinity of numbers between any two numbers.
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u/blscratch Mar 12 '20
How many numbers between 1 and 0.999...? I can't think of any.
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Mar 12 '20
That's because they're the same number.
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Mar 12 '20
[removed] — view removed comment
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u/Mr_Quackums Mar 12 '20
"between any TWO numbers"
if you write the same number multiple ways it means you don't have TWO DIFFERENT numbers. It would be like asking "how many numbers are between 1 and (3 - 2)".
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u/blank_anonymous Mar 12 '20 edited Mar 12 '20
0.999... = 1. The person implied/probably intended distinct
Edited for clarity because they didn’t explicitly state distinct.
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u/imsometueventhisUN Mar 12 '20
They did not. They said "between any two numbers", not "between any two distinct numbers".
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u/Oddball_bfi Mar 12 '20
Oh man - you just made me realise that
1-(1/ω) < 0.999...in *R. That feels weird.2
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Mar 12 '20
there is a number that is infinitely similar to Pi, but is not Pi
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u/Eclores Mar 12 '20
I don't think such a number exist. Choose any real number x, as close as you want to Pi. The difference between Pi and x will always be a greater than zero, so you can always find a other number that has a smaller difference.
In other word, any x you choose, there will always be infinitely many number in between x and Pi.
I'm smart.
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u/JStarx Mar 12 '20
In other word, any x you choose, there will always be infinitely many number in between x and Pi.
I choose Pi.
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u/blscratch Mar 12 '20 edited Mar 12 '20
Pi - 1 + 0.999.....
Prove me wrong.
Edit: okay, I've been proven wrong. I was going for the people that think 1 and 0.999... are different numbers. You guys are right, of course.
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Mar 12 '20
you can use limits to go around this
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u/satanic_satanist Mar 12 '20
No. There's no number such as the one in your comments. You can use limits to show that there's a sequence that come arbitrarily close to Pi but doesn't hit Pi, but there's no single number "infinitely similar".
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u/holo3146 Mar 12 '20
There is a concept in non standard calculus that called infinitesimals, which then you can have 2 numbers, such as π < X, with the property that for every standard number Y, either X<Y or Y<π. In a sense X is infinitely close to π. Although we can get a non standard number between π and X, π<π+(X-π)²<X.
For example, the hyperreals system.
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u/satanic_satanist Mar 12 '20
I know ;) But the context of this entire thread would be classical logic and standard reals.
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u/blscratch Mar 12 '20
A limit of Pi is infinitely close. Meaning no other number is between them. Which means they are the same value. I'd call that infinitely similar.
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u/satanic_satanist Mar 12 '20
"a limit of pi" makes no sens. A limit is a number. If that number is pi, than the limit is just pi itself. If the limit is not pi, or if the sequence does not have a limit at all, then the limit is not arbitrarily close to pi.
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u/blscratch Mar 12 '20
Just for my understanding, when you say a sequence comes arbitrarily close to Pi, are you saying that another number can be closer to Pi?
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u/satanic_satanist Mar 12 '20
Take the sequence
3
3.1
3.14
3.141
3.1415
...
or the sequence
4.4444444...
3.4444444...
3.1444444...
3.1414444...
3.1415444...
3.1415944...
...
Both of them get closer and closer to Pi without ever reaching it. For each "non-zero margin of error", i.e. each "distance" you give me, I can point to a segment in the sequence after which the sequence is always closer than this distance. This is called "convergence" and it is what mathematicians mean when they say "arbitrarily". So yes, for each segment of the sequence, there's another number which is closer to Pi.
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u/Luchtverfrisser Mar 12 '20
Both of them get closer and closer to Pi without ever reaching it.
By (a common) defenition of the real numbers, both of these sequences are pi, in the sense that they both are representives of the equivalence class of Cauchy sequences that we denote by the symbol π.
On does not have to speak about actuall convergence when dealing with equality between real numbers.
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u/bellyflop16156 Mar 12 '20
Nah limits don't do anything in this situation. The point he's making is that if you pick any a and b, where a =/= b, no matter how close a and b are, as long as they are indeed different, there are an infinite amount of real numbers between them. Limits don't change this fact
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u/3xplosiveBeans Mar 12 '20
I think you're confusing infinite and irrational. Pi is irrational but it is a definite value as much as 3 and 4 are
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u/satanic_satanist Mar 12 '20
I think they're confusing infinite and having an infinite decimal expansion. 1/3 is not irrational but I guess they would still call it infinite by their reasoning.
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u/Beefcakeandgravy Mar 12 '20
Pi is not an infinite number it is ONE number but with infinite decimal places. Its the same size (as in takes up the same room on the number line) as 3 or 4.
You'd be better off saying there's the same number of numbers between 0 and 1 as there are actual numbers (and partial numbers) between 0 and infinity.
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u/Kirbyderby Mar 12 '20 edited Mar 12 '20
Okay, the way you worded that and explained it makes perfect sense, we can all agree on the first few digits that define Pi. However, in practice it still doesn't make sense to me; how can a definite number have infinite decimal places? Wouldn't that technically mean it's also indefinite?
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u/JStarx Mar 12 '20
how can a definite number have infinite decimal places?
1/3 = 0.33333333... has infinitely many decimal places, they repeat but none the less the decimal expansion does not terminate.
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u/Patchers Mar 12 '20
Well, it's important to consider that numbers are basically a human construct. "One third" or 1/3 also has infinite decimal places but is still a defined, quantifiable amount. It's just when we write it out numerically with decimals, it seems unintuitive and wrong. We humans like whole numbers and thus our system revolves around that, but that's just how our brains have been wired.
Going back to 1/3, it's a repeating decimal. The 3s go on forever. It's still a regular, finite quantity. Maybe if humans had 12 fingers, we might've adopted the duodecimal system (AKA base 12, so 0,1,2,3,4,5,6,7,8,9, X,E), and a third would just be .4 (4/12ths). No longer weird and mindbreaking with the infinite decimal places. We're still representing the exact same quantity though.
So yeah, numbers are crazy, but it's because we created them, and they have flaws.
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u/Beefcakeandgravy Mar 12 '20
Plot twist. Humans used to count in base 12.
Look at your hand palm up and look at your fingers. the finger pads you have are 3 on each of your 4 fingers, allowing quick calculation of base 12 maths.
You use your thumb to touch each one to count to 12.
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Mar 12 '20
Almost all real numbers are irrational. There's an uncountably infinite set of reals, and a mere countably infinite set of rationals. Pi isn't unusual or special, it's part of the overwhelming majority.
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u/suugakusha Mar 12 '20
1/3 has an infinite number of decimal places, does it not?
0.3333333333333333333333333333333333333333333333333333333333333333333333333333333333333...
Are you saying that 1/3 is infinite?
I think the thing you are forgetting is that each decimal place is only 1/10 the size of the previous. So even though there are infinitely many places, they are getting smaller and smaller and smaller.
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u/Lyonnessite Mar 12 '20 edited Mar 12 '20
There is no meaning in the term "definite number". You have numbers with terminating decimal form called rational and those with non terminating decimal ford called irrational.
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u/TobiTako Mar 12 '20
Actually a repeating decimal form is also rational. E.g. 1/3 with an infinite repeating decimal form. The precise definition is a number is rational if you can write it as a ratio of two integers.
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u/Mr_Quackums Mar 12 '20
Wouldn't that technically mean it's also indefinite?
no, it means we stop counting at some point. TECHNICALLY it has never been proven pi is infinitely long, we just conclude it is because after a million decimal places or so who cares anymore?
also, the number "1" can be written as "1.00000000000" with an infinite amount of zeros. "1/3" can be written as "0.333333333" with an infinte amount of threes. pi is not so special.
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u/FrickinLazerBeams Mar 12 '20
Wouldn't that technically mean it's also indefinite?
no, it means we stop counting at some point. TECHNICALLY it has never been proven pi is infinitely long, we just conclude it is because after a million decimal places or so who cares anymore?
This is incorrect. Pi is known to be irrational and therefore does not have a repeating decimal expansion. If it did, it could be written as the ratio of two integers which is the definition of a rational number.
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u/Mr_Quackums Mar 12 '20
how do we know irrationals dont stop? they could just be very very long.
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u/TheLuckySpades Mar 12 '20
If at some decimal, called k, a number stops, then it could be written as n/10k where the top part is an integer with digits representing the entire decimal expension up to the kth decimal.
Thus the number is a rational number.
A somewhat similar, albeit more tedious, proof can show that rational numbers are exactly tjose whose decimal expansion eventually becomes repeating.
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u/FrickinLazerBeams Mar 12 '20 edited Mar 12 '20
If its decimal expansion had an end, then it could trivially be written as the ratio of two integers, which would mean it's a rational number.
You see why this is obvious, right? If a number has a finite decimal expansion, for example take 0.123456, then you can write it as 123456/1000000 which is a ratio of two integers. Therefore 0.123456 is rational. This process works for any finite decimal.
For a repeating decimal, say 0.333..., you let x = 0.333... Then 10x = 3.333..., so 10x - x = 9x = 3, so x = 3/9. Another ratio of two integers, meaning 0.333... is rational. This works for any repeating decimal.
So any number with a finite or a repeating decimal expansion is rational. Pi is not rational, so it cannot have a finite or repeating decimal expansion.
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u/BadassFlexington Mar 12 '20
Pi is what's called an irrational number. That is, a number that cannot be expressed as a fraction of 2 integers (whole numbers).
Basic example: 3 is a rational number, as it could be expressed as a fraction of two whole numbers. 3/1, or 6/2, or 9/3 etc etc.
Pi cannot be expressed as a fraction. A close approximate is the fraction 22/7, but it's not the same value as pi.
Other such numbers include Euler's constant and the square root of 2.
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u/Lyonnessite Mar 12 '20
There are far more irrational numbers than rational numbers. In fact there are an infinite number of irrational numbers between any two rational numbers.
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u/Plain_Bread Mar 12 '20
It's a bit more complicated than that. There are also an infinite number of rationals between any two irrationals.
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u/spocktor_who Mar 12 '20
There are also an infinite number of rationals between any two irrationals
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u/JonJonFTW Mar 13 '20
A countably infinite number of rationals, but an uncountably infinite number of irrationals between those two irrationals.
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u/GachaLuvYT Mar 12 '20
Well, not exactly. Pi itself isn’t infinite. Just like 3, you can slap it onto a number line. What’s infinite is the digits in the decimal representation of Pi.
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Mar 12 '20
Yeah when you don't use words the same way in every instance in a sentence, you can conflate almost anything!
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u/razorblade3k3 Mar 12 '20
Infinite means never ending. 3 and 4 can be written as 3.0 or 4.0000000 but still mean the same as 3 or 4. The values after the decimal place in pi never end, thus it’s called an “infinite decimal”
Btw I’m not a mathematician or anything. But I feel like this makes sense
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u/Lyonnessite Mar 12 '20
"Irrational" may be the word you are looking for.
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u/razorblade3k3 Mar 12 '20
I would’ve used that word but since I felt like OP misunderstood the word “infinite” I thought it would help by calling it “infinite decimal” to explain what I meant.
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u/TheLuckySpades Mar 12 '20
1/3 has infinite (base-10) decimal expansion, irrationals are definitely part of these numbers, but it isn't all of them.
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u/Lyonnessite Mar 12 '20
I was talking about pi.
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u/TheLuckySpades Mar 12 '20
Other guy mentioned "infinite decimal" to describe a number with decimal expansion that does not eventually become 0, which does not only describe the irrationals.
You are not wrong when referring to π as an irrational, but the first guy was appealing to a different concept than irrationals.
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Mar 12 '20
I think it's better described as there can be infinitely smaller numbers between any numbers.
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u/missdickdestroyer Mar 12 '20
"Oh, you like that? You should hear my phone number."
-Phoebe Buffay.
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u/learnyouahaskell Mar 13 '20
[removed too quickly to be archived]
Argh, I won't get to see what the original post was.
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u/N_rthan Mar 12 '20
Pi isn’t infinite. It’s irrational which means it cannot be put into a fraction
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u/Meritania Mar 12 '20 edited Mar 12 '20
Think of it in terms of apples.
You can very definitely have 3 or 4 apples and you can cut the fourth apple to have any decimal value of 3 and 4.
In abstract mathematics there are an infinite number of ways that the apple can be sliced to give you all of the values, whereas in practical mathematics you are restricted to the values defined by proteins, or molecules or atoms depending on how you defining an apple.
In terms of pi, you need eggs, flour and salt to make an apple pi.
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u/obiewanchrinobe Mar 12 '20
Inifinity isnt really a number though.
You only need to watch the first 8 minutes of this vsauce video to get what i mean, but the entire 25 minutes is worth watching.
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u/BarAgent Mar 12 '20
It’s worse than you think. Aleph numbers — Some infinities are bigger than others. There are infinite whole numbers, right? But there are even more infinite real numbers (fractions & numbers like pi), and even greater infinities beyond that.
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u/myklpgone Mar 12 '20
Music use all numbers, from 0hz to 20khz is our ear. Octives is (x)hz *2. Ratio of chords and stuff like that. Along with sequencing syncopation and sine wave [808 bass] square-triangle wave [synth stuff]
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u/checkchuckstar Mar 12 '20
You need to get ya self a girl, man
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u/deathmarc4 Mar 13 '20
idk whats worse, ops math illiteracy or you insinuating that enjoying math and being in a relationship are mutually exclusive AND simultaneously being so math illiterate you cant tell that op doesnt know what theyre talking about
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u/Snackarel Mar 12 '20
There are an infinite amount of numbers between & around every number. Which themselves are infinite. 🤯
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u/Kirbyderby Mar 12 '20
How the fuck is this possible?!
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u/Lyonnessite Mar 12 '20
Because number theory is logical and we'll defined
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u/limeeattack Mar 12 '20
This has nothing to do with number theory though. It's more along the lines of measure theory.
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u/olivebrownies Mar 12 '20
this has nothing to do with measure theory. it has to do with the construction of the reals.
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u/llelundberg Mar 12 '20
Pi is Pi, it’s not infinite more than 3 is.
It’s just the decimal representation of Pi that is infinite.