Submitted by Resinate1 t3_zyzi9w in Showerthoughts
Comments
M8dude t1_j29dbmx wrote
well said, although it should be the bijection 1/(x+1) for OP's claim, but that's me nitpicking.
also i think it's natural to assume that OP is talking about the real numbers and the 'counting measure'.
jaydfox t1_j29ejxx wrote
I think they meant (1/x)-1, not 1/(x-1)
M8dude t1_j29f874 wrote
aaaah (1/x)-1 is the inverse of 1/(x+1), i mixed up the sets and thought there'd be a mistake, my bad.
alukyane t1_j29eswo wrote
(1/x)-1 is correct for going from (0,1) to (0,infty).
Your function would send the interval (0,1) to (1/2,1) in a weird reversed/distorted way (check endpoints to confirm).
And the op is most likely talking about cardinalities, not the counting measure, if we're nitpicking. :)
mrezar t1_j29h13o wrote
Yes but you have to define your domain. For Z its just false, for R it's true.
ThePhilosofyzr t1_j29ff67 wrote
Not a mathematician here, is this how 2+2 = 5 when the limit of 2 goes to infinity?
alukyane t1_j29fxza wrote
Sure, why not!
Fun fact: 2 * 0+2 * 0=5 * 0. Also, 2 * infty+2 * infty=5 * infty.
ThePhilosofyzr t1_j29gc4j wrote
Nifty=infty
Aki_The_Ghost t1_j29gh9u wrote
No, the limit of x going to infinity of 2 is just 2.
2=2×1=2×X^0 , so as X goes to infinity, X^0 is still 1, so 2 is still 2 and doesn't change.
"the limit of 2 goes to infinity" doesn't make sense, 2 can't go to infinity, 2 is just 2.
alukyane t1_j29h3oy wrote
Who said anything about x?
Aki_The_Ghost t1_j29yjfr wrote
We need to know what a limit is. It is the value that some variable approaches but never reaches if one of their parameters approaches a certain point. Example : When X approaches but never quite reaches 0, X^2 approaches but never quite reaches 0. It's the limit of x going to 0 for X^2 . The Limit. A limit implies the constant movement of some variable towards a certain point. So we NEED to talk about X, or a variable of any kind, if we want to calculate a limit. He just said the number 2, a constant number, not a variable, we can't work with it.
alukyane t1_j29ys77 wrote
They were joking about 2 going to infinity...
Aki_The_Ghost t1_j29zaav wrote
Really ?
[deleted] t1_j29i7nt wrote
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seboll13 t1_j29lr6k wrote
But this would be assuming both sets are countable, right ? But here they are not, so what can we say ?
alukyane t1_j29m97s wrote
Neither of them is countable (in the sense that neither of them can be matched up with the natural numbers).
navetzz t1_j28qm4u wrote
No! There is infinitely less of them. (Also, some people in the comments are mixing numbers and integers)
farineziq t1_j292xsd wrote
Op is right. Cardinality of the Continuum: "Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with {R} ."
Angry_Guppy t1_j29aopr wrote
How can that be true? For each real number between 0 and 1, there is a real number that is that value +1, +2, +3, etc. all the way up to infinity?
hacksoncode t1_j29byu4 wrote
Ultimately it's a semantics question about how you measure the "number" of things in an uncountably infinite set.
Consider "every real number" x in (1, +∞). There is a corresponding real number in (0,1) which is 1/x. So the sets are exactly the same size.
JoePoe247 t1_j29qncu wrote
How is that true though? For every number between 1 and 3, there is only one correlating number between 0 and 1 (1/x). But for every number between 0 and 1, there are two correlating numbers (x+1 and x+2)
M8dude t1_j29t6r5 wrote
you have to use a different 'correspondence' function for (0,1) and (1,3), which in this case is 2*(x+1/2).
notice how __every__ number in one set has to be matched with __exactly__ one of the other set.
JoePoe247 t1_j2ayejy wrote
Ok thanks, didnt realize that with the different corresponding function. My logic would be that y=2*(x+1/2) can be used to define every number in both sets (where y is 1-3 and x is 0-1). But there are numbers outside of the set 1-3, say 15. So if I made a different set, inclusive of 1-3 and 15, then there are more numbers in this new set.
I guess I understand that I'm wrong since mathematicians smarter than me have come up with theories/proofs to what you're saying, but I think there's logic in my argument.
M8dude t1_j2bv9x8 wrote
yeah, there's plenty logic in your argument, there are many different so-called 'measures' to quantify sets of numbers, example the 'distance measure' (i think) of an interval [a, b] is just denoted by b - a.
This is makes the sets have different measure (and more useful ones than just "infinite"), even though they have the same number of numbers.
the measure we would have used before is called the 'counting measure', telling us we'd have to count to infinity for both sets, but that doesn't mean they have the same number of elements (see cantor's second diagonal argument, or yours with the 15), so it has to be shown using a so-called 'bijective function' (our correspondence function), which thank god is pretty easy to construct for any two intervals.
But anyway, good thinking and yep you are right about the example set including [1, 3] and 15, for the counting measure.
hacksoncode t1_j2adzng wrote
> But for every number between 0 and 1, there are two correlating numbers (x+1 and x+2)
There are an uncountably infinite number of ways to correlate numbers in any open set of real numbers... they all have the same "count" in that they can all be correlated to all of the other reals with unlimited simple relationships.
Example: 1/2x is also in (0,1) just like x+2 and x+3 exist outside of that range. So is 1/(pi*x)...
[deleted] t1_j29frnv wrote
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moomerator t1_j29chbu wrote
So as somebody relatively well versed in math but definitely not full blown mathematician (engineer with a math minor) - I accept that this is a conclusion that a lot of people smarter than myself have agreed on but it still always bothered me.
I’m not disagreeing that it’s how the math falls out but I feel like there’s something fundamentally wrong with it. It’s like (and more or less related to) our understanding of quantum mechanics.. it feels like our current understandings are a case of the kid who got the right answer despite solving it incorrectly and like there’s a significant error/piece missing.
farineziq t1_j29u4qj wrote
I think finding a one to one relation between elements of one set to elements of another set is a convincing way to know if they have the same amount of elements without counting them. Like if you have two heaps of rocks, you could make two lines of rocks where rocks are side by side and if both lines end together, both sets have the same amount of rocks.
Doing the same with infinite sets is useful because we can't count the elements, but might be able to find a function that associates each element of one set to one and only one element of the other. (Regarding the rocks example, every possible way to put them side by side is such a function.)
For example, there are as many positive integers as positive and negative integers because f(x) = (x % 2) * (-x / 2 - 1 / 2) + (1 - x % 2) * (x / 2) from 0 to ∞ gives 0, -1, 1, -2, 2, -3, 3, ...
Now regarding uncountable infinite sets like the real numbers, someone else in the comment showed that ]0, 1] has the same amount of elements as [1, ∞ [ using f(x) = 1 / x. And regarding what we're actually looking for, let's say [0, 1] has the same amount of elements as ℝ, I didn't really pay attention but this seems convincing.
DooDooSlinger t1_j298q5t wrote
No. All real intervals have the same cardinality
Ok_Substance_1560 t1_j28x5jd wrote
Yes. It would be infinitely less. There are different measures of infinity too!
7h3_70m1n470r t1_j297xff wrote
Remember kids, some infinities are bigger than other infinities
Future_Seaweed_7756 t1_j28m8y1 wrote
There’s actually more between 1 and 0 than 1 and infinity, I’m not exactly sure on the proof but it’s something to do with how infinite numbers work.
liarandathief t1_j28rfot wrote
Only if, when they say from 1 to infinity they mean integers. There are countably infinite integers, where there are uncountably infinite real numbers.
Phr3nic t1_j2941o3 wrote
There's less Integers between 1 and "infinity" than there are Real numbers between 0 and 1. But there are equally many real numbers between 0 and 1 as there are real numbers between 1 and "infinity"
canucky55 t1_j28nl2k wrote
If I remember correctly, it has to do with rational numbers from one to infinity being countable infinite and the real numbers between 0 and 1 being uncountable infinite. The trick to the proof comes with being able to count rational numbers from smallest to largest (easy to think about with integers but even with rational numbers it's just integers in the numerator and integers in the denominator so just assign a count to the numerator first and the next count to the denomator and it works). for real numbers if you try to count from one number to the next, there will ALWAYS be an number in between those that you missed and should have counted. blew my brain when the professor showed the proof in class.
M8dude t1_j28scze wrote
lemme try.
for every x in R such that 1 < x < infty, there is exactly one number (1/x) in R, s.t. 0 < (1/x) < 1.
also vice versa.
there's a bijection between the two sets, therefore they are the same size.
am i missing something?
unpopular_tooth t1_j28vqfs wrote
TIL the awesome new word “bijection.”
[deleted] t1_j28wghl wrote
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unpopular_tooth t1_j291m1g wrote
Oh for fuck sake… Yeah, obviously the word didn’t just get invented. New TO ME, okay? I really didn’t think my wording would confuse anyone.
M8dude t1_j292gvo wrote
oh i thought you were sarcastic, sorry about that..
unpopular_tooth t1_j2af181 wrote
No problem. Jeez, now I feel like a big bijection for blowing up about it.
DooDooSlinger t1_j298nzr wrote
No. The cardinality of any real interval, bounded or unbounded, is the same. You can find a bijection between any of them.
101_210 t1_j28vuun wrote
There is indeed an higher order of infinity for the real numbers between 0 and 1 than integer between 0 and infinity.
The proof (simplified) goes as follow:
For each integer between 0 and infinity (let’s call that number x) you can match it with a number between 0 and 1 that contains a number of 1 after the dot equal to our integer x. So you get:
1 -> 0.1
2->0.11
3->0.111
4->0.1111
etc.
As you can see, going to infinity, we will have two matched sets where every element is different. However, if we add a 2 just after the dot in the real number set, we get 0.21, 0.211, etc, an entirely NEW set, of which no elements were contained in any of our previous sets. There is actually an infinite number of these transformations that can be made to the real set, and none that can be made to the integer set.
The integer set is named a countable infinity, where although there is an unlimited number of element, if you choose two different elements, there is a finite number of elements between them.
The real numbers are an uncountable infinity, where if you chose any two elements, there is still an infinite number of elements between them.
Future_Seaweed_7756 t1_j2e7srg wrote
This makes a lot of sense thank you
Resinate1 OP t1_j28nw3p wrote
Yeah I should’ve said 1 to infinity! Title typo
HereIAmSendMe68 t1_j29ce1l wrote
cantor's diagonal
ExplorerDisastrous38 t1_j28rvik wrote
And that, my good friend, is why we will be contemplating life for the next couple of hours staring at a Reddit page and generally experiencing the symptoms of brain damage
ThePhilosofyzr t1_j29f8ky wrote
It’s wild that reality flexes with the ways in which we find to quantify it.
Capnreid t1_j29gyp2 wrote
And almost as many times as this uninteresting stolen shower thought has been posted in this sub
Showerthoughts_Mod t1_j28jwar wrote
This is a friendly reminder to read our rules.
Remember, /r/Showerthoughts is for showerthoughts, not "thoughts had in the shower!"
(For an explanation of what a "showerthought" is, please read this page.)
Rule-breaking posts may result in bans.
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xFearful425 t1_j29b4iw wrote
Friendly reminder OP did not specify whole numbers. They are correct.
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orbital0000 t1_j29afvi wrote
People have already lost their minds thinking about infinity.
UnhingedCringeLord t1_j296nfc wrote
Hmm...the OP didn't specify which type of number he's referring to. If any kind of number goes, I would say that there are more numbers from 0 to infinity as those numbers would also contain fractional numbers. Or maybe I'm just dumb...in that case, ignore my opinion lol
M8dude t1_j299cw4 wrote
you're not dumb, it's just counterintuitive.
any number x between 0 and infinity can be paired with a unique number 1/(1+x), which is between 0 and 1.
any number y between 0 and 1 can be paired with a unique number (1/y)-1, which is between 0 and infinity.
there are no exceptions, so none of the two sets of numbers have 'more' numbers in them.
Talking about the 'size' of these sets of numbers is a whole different story.
Dorito_flames t1_j297pyt wrote
Weird to explain, but there are bigger and smaller infinites. Aleph null is the set of natural numebrs, which is infinite, but aleph one is the set of all ordinal numbers, which is infinite, yet bigger. Hopefully this makes sense??
JKIE1998 t1_j299flp wrote
As 0 to 1 is ∈ of 0 to infinity, but 0 to 1 is ≠ 0 to infinity, there are less numbers between 0 to 1 than in 0 to infinity.
asdsav t1_j29iaa4 wrote
You cant compare infinite things. They just both infinite and you cant claim they are same amount.
Se7enLC t1_j28w0qi wrote
Nope
Incredibly relevant video: https://youtu.be/HeQX2HjkcNo
Edit: I don't even understand, why would people downvote a Veritasium video that exactly covers the topic? (Starts around 3:40)
M8dude t1_j2a7b7p wrote
it's because the statement in the video maps (0,1) to the natural numbers, not the real numbers.
Veritasium is right with what they said, but since we're talking about 'numbers', generally we're talking about real numbers, not natural ones.
it's a completely different statement.
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Herkfixer t1_j293yxn wrote
Can't be true as stated because, for any "number" between 0 and 1, you will have the same amount between 1 and 2, and 2 and 3, and 3 and 4, and so on. So you would have an infinite more numbers "after" 1 than you have between 0 and 1. Unless you specify whole numbers, integers, rational numbers, etc.
M8dude t1_j295vsc wrote
It's counterintuitive, but it's true for real numbers.
the argument is that you can assign a unique real number between 0 and 1 to any real number etween 0 and infinity and vice versa.
Herkfixer t1_j2963q7 wrote
Yes, for real numbers. My point is that the OP not specifying the two classes of numbers makes it untrue.
M8dude t1_j29dx0r wrote
i think it's fair to assume that OP is talking about the real numbers.
Haven_Stranger t1_j29vkgd wrote
Two classes? What two classes? OP specified "numbers between 0 and 1" and "[numbers] from 0 to infinity". The class involved (entailed, even) is "numbers on the number line". In other words, reals. That's one class. That's the basis of the comparison.
Both specified ranges are uncountably infinite.
Here's an easier comparison: There are as many numbers between 0 and 1 as there are between 1 and infinity. It's an easier comparison because now both sets are strictly between their bounds, the two sets don't overlap, and the bijection formula is simpler.
If A is a real between 0 and 1, and B is the matching real greater than 1, then the mapping is:
A -> 1 / B
Also
B -> 1 / A
That's it. For every real number larger than one, there exists exactly one matching number between zero and one, and vice-versa. No exceptions, no excuses, nothing left unaccounted.
The size of the two sets are exactly the same, even though the extents of the two sets are wildly different.
Also also, that's the comparison OP meant to express. Even so, the comparison posted still holds true. It just that, instead of mapping A to the inverse of B, we map A to the inverse of one more than B.
So, no, we don't have to compare reals on one side and rationals on the other, or anything else where we'd have to specify two classes. Those comparisons can be made, of course, but they're not relevant to the post.
It takes different real numbers to make it all the way out to infinity, but it doesn't take more of them. There are exactly as many real numbers greater than 1 as there are between 0 and 1.
DevilishDiamond1 t1_j28mymf wrote
This is essentially saying the set of real numbers have the same amount of numbers as the set of natural numbers, which is untrue. Take any random number from a list of numbers between 0 and 1, for instance 0.7362957293749274. Now add 1 to each number and if it’s a 9 roll it back to an 8. The number is now 0.84738683858385. This number will not be anywhere in your list.
xSteee t1_j28na4e wrote
It doesn't say this, op doesn't specify real, natural or whatever set of number even though by same "number from one to zero" we can think of fraction and irrational number
eegocentrik t1_j28l9tj wrote
Disagree.
Fractions are portions of numbers, not new numbers themselves.
1 pie, cut in half does not yield 2 pies.
It yields 2 portions of the same pie. This does not create a new pie.
xSteee t1_j28mwko wrote
Are you saying that 0.1 and 0.2 are not different numbers?
eegocentrik t1_j28t507 wrote
I'm saying they are two different quotients of the same number.
eegocentrik t1_j28nbj6 wrote
Nope portion of the same number, the number 1.
xSteee t1_j28nh6u wrote
0.1 is 1/10, a portion out of 10 of 1. 0.2 is 2/10, two portion out of 10 of 1. 0.1 and 0.2 are two different number.
Or am I misinterpreting something?
eegocentrik t1_j28nwgv wrote
A piece cut into ten pieces.
.1 is one PIECE of the original pie, not an entirely new pie.m.1 describes the piece, not the pie.
You could have .9 grams of the .1 piece, still not a new pie. There are no new numbers created by dividing the unit.
xSteee t1_j28o5od wrote
As someone said, numbers are not pies. It's like you are trying to demolish hundreds of years of maths by saying that fractions are not different numbers ahah
eegocentrik t1_j28ocaf wrote
Define fraction.
xSteee t1_j28ou9b wrote
Fractions are another way to represent finite and infinite decimal numbers
eegocentrik t1_j28p4xp wrote
Not a definition.
Please define fraction.
xSteee t1_j28pjie wrote
A fraction is a numeral which represent a rational number. It is composed by a numerator and a denominator.
eegocentrik t1_j28qg64 wrote
Definition:
fraction, In arithmetic, a number expressed as a quotient, in which a numerator is divided by a denominator.
The original number being expressed as a quotient. Fractions are quotients of the original number but not new numbers in themselves.
xSteee t1_j28qy78 wrote
So 1/4, or 0.25, is not different from 3/4, or 0.75? Or are you saying that 0.25 and 0.75 are not numbers at all?
eegocentrik t1_j28s8lc wrote
.25 and .75 are quotients of 1 and not independently new numbers.
1/4 == .25
xSteee t1_j28sm8e wrote
Does it extends to number above 1? 2.25 and 2.75 are not different number?
eegocentrik t1_j28u8wa wrote
Sure.
The .25 in 2.25 is the quotient of from 2 to 3 (1). 2+.25
Also, 2.25 is the quotient of 45 ÷ 2 or 45/2.
However, the OP said from 0 to 1, this is where my argument lies.
There are NOT the same amount of numbers from 0 to 1 as there are 0 to infinity.
There ARE the same number of ELEMENTS.
xSteee t1_j28vjyh wrote
And the elements of a set of number ("all the numbers between 0 and 1" is a set of numbers, right?) are called...?
eegocentrik t1_j28vo6x wrote
Quotients.
xSteee t1_j28vv0o wrote
And a quotient, by definition, is "the number we obtain by dividing a number by another"
eegocentrik t1_j28wc0a wrote
It's not a new number. It's a fractional representation of the numerator and does not exist outside of it.
There are not infinite NUMBERS between 0 and 1 only elements.
xSteee t1_j28xebs wrote
Have you ever heard of real numbers? Irrational numbers?
eegocentrik t1_j28xw3p wrote
Yes, irrational numbers do not apply here.
This is not a math issue, this is a semantic issue.
Fractions are quotients of rational numbers.
1 is the smallest unit of count.
Half of 1 can be represented by the quotients 1/2 or .5
Half of one can also be expressed as 2 new things that are not 1.
xSteee t1_j290guf wrote
And we, as as species, for hundreds of years have decided to call them number too, so 0.5 is a number
Nilonik t1_j28p48j wrote
A fraction is a number which can be written as a/b, where a and b are integers, while b is unequal to zero.
eegocentrik t1_j28pvt0 wrote
And b. is the number, and a. is its fraction.
a. cannot be a unit of its own and does not exist without b. in this example.
lt_Matthew t1_j28omf9 wrote
So if someone says they're eating pie, you correct them and say it's 'a piece of pie'?
eegocentrik t1_j28phh8 wrote
Only if they are counting it while they eat it.
Do you say that you plated 8 pies for dessert?
Or are there 8 pieces of pie on the counter?
liarandathief t1_j28qmgj wrote
We're upping your tax rate to .99, don't worry, it's basically the same thing.
symmetra__main t1_j28ydqz wrote
Numbers do not have to be integers to be numbers.
eegocentrik t1_j28ytru wrote
I didn't say that.
I said there are no numbers between 0 and 1, only quotients (Elements)
symmetra__main t1_j28zbhq wrote
Oh, you're doubling down. Very well, have a blessed day.
eegocentrik t1_j28zu8k wrote
No evidence for gods either.
symmetra__main t1_j29113h wrote
AKTUALY I was blessed with knowledge of 8th grade math by my 8th grade math teacher
imregrettingthis t1_j28nh4q wrote
Numbers are not pies and it doesn’t work that way.
M8dude t1_j28un4y wrote
yet pi is a number.
checkmate, atheists.
eegocentrik t1_j28nyx6 wrote
Numbers are constructs and don't really exist, and yes it does.
imregrettingthis t1_j28onbx wrote
Lol. Exactly!
Numbers are a construct and now you’re trying to define them in some real world way like pies. Thanks for helping me prove my point while trying to prove yours I guess.
If you keep wanting to disagree I obviously won’t mind since you’re actually just agreeing but feel free to look up this very established and agreed on mathematical concept that is again... agreed on by humans... the people that constructed it. As you so helpfully pointed out.
eegocentrik t1_j28p72x wrote
Pies are constructs.
imregrettingthis t1_j28pdg6 wrote
Pies are pies my friend. You were so close. Do you have a math teacher who can explain this? Or Google?
eegocentrik t1_j28pjvz wrote
Pies are constructed
imregrettingthis t1_j28psjo wrote
You do have me here.
eegocentrik t1_j28pp5a wrote
Please provide to me a source explaining that fraction are new individual numbers separate from their whole.
imregrettingthis t1_j28q9f7 wrote
A pie is 1 pie. A half a pie is a either .5 a pie or 1 half pie. It’s arbitrary. The only difference is representation. You nailed it at the beginning. Math is a construct. I’m only smart enough to understand this not explain it well perhaps since I didn’t look up a more educated way to reply.
eegocentrik t1_j28qmr4 wrote
.5 represents the PIECE.
I nailed this in every comment.
imregrettingthis t1_j28qxp2 wrote
To you. That’s arbitrary. Again. This shower thought isn’t a shower thought but an incredibly established mathematical concept.
If you want to be a smug ass and think you have nailed it go on. But according to all mathematicians on earth you are wrong and this shower thought is right.
If you want to disagree you might as well be a flat earther.
Again feel free to completely disregard me and look it up.
eegocentrik t1_j28slb5 wrote
To you.
Again you're arguing on shower thoughts, and if you want to be an incorrect smug ass while doing so that's your right.
Again feel free to provide your source as I have looked it up and I am still representing the concept accurately.
Please provide evidence for you claim
Strawman, equivocation fallacy.
imregrettingthis t1_j28t4vi wrote
not to me. to everyone on earth.
​
my source is actually just my basic high school level education. but if you need a different one.
https://www.cantorsparadise.com/number-of-numbers-infinite-weirdness-9387faa58368
here is a link. let me know if you want thousands more. or just google it.. or ask anyone with a brain.
eegocentrik t1_j28tjlz wrote
From youur source:
An infinite set (e.g. integers) and an infinite proper subset of the set (e.g. natural numbers) can have the same number of elements. In fact, all the following infinite sets have the same number of elements: natural numbers, whole numbers, integers, even numbers, odd numbers, prime numbers, etc."
Elements, not numbers.
There are the same number of elements between 0 and 1 and 0 and infinity, not NUMBERS.
imregrettingthis t1_j28u9gn wrote
again you could just google this. I am done, please feel free to leave this convo thinking you are right. you should even tell a bunch of people about how big an idiot I am... please also tell them about this concept and why I am a moron.
eegocentrik t1_j28urvo wrote
Your Google search didn't work.
Your evidence failed.
Thank you for finally agreeing with me.
Sometimes you just need to check it out for yourself and learn something new.
So don't think that you are a moron or an idiot. We all start from scratch.
imregrettingthis t1_j28vg2d wrote
sure, as I said please tell everyone about this concept and your point of view.
eegocentrik t1_j28vlur wrote
I appreciate your support in this cause.
M8dude t1_j2a5nap wrote
any element of a set of numbers is a number.
of course, how would you know, never having heard of set theory.
SaraF_Arts t1_j29eyat wrote
So, let's make an example here. I have a nice picture on my phone, but I want it to be a quater size smaller. What do I do? Multiply the pictures' dimensions for a "piece" of 0.5 each. What do I get at the end? A "piece" of the picture"? No, the SAME picture, but... Oh my god, SMALLER!
From your science perspective how do you explain that a "full" times a "piece" makes a smaller full and not a piece?
(Also, obv you study business, no brains to be found there)
liarandathief t1_j28qsw9 wrote
Define number
eegocentrik t1_j28t06q wrote
A number is an arithmetic value used to represent quantity, fractions are quotients of theses quantities.
M8dude t1_j28wbfx wrote
you're thinking of natural numbers.
fractions of natural numbers are still numbers, as anyone who knows any actual maths will confirm.
eegocentrik t1_j28x6o7 wrote
Nope.
The numbers from 0 to 1 are elements of 1, quotients, not numbers.
Does this contradictory statement make more sense to you?
M8dude t1_j2902dz wrote
no.
you're making your own definitions, which is fine, but there are conventional definitions for these things in maths already, and (almost) everyone uses those.
eegocentrik t1_j290ffx wrote
Look up the definition of mathematical fraction.
M8dude t1_j2927ms wrote
Look up the definition of 'Number'.
here, i'll help you:
> https://en.m.wikipedia.org/wiki/Number
and if that's too difficult, here's the relevant part:
> In mathematics, the notion of a number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ...
yellowistherainbow t1_j28lhha wrote
Maybe digits is a better word
Zeelacious t1_j28mbew wrote
No because an infinite fraction of one whole is just 1/10th of infinity. For every number of infinity there is an equal infinite fraction of zero to one.
yellowistherainbow t1_j28um97 wrote
Yeah I was just trying to lighten the mood, sorry
M8dude t1_j28tkip wrote
yeah clearly the set of rational numbers consists of almost no numbers. /s
eegocentrik t1_j28utwg wrote
From 0 to 1.
alukyane t1_j2998zx wrote
Mathematician here. The op is correct, at least for one common interpretation of "as many".
The usual meaning of "as many" is that you can match up the sets. For example, the interval (0,1) has as many points as the interval (2,3) because I can match x up with x+2.
(0,1) also has as many points as (1,infinity) because I can match x up with 1/x. Or we can match x up with 1/x-1, for the op's claim.
The weird thing is that (0,1) is definitely smaller than (0,infty), in the sense that there are points in (0,infty) that are not in (0,1)... infinity is weird.
The other weird thing is that there are other ways of measuring size that aren't based on cardinality (the pairing up of points). For example, the interval (0,1) has the same cardinality as the interval (5,7), but the two intervals have different total lengths So in that sense (5,7) is bigger... and of course (0,infty) is bigger yet...
So, in "practice" it matters what measure of "more points" makes sense for the particular comparison.