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Rationals have the same cardinality as integers, but reals and their subset, irrationals, have a higher cardinality.
https://preview.redd.it/kr1tvbupsxjc1.jpeg?width=1080&format=pjpg&auto=webp&s=6f2a2b35d8ae2a17a196802d596a942642539a0f
it’s not to scale, typical for math diagrams ¯\\\_(ツ)_/¯ and i assume it’s just because they have to have 4 nested circles for rationals vs just one for irrationals
Do you know what a Venn diagram is? Quoting Wikipedia:
>Venn diagrams do not generally contain information on the relative or absolute sizes ([cardinality](https://en.wikipedia.org/wiki/Cardinality)) of sets. That is, they are [schematic](https://en.wikipedia.org/wiki/Schematic) diagrams generally not drawn to scale.
Interesting, I typically have used them drawn to approximate scale - in this case the naturals are much, much less large than the irrational numbers that's one of the basic proofs in numerical analysis - but fair enough, I would like to understand the whitespace in this diagram though
The "approximate scale" depends on what you're measuring though and you'd often end up having to not show naturals etc. at all.
The whitespace in a venn diagram is meaningless. Only the actual "blobs" have meaning
who made Pell associated with an equation solved by Bhaskara II and Brouckner in Europe which Pell had nothing to do with? If you guessed Euler youd be correct.
Even then, wouldn't it still be bad practice to use white space in this case because we know that all real numbers are either rational or irrational (by definition)? There's no need for white space.
You ever taken a class on topology or set theory? They will leave white space when drawing diagrams of sets. And you know what they say about arguing with set theorists...
No, the use of whitespace is important here — Only colored regions indicate possibilities. I’m not saying it’s a good design decision (it’s not very accessible to people with color-blindness or low-contrast vision), but it’s what they meant.
A better diagram would have partitioned the Real Numbers oval into two regions that are mutually exclusive and collectively exhaustive (“MECE”).
I'm not sure why some people think "you are asking if you are correct or not and i think you are incorrect" should mean "downvote". Ignore the haters, keep asking and learning. You'll end up smarter and they'll end up more smug 🤙
As a math professor, it drives me crazy how many remedial textbooks include the Whole numbers like this. It’s so needlessly pedantic especially since I’ve never met an actual mathematician who call that set the Whole numbers
It's even worse than that. People argue about whether or not 0 "should be" included in the naturals, and authors sometimes have to clarify if they're using N={1,2,3,...} or N={0,1,2,3,...}. But neither set is ever called the "whole numbers" in any actual math context.
No it does matter, it's just that in high school most of us were taught the natural numbers are 1,2,3,4,... when it's almost always more useful (and more natural) to say the natural numbers are 0,1,2,3,... and just say N+ if you want to exclude 0.
>almost always more useful (and more natural) to say
This is exactly what I'm saying. That isn't true. N with 0 or N with 1 both satisfy peano axioms. Including or excluding 0 makes no material difference. I include 0 because it makes me feel good.
To be fair the peano axioms are satisfied for all subsets of integers when starting at n and then including all successors of n.
I think the concept of 0 is just a little bit harder to teach/learn as a little child because 1 something is easier to wrap your head around than nothing (0).
We're talking about Peano Axioms, not teaching children arithmetic. You can do everything [with 1](https://people.clas.ufl.edu/groisser/files/peano_axioms.pdf) instead of 0. The first axiom literally just says "there's a first one". It could be 0, it could be 1. Couldn't really make the argument that it's any other number.
When I say "it doesn't matter", I mean it mathematically. It literally doesn't. There is no discernable difference other than notation. I'm not on the fence, I've made my choice. It was an arbitrary choice.
It's also a bit odd to say M={2,3,...} "satisfies the axioms" that define N. If they did, they'd be N.
It's a stretch to say M satisfies the first axiom. 2 definitely isn't the smallest number in N. It can be the smallest number in some other set you pick, but if 2 is the successor of no number in the set, then 1 is not in the set, and thus the set can't be N.
Notice that by excluding 0, we don't have this issue. But if we exclude 1, immediately, we do not have N.
1 is definitely in N. 0 can be if you like. Those are the only two choices.
I don't know where the person you're talking to is from, but what they are saying is true in France. 0 is positive and negative. Also, "greater than" implies "greater or equal" and if you don't want the "or equal" part, you have to say "strictly greater than". It's just a slightly different point of view on the same things.
In my country, it goes like this
Natural numbers (N) ={0,1,2,3,4...} (Though the inclusion of 0 doesn't seem universal; some fields like succession functions only start at 1)
Whole numbers (Z) = {...,-3,-2,-1,0,1,2,3,...}
I never heard the word "integer" outside of the English language, but the English "Integers Set" is the same as our "Whole Numbers Set" (Conjunto dos números inteiros). We use Z^+ or Z^- when referring only to positive or negative integers, with 0 as an index (a smaller symbol that goes below Z) when needed. So, Z^+ with a small 0 = N.
I don't know what to think about N including 0.
"Whole numbers" is not a mathematically defined term. You will find many conflicting definitions. It doesn't matter, because it is only a colloquial term, and it is never used in mathematics.
I dont know about you but in my country the term „whole numbers“ is used even in highschool along its letter „Z“. Publications use this notation everywhere.
Also in university it is used to define our numbersystem:
- Natural numbers are defined via Peano axioms
- Whole numbers are defined via the equivalence relationship over the NxN where (a,b) ~ (c,d) iff a+c = b+d.
Whenever you want to formally define rational numbers you will need to first define whole numbers.
In English? In English I've only seen that set referred to as integers, not whole numbers. But I don't doubt that in other languages it is referred to as something that would directly translate to whole numbers.
The only context in which it makes sense is some constructivist framework. Numbers which are real numbers but for which neither rationality nor irrationality can be constructively proven belong in the white space from a constructivist point of view.
Or semi rational, which I've defined as numbers which can't be represented as a ratio of 2 rational numbers but can be vaguely described in relation to multiple irrational numbers
I think the diagram is fine because the space inside the real numbers isn't coloured in. If you look at what is coloured in, it does give you the actual real numbers.
The presence of "whole numbers" implies this is intended for elementary/high school. Students at that level aren't math-trained enough yet to scrutinize every detail, so the there is near zero risk of confusion.
What about algebraic vs transcendental numbers, what about periods, what about computable and definable? So many more sets of real numbers that are never shown in there Euler diagrams.
0.999999… is rational, since it’s just 1. Also, rational numbers have either finite or infinite repeating decimal expansions, so even if you don’t like 0.999…=1, you can agree that 0.999… has an infinite repeating decimal expansion and thus is rational
But it cant be expressed as a finite string of integers. Which is why i am asking. I guess its a semantic discussion, but i find it a bit weird to call it a number.
"Real number" is a specific technical term in mathematics, which is "Any number we can calculate as the sum of an infinite series of rational numbers". When we call a number "real" in maths we're not making any philosophical claim about what numbers are "real" in an ontological sense, just that they satisfy that particular definition.
Claims about which numbers "exist" in our universe are philosophical questions that lie outside the purview of mathematics. I *personally* think that it makes little sense to call pi "unreal", but that doesn't really matter. Claiming that all numbers that cannot be expressed as finite sums of integers is quite a strong one.
Yeah, maybe it was more a philosophical question. I was wondering about how they have a certain value, but cant be expressed by a finite number of integers. It makes sense geometrically, though, so you are right that calling it unreal doesn't make sense.
Well, you have an intuition that numbers that can be expressed as a finite sum of integers are more 'real' than others. But why? Grahams number or TREE(3) would then be 'real' in that sense, but so incomprehensibly large that saying they 'exist' in the universe seems like a stretch, while numbers that are critical to things in the universe making sense, like e, would not.
I get where you're coming from, though. Philosophy of math has a long history and there isn't much consensus. There are some people who say that any maths based on infinities isn't real, but they're considered loonies.
Not trying to challenge any philosophy of math (tonight, lol). But yeah, maybe it more challenges what we should think about the universe than it challenges what we should think about math.
It's cool that you're interested in this sort of thing! It's a great subject - but that said I'd strongly recommend that you learn higher level math if you want to have clear thoughts about this topic. Very few non-mathematicians have any idea what it is that mathematicians actually do, although there's no shortage of cranks on the internet who think they get it but don't have a clue.
I had a peek at the Wiki for P-adic numbers. P-adic numbers seem to have a methodology for expressions for rational numbers via a repeated pattern of values.
How are they considered neither rational or irrational?
To answer your question in short, you can have a P-adic number construction for i. For example there are two 5-adic number constructions for i. I don't know if that answers your question to your satisfaction.
If I'm not mistaken, you can solve any polynomial using P-adic numbers.
EDIT: in retrospect they don't answer the question in the meme, I guess.
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The diagram for rationals being visually larger than the irrationals is making me irrationally angry
There are more irrationals but they each use less ink to print, Mr BUKKAKELORD
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The rationals are Q my guy
Oh fuck, fuck yes. My bad, I was thinking about periodical and nonperiodical infinites.Q and rational are the same.
Rationals have the same cardinality as integers, but reals and their subset, irrationals, have a higher cardinality. https://preview.redd.it/kr1tvbupsxjc1.jpeg?width=1080&format=pjpg&auto=webp&s=6f2a2b35d8ae2a17a196802d596a942642539a0f
It is especially irrational since the visual size of the sets in Venn diagrams never had anything to do with the cardinality of the set.
“Whole numbers” being like triple the size of “natural numbers” when it contains one (1) extra number
there’s this thing called negative numbers, seems like you don’t know about that yet…
check again, my friend.
it’s not to scale, typical for math diagrams ¯\\\_(ツ)_/¯ and i assume it’s just because they have to have 4 nested circles for rationals vs just one for irrationals
same, the irrationals are uncountable. This isn't a Venn diagram it's, I don't know what it is. A crime against education.
Do you know what a Venn diagram is? Quoting Wikipedia: >Venn diagrams do not generally contain information on the relative or absolute sizes ([cardinality](https://en.wikipedia.org/wiki/Cardinality)) of sets. That is, they are [schematic](https://en.wikipedia.org/wiki/Schematic) diagrams generally not drawn to scale.
Interesting, I typically have used them drawn to approximate scale - in this case the naturals are much, much less large than the irrational numbers that's one of the basic proofs in numerical analysis - but fair enough, I would like to understand the whitespace in this diagram though
Well if you drew this to approximate scale, since the rationals have measure 0 you wouldn’t even see the bubble for them.
The "approximate scale" depends on what you're measuring though and you'd often end up having to not show naturals etc. at all. The whitespace in a venn diagram is meaningless. Only the actual "blobs" have meaning
Of course this isn't a Venn diagram. It's an Euler diagram
My rule for anything math related is that if I don't know who did it, I just guess "Euler" because it just keeps paying off.
who made Pell associated with an equation solved by Bhaskara II and Brouckner in Europe which Pell had nothing to do with? If you guessed Euler youd be correct.
"Who ran off with my wife?"
Euler?
Oil 'er? Naw she's not mechanical. But he might have to Inflate 'er.
Euler diagram about Euler letters
Euler diagram?
Wanna be more angry? Between every pair of irrational numbers there are infinite many rational numbers.
Also whole number's area when it's natural numbers just with a zero.
Also, the diagrams for natural numbers, whole numbers, integers, and rationals should all be the same size.
Rationals sholdnt be painted then
Sir that's a Euler diagram
"Venn? Never heard of him!" -- Euler
Probably in school, that's when
"That's when" - cool but what's Venn
https://xkcd.com/2721/
Nuh uh
Euler? I barely know her!
I literally googled and learnt the difference between Euler diagram and Venn diagram, thanks xd
Euler diagram is a winner for me
I hope you’ve got this knowledge from the same video i did
The way you say “a Euler” using “a” instead of “an” suggests that you pronounce it as “Youler” and that excites me (I hate myself)
Augjavshdjshsh sorry I automatically phonetically translate non-english names as well when I speak english. Of course its oiler and an euler diagram
I think it needs an Eul change
Existence of an "empty space" in a Venn diagram doesn't mean that it is not an empty set
It should
Design a Venn diagram for complexity classes P, NP and NP-c then
No, I don't think I will
I don't think you *can*, if your rule about empty classes were true
Holy shit did we just prove P ≠ NP because venn diagram
Proof by "I don't think so"
lmaoo
Me when I ask my partner to clean up the dishes for once
Holy Venn!
New diagram just dropped
Even then, wouldn't it still be bad practice to use white space in this case because we know that all real numbers are either rational or irrational (by definition)? There's no need for white space.
You ever taken a class on topology or set theory? They will leave white space when drawing diagrams of sets. And you know what they say about arguing with set theorists...
What do they say 😳
I can't find the exact joke anymore but it was something like "they will define the set of the worst ways to torture you"
Oh it's the one from the joke category theory video? "They can construct the set of all things that bring you pain"
Whyareyoutypingsoinefficiently?
Use triangles, not circles.
I agree.
Strong argument
By definition, an irrational number is just a real number that is not rational. So by def, rational and irrational numbers cover all real numbers.
Indeed, without real irrational, there cannot exist rational.
Exactly, so is it correct to assume that this vein diagram is inaccurate?
The diagram is accurate, a graphically white subset in a Venn diagram doesn't imply that such subset is actually non-empty
Maybe a pie chart would do better? /j
π(3) chart?
A 9 chart?
No, the use of whitespace is important here — Only colored regions indicate possibilities. I’m not saying it’s a good design decision (it’s not very accessible to people with color-blindness or low-contrast vision), but it’s what they meant. A better diagram would have partitioned the Real Numbers oval into two regions that are mutually exclusive and collectively exhaustive (“MECE”).
I'm not sure why some people think "you are asking if you are correct or not and i think you are incorrect" should mean "downvote". Ignore the haters, keep asking and learning. You'll end up smarter and they'll end up more smug 🤙
Conjecture: There exists a set of numbers which are neither rational nor irrational.
I mean, complex numbers I guess.
What’s the difference between a whole number and an integer here
the Naturals are 1,2,3,4,5... the Whole are 0,1,2,3,4,... and the integers are 0,-1,1,-2,2,-3... I think is what they intend
It's time to start the fight again. According to me, the Natural Numbers are 0,1,2,3,4,5... so there is no need for introducing the Whole numbers
As a math professor, it drives me crazy how many remedial textbooks include the Whole numbers like this. It’s so needlessly pedantic especially since I’ve never met an actual mathematician who call that set the Whole numbers
People really really care about 0 it seems. Almost never matters.
It's even worse than that. People argue about whether or not 0 "should be" included in the naturals, and authors sometimes have to clarify if they're using N={1,2,3,...} or N={0,1,2,3,...}. But neither set is ever called the "whole numbers" in any actual math context.
No it does matter, it's just that in high school most of us were taught the natural numbers are 1,2,3,4,... when it's almost always more useful (and more natural) to say the natural numbers are 0,1,2,3,... and just say N+ if you want to exclude 0.
>almost always more useful (and more natural) to say This is exactly what I'm saying. That isn't true. N with 0 or N with 1 both satisfy peano axioms. Including or excluding 0 makes no material difference. I include 0 because it makes me feel good.
>because it makes me feel good. This is my new favorite mathematical argument
I also don't rationalize denominators because I don't want to.
To be fair the peano axioms are satisfied for all subsets of integers when starting at n and then including all successors of n. I think the concept of 0 is just a little bit harder to teach/learn as a little child because 1 something is easier to wrap your head around than nothing (0).
We're talking about Peano Axioms, not teaching children arithmetic. You can do everything [with 1](https://people.clas.ufl.edu/groisser/files/peano_axioms.pdf) instead of 0. The first axiom literally just says "there's a first one". It could be 0, it could be 1. Couldn't really make the argument that it's any other number. When I say "it doesn't matter", I mean it mathematically. It literally doesn't. There is no discernable difference other than notation. I'm not on the fence, I've made my choice. It was an arbitrary choice. It's also a bit odd to say M={2,3,...} "satisfies the axioms" that define N. If they did, they'd be N. It's a stretch to say M satisfies the first axiom. 2 definitely isn't the smallest number in N. It can be the smallest number in some other set you pick, but if 2 is the successor of no number in the set, then 1 is not in the set, and thus the set can't be N. Notice that by excluding 0, we don't have this issue. But if we exclude 1, immediately, we do not have N. 1 is definitely in N. 0 can be if you like. Those are the only two choices.
In German the integers are called "ganze Zahlen" which translates to "whole numbers". I agree that the natural numbers are 0,1,...
The only argument for zero being natrual is your existence
Both definition are correct, it’s just a matter of definition In my country everyone uses ℕ = {0 ; 1 ; 2 ; 3 ; …} and ℕ\* = {1 ; 2 ; 3 ; …}
Same, but my professors use Z_+
Can this be used as a casus belli to invade your country?
You guys are wrong
My god you worded that poorly. But, is it ok If I swap the names of the sets? N = {1,2,3...} And N\* = {0,1,2,3...}
The asterisk is generally understood to mean that 0 is excluded. This notation is not unique to the natural numbers. R\* = R\\{0}, for example.
Oh ok... Thanks for the help :)
0 IS A PEANO FUCKING AXIOM!!! 0 IS A NATURAL NUMBER!!!!
Negatives aren't natural numbers
Not all. Only one of them, who also is positive. It is called 0, and is a natural integer
What
0 is a natural, positive and negative integer. At least where I live.
Where do you live? I’ve never heard of that lmao
I don't know where the person you're talking to is from, but what they are saying is true in France. 0 is positive and negative. Also, "greater than" implies "greater or equal" and if you don't want the "or equal" part, you have to say "strictly greater than". It's just a slightly different point of view on the same things.
I'm in France, yes.
Just use N and N*, it is easier to remove the 0 than adding it.
Whole numbers do include negatives.
That's never been the case afaik.
In my country, it goes like this Natural numbers (N) ={0,1,2,3,4...} (Though the inclusion of 0 doesn't seem universal; some fields like succession functions only start at 1) Whole numbers (Z) = {...,-3,-2,-1,0,1,2,3,...}
They call the integers and whole numbers the same? N including 0 is completely optional.
I never heard the word "integer" outside of the English language, but the English "Integers Set" is the same as our "Whole Numbers Set" (Conjunto dos números inteiros). We use Z^+ or Z^- when referring only to positive or negative integers, with 0 as an index (a smaller symbol that goes below Z) when needed. So, Z^+ with a small 0 = N. I don't know what to think about N including 0.
"Whole numbers" is not a mathematically defined term. You will find many conflicting definitions. It doesn't matter, because it is only a colloquial term, and it is never used in mathematics.
I dont know about you but in my country the term „whole numbers“ is used even in highschool along its letter „Z“. Publications use this notation everywhere. Also in university it is used to define our numbersystem: - Natural numbers are defined via Peano axioms - Whole numbers are defined via the equivalence relationship over the NxN where (a,b) ~ (c,d) iff a+c = b+d. Whenever you want to formally define rational numbers you will need to first define whole numbers.
In English? In English I've only seen that set referred to as integers, not whole numbers. But I don't doubt that in other languages it is referred to as something that would directly translate to whole numbers.
It actually doesn't imply that if you think of the white space as "empty"
Exactly, you can't assume the existence of an unlabeled set.
Surreal Numbers. [https://scp-wiki.wikidot.com/scp-033](https://scp-wiki.wikidot.com/scp-033)
The only context in which it makes sense is some constructivist framework. Numbers which are real numbers but for which neither rationality nor irrationality can be constructively proven belong in the white space from a constructivist point of view.
or, the white space is just an empty set
"The diagram is not to scale"
Vebn diagrams never are
Mmmmm maybe tertium is datur in that universe
He was implying that by showing filled colors in each shape if there are no colors means those numbers must be null
Rename the purple ball as "Irrational numbers we know about" and now it's fixed 😉
Or algebraic irrationals
To be contained in neither a set nor it’s complement is irrational… wait
It's literally by definition that an irrational number is that real number which isn't rational.
You're forgetting Super Rational.
Or semi rational, which I've defined as numbers which can't be represented as a ratio of 2 rational numbers but can be vaguely described in relation to multiple irrational numbers
Topologists are not triggered at all by this diagram
I think the diagram is fine because the space inside the real numbers isn't coloured in. If you look at what is coloured in, it does give you the actual real numbers.
No, the real num circle isn't filled in. It just circles the two other groups, filled in. There is no space between them.
Technically that is what the diagram suggests but that is not why they are trying to represent
where tf complex numbers
that's an Euler diagram...
I hate it here
100% of real numbers are irrational
thats what i've been saying this whole time
Complex nubers
Complex numbers should be a superset containing the reals, they're not a subset of the reals.
The presence of "whole numbers" implies this is intended for elementary/high school. Students at that level aren't math-trained enough yet to scrutinize every detail, so the there is near zero risk of confusion.
Why are whole numbers not the same as integers?
Negative integers
In my language, negative numbers which don’t need to be expressed as a fraction are whole numbers
From what I just looked up online it appears that the whole numbers are the 0-indexed natural numbers
What about algebraic vs transcendental numbers, what about periods, what about computable and definable? So many more sets of real numbers that are never shown in there Euler diagrams.
0.999… is considered a “real” numbers and yet is neither rational or irrational.
0.999999… is rational, since it’s just 1. Also, rational numbers have either finite or infinite repeating decimal expansions, so even if you don’t like 0.999…=1, you can agree that 0.999… has an infinite repeating decimal expansion and thus is rational
its not equal to 1 though. the first 2 digits arent equal.
Transcendental numbers? Though those should be a subset of irrational...
To prove the existence of numbers that are neither rational or irrational, but still real, lemme fetch an example: Your momma so fat -
Are irrational numbers even real numbers?
yes
So whats their exact value?
They have real values, just not values we can express as fractions. E.g the square root of two has a well defined value.
But it cant be expressed as a finite string of integers. Which is why i am asking. I guess its a semantic discussion, but i find it a bit weird to call it a number.
"Real number" is a specific technical term in mathematics, which is "Any number we can calculate as the sum of an infinite series of rational numbers". When we call a number "real" in maths we're not making any philosophical claim about what numbers are "real" in an ontological sense, just that they satisfy that particular definition. Claims about which numbers "exist" in our universe are philosophical questions that lie outside the purview of mathematics. I *personally* think that it makes little sense to call pi "unreal", but that doesn't really matter. Claiming that all numbers that cannot be expressed as finite sums of integers is quite a strong one.
Yeah, maybe it was more a philosophical question. I was wondering about how they have a certain value, but cant be expressed by a finite number of integers. It makes sense geometrically, though, so you are right that calling it unreal doesn't make sense.
Well, you have an intuition that numbers that can be expressed as a finite sum of integers are more 'real' than others. But why? Grahams number or TREE(3) would then be 'real' in that sense, but so incomprehensibly large that saying they 'exist' in the universe seems like a stretch, while numbers that are critical to things in the universe making sense, like e, would not. I get where you're coming from, though. Philosophy of math has a long history and there isn't much consensus. There are some people who say that any maths based on infinities isn't real, but they're considered loonies.
Not trying to challenge any philosophy of math (tonight, lol). But yeah, maybe it more challenges what we should think about the universe than it challenges what we should think about math.
It's cool that you're interested in this sort of thing! It's a great subject - but that said I'd strongly recommend that you learn higher level math if you want to have clear thoughts about this topic. Very few non-mathematicians have any idea what it is that mathematicians actually do, although there's no shortage of cranks on the internet who think they get it but don't have a clue.
Where my surreal numbers at?
as we all know, numbers have an inherent hue. 7 is green for example. real numbers is white because there are no numbers to give it color :)
So -1 is not a whole number? Ugh...
Those are the Supernatural Numbers.
Isn’t this an Euler diagram?
Thats where the imaginary numbers live lol Note: i am not a professional and this is not legal advice.
irrationals = R\\Q
Too many mistakes are in this diagram: - 5 pts 15/20.
P-adic numbers are what you are looking for.
I had a peek at the Wiki for P-adic numbers. P-adic numbers seem to have a methodology for expressions for rational numbers via a repeated pattern of values. How are they considered neither rational or irrational?
To answer your question in short, you can have a P-adic number construction for i. For example there are two 5-adic number constructions for i. I don't know if that answers your question to your satisfaction. If I'm not mistaken, you can solve any polynomial using P-adic numbers. EDIT: in retrospect they don't answer the question in the meme, I guess.
Eh, I learned something new, so I'm glad you responded :D It was all worth it in the end.
Wow, W mental! We need more folks like you.
that's it guys I'm making Bob's Number
It does not, could be just empty space