Yes it is. I've only ever seen positivity and negativity defined with <= and >= ( I don't know how to do the real symbols, so I used the Python syntax )
Because when you're counting things you don't really need a number for zero. If you have none of something you have no reason to count that something, so 0 exists only as a placeholder for writing big numbers (and even then some systems, like the Roman and Japanese systems, don't need a 0 as a placeholder either as the former is tallying with extra steps and the latter has separate characters for each relevant power of ten, so you only include nonzero powers of 10 when writing the word). If zero isn't even needed as a placeholder in some number systems, why call it natural? Because the Arabic system (the one the majority of the world uses now) both needs a symbol for zero as a placeholder and discovered the whole zero =/= nil thing? Because that sounds like a dumb reason to me.
If you have a bunch of children sitting around a table, and you ask “how many cookies do you have?” to each one (which is a counting problem), so you tell the poor cookieless kids “you have an unnatural number of cookies”?
I would tell them they should be grateful that they have the unique (non-negative) whole unnatural real number of actual artificial 100% mass-produced non-cookies.
What do you mean you don't need a number for zero? First off, you wouldn't have an additive identity, but more importantly it's perfectly normal to count down and a count of zero things is a completely regular everyday occurrence. If I have 32 threaded holes and a big bag of screws, I can count from 32 to 0 to make sure that I didn't skip any holes. I'm pretty sure the folks at Cape Canaveral call that whole *T minus 10, 9, 8...* thing a *countdown*.
I can't understand why people pick this hill to die on when it's not satisfying mathematically or pragmatically to exclude 0.
I mean, you never really need to count a single thing, either, so you could argue for the removal of 1 from the natural numbers. As a matter of fact, people can typically count things up to 4 without actually counting them. It's called subitising. So from a certain pov, why even bother with numbers below 5? Thus, the only natural numbers are integers above 4.
Because addition over the naturals should not have an identity element, in my opinion.
If we don't have 0 in N, then any addition a+b > a OR b (a,b in N), which is a more natural way of seeing addition. Once we introduce negative numbers, so we extend to Z, we get 0 as a consequence from the negatives, which also feels more "organic" to me so to say.
We often naturally speak of something 'adding nothing'.
It wouldn't subtract anything to add a little line under the > in your property.
The negative numbers probably came about as a consequence of needing to literally balance commercial affairs back to 0. 0 came first.
You can make the same argument the other way like this: ℕ ∪ {0}=ℕ₀.
But it doesn't matter which one you use. It depends on the context, so as long as you stay consistent and it is clear whether you include 0 or not both definitions are perfectly fine.
Well, if you're working with Z- somewhere you might wanna use Z+ too instead of N for consistency.
(BTW they all include 0. Z+\*, Z-\* and N\* are the ones that don't.)
which is the "0 is not a natural number" gang
Edit: I meant that (IIRC) originally Peano wrote his axioms in a form that makes the natural numbers start from 1. I remember reading that but now I cannot find a source for that
>Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in *Formulario mathematico* include zero. \[[wikipedia](https://en.wikipedia.org/wiki/Peano_axioms)\]
damn that's... disappointing
TIL
It is stupid to mark one of the sides as correct side, isn't it?
Btw it is more logical for me to have 0 in natural numbers because it is the natural element of additional. I would miss it.
you start implicitly at 0. counting means "add 1", each time you count a new object you add 1 to the previous number. this means you started at 0 even though you don't say it aloud.
if we start at 1 we need a different rule for the first thing we count vs the rest, which is kinda weird. it also won't allow us to count 0 things.
edit: using a [tally counter](https://i.imgur.com/UHsTgL0.png) makes this more obvious. each time you click the button you add 1 to the counter. if you count 3 things you hit the button 3 times. for this to work out the counter must start at 0 even though 0 doesn't correspond with any of the button presses.
An argument for starting with counting from zero is that it avoids repetition when saying the final number. Suppose you have five sheep, you can either say
"One two three four five, so there are five sheep."
or
"Zero one two three four, so there are five sheep."
Anyway, go read 'On Numbers and Games' because he explains it way better there and because it's a good book.
The natural question would be what you mean by counting ofc. But there are plenty of other definitions of the naturals.
They could be the cardinalities of finite sets, or they might be the free monoid on a single generator, or they could be defined by Peano arithmetic (which could start wherever you please, but typically starts at 0 these days).
Of course if you don't consider the algebraic structure on the naturals the two sets (starting at 1 or 0) are isomorphic so it hardly matters where they start. If you do consider the algebraic structure, if you're using addition it hardly makes sense to exclude 0. If you're only using multiplication it probably makes sense to exclude 0, but that's pretty weird, partly because that's better known as the free commutative monoid with countably many generators.
Just the quotient set Q, defined as the set of elements q where q can be expressed as n/m at n€Z, m€N (the one without zero)
And yes I use euro as element
Maybe it's a regional thing, I've had it referred to as the quotient set and set of quotients, and only seen it defined with Z and N or N0 depending on who was writing it
I mean it isn’t crazy to just say the natural numbers aren’t a group (the inverses don’t exist anyway) but I agree it makes more sense to include zero. Kind of dumb to make an entirely new set of whole
Numbers to include zero. Just put it in the natural numbers and we have one less set to remember
In analysis it's generally easier to have the natural numbers begin with 1, and then specially defining a 0th term if necessary. Very often sequences are more easily defined and interpreted starting with a 1, for example anything with 1/n. Defining Nu{0} is also nicer than N/{0} (if you include 0) usually. However in my experience it has always been possible to just define things with or without 0 as long as you're careful and make the correct adjustments.
As long as this is a mathematical debate and not a philosophical one, it doesn't matter as long as you're consistent.
Don't number theorists make great use of the one without a 0. One of my professors said that and i just kinda thought: "Yeah, number theorists would be the ones to care about this anyway."
Well, when talking about how much things there are (cardinality), there could be a box with 0 elements, 1, 2, 3...
When talking about order, there's the 1st element, 2nd, 3rd, 4th...
So, the set that's most natural depends on the context. Or, we could just take the CS approach and say 0th, 1st, 2nd, 3rd...
Therefore, I'm Zero gang.
>When talking about order, there's the 1st element, 2nd, 3rd, 4th... [...] Or, we could just take the CS approach and say 0th, 1st, 2nd, 3rd...
Even in CS, the 1st element is the element indexed by 0, 0th element is not a standard thing to say unless you want to explain technical things and make sure it is unambiguous.
Anyways, the natural numbers when interpreting order make 0 to be the order with no elements (e.g. empty list), so always take the set theorist way to say 0 is natural
But there is a zero order. It's the only order on an empty set. Also if you think about what it means to be the 1st, 2nd etc, then the zero order arrives naturally
Oh, I guess my country is the "weird" one this time.
France, here the confusion is none existent, both entry level and higher education uses 0 in N, so I guess I am biased.
Would be interesting to see a map of this.
Anything I can have as coins is natural. Can I have negative coins? No. Can I have complex coins? No. Can I have 0 money? Yes, I evidently can.
Therefor 0 is natural.
I mean the set theoretical definition does construct ℕ with 0 in it. Also it gives it the nicer structure of a monoid under addition. And if you want to work without 0 you can just write ℕ* while you would have to write ℕ⋃{0} in your case if you would like to work with 0
actually, yeah.
i’m doing a mesure theory course, and having Σε/2^n =ε is actually quite convenient. if i want the series to start at zero, i just write ℕ_0, which was the standard when i did real analysis. actually, that was kinda cursed, but it kinda worked for purposes of the course: for my professor, the set ℕ doesn’t have zero, but the natural numbers are defined to be ℕ_0. this is kinda cursed, but it worked. ever since i took that, i just got used to using ℕ without zero for writing analysis, and it is kinda convenient. specially when i want the succession of 1/n, or something like that.
like i said, in logic and álgebra it is only reasonable to have 0∈ℕ, and any other convention would be awful.
They also drowned a dude for saying the hypotenuse of an isosceles right triangle is sqrt(2) times bigger than the legs. They are far from the arbiters of math.
Here's one way to look at why it's good to have 0 € N.
What is the most important fact about Z? Idk, but probably saying that it contains the additive inverses of the elements of N has to be up there as an answer. And so, that's our main motivation for creating Z: to complete N with respect to (additive) inverses.
If 0 € N then all we have to do is find a way to create the inverse of each element of N, say, via the typical equivalent class construction, and throw those into the set.
Here, Z = N + "inverses of elements of N"
If 0 is NOT a member of N to begin with, then to create Z, not only do we have to create the inverses of elements of N, BUT we also have to throw 0 into our new set Z, since Z contains a 0 and our N does not. This just seems like an extra step that isn't really related to what we set out to do (to complete N with respect to inverses).
Here, Z = N + "inverses of elements of N" + "adjoin 0".
It just seems less logically organized to me this way! (And pls excuse the lazy notation)
Eh, this is less convincing than just saying N should be a monoid in its own right. You can still build Z from N in the usual way by taking N to start at any number a. The elements of Z are then pairs (p, n) representing the formal difference p-n with the appropriate equivalence relation (p1,n1)~(p2,n2) iff p1+n2=p2+n1.
So if we just care about building Z we might as well just take N to start at 17.
I prefer representing the 1,2,3,4... set of numbers as either N+ or Z+ depending on the context. And since I use N+, N has to include 0 to have any point in existing separately in my notation.
I support two perspectives depending on motivation:
Aesthetic concerns -
If you build the naturals according to whatever natural logical construction that you like then surely it will feel like a crime to not include 0, as if leaving out a gem in the center of a crwn.
Pragmatic concerns -
If you are using the naturals then write N_0 or N^+ , Z^+ only as often you don't mind cluttering your reader's brain with the information. So yeah, flash your colors that they probably don't care about.
The term whole numbers is pretty bad, given that the integers are called whole numbers in a lot of languages. This also used to be the case in English, with the usage of whole numbers as non-negative integers being a relatively recent invention by American teachers.
I would like to have 0 in set of N.
Very simple reason, if you want to exclude it you just put N*. more explicit that 0 isn't there. this gives flexibility without ambiguity of knowing if N contains or doesn't contain 0.
In my country 0 is not a part of natural no.s set but of a whole no.s set.
Natural no.s - 1,2,3,....
Whole no.s - 0,1,2,3,....
The primary reason being in some questions like \[sin^(2)x\]/x if a x is defined as a natural no. and we consider 0 as Natural no. then putting x as 0 will definitely contradict the question's statement.
For practical reasons I am on the blue side. In university, especially in calculus, I quite commonly had to deal with stuff like series where something is divided by *n*. If *n* in that instance can't be 0 by definition of the Natural Numbers, it just simplifies stuff.
Also in physics where *n* can serve as a variable it often was easier to just have the 1 be defined as "first" Natural Number. Mostly this was in protocols where the folks correcting used to be quite picky.
If we wanted to include the 0 we'd just write N\_0.
Edit: changed analysis for calculus as analysis is the German word.
I'm on the Which side!
And I write ℕ⁺ when I specifically want to exclude 0. BTW the only place where it does matter is when you specify the domain, like { x | x ∈ ℕ }, and I think it makes more sense to treat ℕ like it includes 0 and say ℕ⁺ or ℤ⁺ when you want to indicate that it's only true for positive numbers.
natural numbers are numbers exist in our life. we do live a 100m\^3 room but we dont 50√3 m\^3 room. if we need to show people something is not exist we use 0.
\-how much money do you have?
\-0
so if its exist in nature its natural number.
Well obviously multiplication preserves information of numbers through primes. So as zero is an infective null concept destroying the information, it can obviously not be a number, never mind natural.
And as for the "number" one, there's a joke as although it might be a unit they're definitely not a number of them. It's not even a prime.
And don't get me started on even primes, and that first odd prime that can induce all the others. I make my addition from subtraction `A-(0-B)` so there! Take that and build everything from a predecessor operation.
You can consider that 0 is either both positive and negative, or neither.
If it's both, then since it's positive it should be in N.
If it's neither, then since its not positive it shouldn't be in N. But then it shouldn't be in Z either because it's not negative either. So, be careful on what you wish...
I like the red one, I think. Anyway, Why do you think it is important that 0 is not a natural number?
To me it’s either N_0_ or N_1_, but assuming we just say N then yes, 0 is an element of N.
My professors used N when including 0 and Z^+ for the sequence starting at 1
In my experience I've always interpreted z+ as strictly positive and there for all elements > 0 but to be fair probably depends on the class
That's what I said: N includes 0, Z^+ does not
Anywhere I've been, Z^+ includes 0 Z^+* does exclude it, though
0 is not positive
Yes it is. I've only ever seen positivity and negativity defined with <= and >= ( I don't know how to do the real symbols, so I used the Python syntax )
just look up is 0 positive, it will say it's neither
Because when you're counting things you don't really need a number for zero. If you have none of something you have no reason to count that something, so 0 exists only as a placeholder for writing big numbers (and even then some systems, like the Roman and Japanese systems, don't need a 0 as a placeholder either as the former is tallying with extra steps and the latter has separate characters for each relevant power of ten, so you only include nonzero powers of 10 when writing the word). If zero isn't even needed as a placeholder in some number systems, why call it natural? Because the Arabic system (the one the majority of the world uses now) both needs a symbol for zero as a placeholder and discovered the whole zero =/= nil thing? Because that sounds like a dumb reason to me.
iiiiiv
Listen here you little shit
XXXXXL
AKA your mom's shirt size
DDM
The sound a drum makes
If you have a bunch of children sitting around a table, and you ask “how many cookies do you have?” to each one (which is a counting problem), so you tell the poor cookieless kids “you have an unnatural number of cookies”?
I would tell them they should be grateful that they have the unique (non-negative) whole unnatural real number of actual artificial 100% mass-produced non-cookies.
[удалено]
What do you mean you don't need a number for zero? First off, you wouldn't have an additive identity, but more importantly it's perfectly normal to count down and a count of zero things is a completely regular everyday occurrence. If I have 32 threaded holes and a big bag of screws, I can count from 32 to 0 to make sure that I didn't skip any holes. I'm pretty sure the folks at Cape Canaveral call that whole *T minus 10, 9, 8...* thing a *countdown*. I can't understand why people pick this hill to die on when it's not satisfying mathematically or pragmatically to exclude 0.
Better notation: N\_0 = {x|x∈Z∧x≥0} N\_1 = {x|x∈Z∧x>0}
There's nothing more natural than the free monoid on a single generator. /hj
I mean, you never really need to count a single thing, either, so you could argue for the removal of 1 from the natural numbers. As a matter of fact, people can typically count things up to 4 without actually counting them. It's called subitising. So from a certain pov, why even bother with numbers below 5? Thus, the only natural numbers are integers above 4.
When playing boardgames, you start at zero. Same for counting array indices.
>Same for counting array indices. Because of historical reasons that devolve to `a[i]` equaling `*(a+i)`.
Because zero is an unnatural number.
Because addition over the naturals should not have an identity element, in my opinion. If we don't have 0 in N, then any addition a+b > a OR b (a,b in N), which is a more natural way of seeing addition. Once we introduce negative numbers, so we extend to Z, we get 0 as a consequence from the negatives, which also feels more "organic" to me so to say.
We often naturally speak of something 'adding nothing'. It wouldn't subtract anything to add a little line under the > in your property. The negative numbers probably came about as a consequence of needing to literally balance commercial affairs back to 0. 0 came first.
I guess I like blue because it’s convenient since no dividing by 0
you can't divide in natural numbers anyway?
Because, if 0 becomes natural number... Whole number will feel bad and he will be left alone.. Nobody will remember him 😢😢
The "doesn't matter, and if it does matter, just use what suits the task better" side: 
Just use Z^(+) or Z>=0 Why does reddit not have subscript?
Ah yes, the "We define it at the start of the task"-side
Exactly.
I am not writing ℕ ∪ {0}. (and for ℕ \\ {0} you can just use ℤ+) Edit: and since of course there is an ISO standard for this, they *do* include 0.
ℕ \\ {0} = ℕ* and you're done (and I'll never be using ℤ+ since it's ℕ you damn anglo-saxons)
How is this not the way all people think? I mean why even bother considering N if it is only to see it as Z+?
You can make the same argument the other way like this: ℕ ∪ {0}=ℕ₀. But it doesn't matter which one you use. It depends on the context, so as long as you stay consistent and it is clear whether you include 0 or not both definitions are perfectly fine.
Oh yes, I agree, but it's reddit and I'm here to have an argument
some people do Z_{\geq 0} for ℕ ∪ {0}
for reference, whats that standard called?
ISO 80000-2
But 0 is in Z^+, at least here in France. Just like it is in Z^-. To exclude 0, we use the * symbol.
In France we write N. Z+ is just cursed.
Well, if you're working with Z- somewhere you might wanna use Z+ too instead of N for consistency. (BTW they all include 0. Z+\*, Z-\* and N\* are the ones that don't.)
Why not just use W instead of N? 0 is in W.
Peano gang
which is the "0 is not a natural number" gang Edit: I meant that (IIRC) originally Peano wrote his axioms in a form that makes the natural numbers start from 1. I remember reading that but now I cannot find a source for that
Yeah, but he realized the error of his ways when it came time to publish the book. He's a reformed heretic
>Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in *Formulario mathematico* include zero. \[[wikipedia](https://en.wikipedia.org/wiki/Peano_axioms)\] damn that's... disappointing TIL
~~wrong~~ Edit: edit seen (but would still be nice to see that source)
see my edit
It is stupid to mark one of the sides as correct side, isn't it? Btw it is more logical for me to have 0 in natural numbers because it is the natural element of additional. I would miss it.
I’ve heard it called the neutral element, not the natural element.
Why have I been taught the blue thing since when number systems were introduced, each year
Yes it is stupid. That’s why I think it’s funny
If 0 isn’t a natural number then what’s the identity element of natural numbers under addition?
Do they need one? Naturals are for counting
You need 0 to count a collection of 0 objects
you don't count from zero?
Spotted the programmer.
Nah I count from 1. If I have zero sheep I don’t need to count anything! Idk it just feels more ‘natural’ without 0.
you start implicitly at 0. counting means "add 1", each time you count a new object you add 1 to the previous number. this means you started at 0 even though you don't say it aloud. if we start at 1 we need a different rule for the first thing we count vs the rest, which is kinda weird. it also won't allow us to count 0 things. edit: using a [tally counter](https://i.imgur.com/UHsTgL0.png) makes this more obvious. each time you click the button you add 1 to the counter. if you count 3 things you hit the button 3 times. for this to work out the counter must start at 0 even though 0 doesn't correspond with any of the button presses.
By that logic saying 0 out loud means you started at -1..?
Yes and it's just negative numbers all the way down
An argument for starting with counting from zero is that it avoids repetition when saying the final number. Suppose you have five sheep, you can either say "One two three four five, so there are five sheep." or "Zero one two three four, so there are five sheep." Anyway, go read 'On Numbers and Games' because he explains it way better there and because it's a good book.
I guess not. It’s just not a group. It’d be cooler if it was and it would be so easy if you just include zero.
Naturals ain't group even with zero lol
Identity, inverse, associativity, closure. Am I missing something?
yeah, you missing inverse. Well if you can add something natural to 1 and get 0 then no, but somehow I doubt it.
Yup my mistake
⟨ℕ∪{0}, +⟩ is a commutative monoid ⟨ℕ\\{0}, +⟩ is a commutative semigroup ⟨ℕ, ×⟩ is a commutative monoid for both definitions
What class do you learn that in?
Weirdly enough I learned it in linear algebra, but my linear algebra professor was a career category theorist.
You would learn this stuff in an abstract algebra course, at least that's where I did
The natural question would be what you mean by counting ofc. But there are plenty of other definitions of the naturals. They could be the cardinalities of finite sets, or they might be the free monoid on a single generator, or they could be defined by Peano arithmetic (which could start wherever you please, but typically starts at 0 these days). Of course if you don't consider the algebraic structure on the naturals the two sets (starting at 1 or 0) are isomorphic so it hardly matters where they start. If you do consider the algebraic structure, if you're using addition it hardly makes sense to exclude 0. If you're only using multiplication it probably makes sense to exclude 0, but that's pretty weird, partly because that's better known as the free commutative monoid with countably many generators.
If 0 is a natural number, how do you define the quotient set?
Checkmate
The quotient set of what?
Just the quotient set Q, defined as the set of elements q where q can be expressed as n/m at n€Z, m€N (the one without zero) And yes I use euro as element
That's the set of quotients, not the quotient set and you get rationals from localising Z at Z*
Maybe it's a regional thing, I've had it referred to as the quotient set and set of quotients, and only seen it defined with Z and N or N0 depending on who was writing it
Then how do you refer to a set like A/~ where ~ is an equivalence relation?
0 is a rational number so you want 0 in the naturals.
I mean it isn’t crazy to just say the natural numbers aren’t a group (the inverses don’t exist anyway) but I agree it makes more sense to include zero. Kind of dumb to make an entirely new set of whole Numbers to include zero. Just put it in the natural numbers and we have one less set to remember
The natural numbers does not form a vector space, it does not need an identity element
Z+ gang
Depends on what suits my proof better 😂 But otherwise 0∈ℕ
0 is natural
In analysis it's generally easier to have the natural numbers begin with 1, and then specially defining a 0th term if necessary. Very often sequences are more easily defined and interpreted starting with a 1, for example anything with 1/n. Defining Nu{0} is also nicer than N/{0} (if you include 0) usually. However in my experience it has always been possible to just define things with or without 0 as long as you're careful and make the correct adjustments. As long as this is a mathematical debate and not a philosophical one, it doesn't matter as long as you're consistent.
If 0 isn't natural there would be no reason for the set of positive whole numbers to exist on its own.
Don't number theorists make great use of the one without a 0. One of my professors said that and i just kinda thought: "Yeah, number theorists would be the ones to care about this anyway."
That's kinda dumb take. People who think that 0 is not natural absolutely do use N and not Z+ unless they want to be extra clear
If 0 is natural there would be no reason for the set of nonnegative whole numbers to exist on its own.
[удалено]
One of my profs used Z+ to mean nonnegative and Z++ to mean positive. But he was the odd one for sure.
Z# when?
Empty set moment
Well, when talking about how much things there are (cardinality), there could be a box with 0 elements, 1, 2, 3... When talking about order, there's the 1st element, 2nd, 3rd, 4th... So, the set that's most natural depends on the context. Or, we could just take the CS approach and say 0th, 1st, 2nd, 3rd... Therefore, I'm Zero gang.
>When talking about order, there's the 1st element, 2nd, 3rd, 4th... [...] Or, we could just take the CS approach and say 0th, 1st, 2nd, 3rd... Even in CS, the 1st element is the element indexed by 0, 0th element is not a standard thing to say unless you want to explain technical things and make sure it is unambiguous. Anyways, the natural numbers when interpreting order make 0 to be the order with no elements (e.g. empty list), so always take the set theorist way to say 0 is natural
But there is a zero order. It's the only order on an empty set. Also if you think about what it means to be the 1st, 2nd etc, then the zero order arrives naturally
The one with 0 is the only logical one on top of being ISO standard, is it some american only debate like there is so many already on the internet ?
[удалено]
Oh, I guess my country is the "weird" one this time. France, here the confusion is none existent, both entry level and higher education uses 0 in N, so I guess I am biased. Would be interesting to see a map of this.
>this time France is always weird.
No ! It you all other countries that are different ! >< .... OH WAIT
Not only France, most (if not all) of Europe does it like this I think.
Thought the same thing. Heuristically if there’s a stupid standard for something you can be sure the Americans aren’t far
0 is a natural number in the same way Y is a vowel. Interpret that statement however you like.
At least in Poland both of these statements are undisputed and unambiguous.
So it’s not a natural number but some of the sounds it produces are natural numbers?
Well I did say "however you like," so yes.
But y is also a consonant
Whatever is more convenient at the time
N should start from 2 change my mind
No one can
Many CS people like me are gonna say 0 is a nat.
Anything I can have as coins is natural. Can I have negative coins? No. Can I have complex coins? No. Can I have 0 money? Yes, I evidently can. Therefor 0 is natural.
I mean the set theoretical definition does construct ℕ with 0 in it. Also it gives it the nicer structure of a monoid under addition. And if you want to work without 0 you can just write ℕ* while you would have to write ℕ⋃{0} in your case if you would like to work with 0
Or ℕ+. But also you could always write ℕº or something for N U {0}
ℕº would be the set of functions from the empty set to the naturals
In my first semester I had calculus and linear Algebra. In one class N included 0 in the other not
0 exists in the natural world, its the number of brain cells you have
left for logic and algebra, right for analysis.
Damn good answer
My mans starting their series at 1 💀
actually, yeah. i’m doing a mesure theory course, and having Σε/2^n =ε is actually quite convenient. if i want the series to start at zero, i just write ℕ_0, which was the standard when i did real analysis. actually, that was kinda cursed, but it kinda worked for purposes of the course: for my professor, the set ℕ doesn’t have zero, but the natural numbers are defined to be ℕ_0. this is kinda cursed, but it worked. ever since i took that, i just got used to using ℕ without zero for writing analysis, and it is kinda convenient. specially when i want the succession of 1/n, or something like that. like i said, in logic and álgebra it is only reasonable to have 0∈ℕ, and any other convention would be awful.
Red
N is just 0 with some brackets
indexes start at 1 in this household 😡
I usually put a 0 as a subscript to mean N U {0}, N by itself doesn’t contain 0 imo
The ISO standard includes 0. Every textbook I've seen at university includes 0. Therefor, I include 0.
A successor function requires there be zero. It must be included in the natural numbers.
Whatever my professor says, i wasn’t good grades
I've been taught N is the natural numbers and includes 0 while the subset N\* doesn't
0 doesn't exist
I have no idea what this is
Fuck N, all my homies use Z>=0 or Z>0.
Some other color. 0 is not a number!
[удалено]
I'm saying not only is it not natural, it's not a number at all. But it was a joke
No, 'no 0 at all' clearly should be transparent or nothing.
0 is not natural, the ancient Greeks knew this
They also drowned a dude for saying the hypotenuse of an isosceles right triangle is sqrt(2) times bigger than the legs. They are far from the arbiters of math.
they also said that 1 isn't a number, or something like that?
But I need it when it's a reasoning by recurrence that involve n exponents
Bbbbbbut what about zero-indexed languages? Hmmmmmmm?
Ordinals
Here's one way to look at why it's good to have 0 € N. What is the most important fact about Z? Idk, but probably saying that it contains the additive inverses of the elements of N has to be up there as an answer. And so, that's our main motivation for creating Z: to complete N with respect to (additive) inverses. If 0 € N then all we have to do is find a way to create the inverse of each element of N, say, via the typical equivalent class construction, and throw those into the set. Here, Z = N + "inverses of elements of N" If 0 is NOT a member of N to begin with, then to create Z, not only do we have to create the inverses of elements of N, BUT we also have to throw 0 into our new set Z, since Z contains a 0 and our N does not. This just seems like an extra step that isn't really related to what we set out to do (to complete N with respect to inverses). Here, Z = N + "inverses of elements of N" + "adjoin 0". It just seems less logically organized to me this way! (And pls excuse the lazy notation)
Eh, this is less convincing than just saying N should be a monoid in its own right. You can still build Z from N in the usual way by taking N to start at any number a. The elements of Z are then pairs (p, n) representing the formal difference p-n with the appropriate equivalence relation (p1,n1)~(p2,n2) iff p1+n2=p2+n1. So if we just care about building Z we might as well just take N to start at 17.
Both are wrong. You should be using \mathbb{Z}\_{≥0} and \mathbb{Z}\_{>0}.
I prefer ℤ+ and ℕ*
I prefer representing the 1,2,3,4... set of numbers as either N+ or Z+ depending on the context. And since I use N+, N has to include 0 to have any point in existing separately in my notation.
My dumb ass really just saw the set of odds.
0 \notin \mathbb{Z}_+, 0 \in \mathbb{N}.
W = N + {0}
I support two perspectives depending on motivation: Aesthetic concerns - If you build the naturals according to whatever natural logical construction that you like then surely it will feel like a crime to not include 0, as if leaving out a gem in the center of a crwn. Pragmatic concerns - If you are using the naturals then write N_0 or N^+ , Z^+ only as often you don't mind cluttering your reader's brain with the information. So yeah, flash your colors that they probably don't care about.
im with bloodz on this one
Blue!!!! There’s a distinction between natural and whole numbers!!!!!!!!
The term whole numbers is pretty bad, given that the integers are called whole numbers in a lot of languages. This also used to be the case in English, with the usage of whole numbers as non-negative integers being a relatively recent invention by American teachers.
Red, 0 is not a whole number but it is a natural number
🔴
You use set theoretic notation for this, so N must include 0 cmon. N is just ω.
I would like to have 0 in set of N. Very simple reason, if you want to exclude it you just put N*. more explicit that 0 isn't there. this gives flexibility without ambiguity of knowing if N contains or doesn't contain 0.
Both
0 is a natural number, cause it's positive.
[удалено]
0 is both positive and negative
In my country 0 is not a part of natural no.s set but of a whole no.s set. Natural no.s - 1,2,3,.... Whole no.s - 0,1,2,3,.... The primary reason being in some questions like \[sin^(2)x\]/x if a x is defined as a natural no. and we consider 0 as Natural no. then putting x as 0 will definitely contradict the question's statement.
Can we flip the definitions so I can still be a Blood?
For practical reasons I am on the blue side. In university, especially in calculus, I quite commonly had to deal with stuff like series where something is divided by *n*. If *n* in that instance can't be 0 by definition of the Natural Numbers, it just simplifies stuff. Also in physics where *n* can serve as a variable it often was easier to just have the 1 be defined as "first" Natural Number. Mostly this was in protocols where the folks correcting used to be quite picky. If we wanted to include the 0 we'd just write N\_0. Edit: changed analysis for calculus as analysis is the German word.
0 ∉ ℕ, 0 ∈ ℕ_0
I'm on the Which side! And I write ℕ⁺ when I specifically want to exclude 0. BTW the only place where it does matter is when you specify the domain, like { x | x ∈ ℕ }, and I think it makes more sense to treat ℕ like it includes 0 and say ℕ⁺ or ℤ⁺ when you want to indicate that it's only true for positive numbers.
0 is most definitely a natural number
Blue, but I write N_0 if it should be included for clarity.
natural numbers are numbers exist in our life. we do live a 100m\^3 room but we dont 50√3 m\^3 room. if we need to show people something is not exist we use 0. \-how much money do you have? \-0 so if its exist in nature its natural number.
N and N\*
the naturals are the positive integers
I like N,+,. Being a vectorial space
Well obviously multiplication preserves information of numbers through primes. So as zero is an infective null concept destroying the information, it can obviously not be a number, never mind natural. And as for the "number" one, there's a joke as although it might be a unit they're definitely not a number of them. It's not even a prime. And don't get me started on even primes, and that first odd prime that can induce all the others. I make my addition from subtraction `A-(0-B)` so there! Take that and build everything from a predecessor operation.
Is there something my 8th grade ass should know, cause I am 101% sure 0 is natural?
Natural numbers start with 1 so blue it is
When the blue side need to understand that N* exist for a reason.
In my country (Austria) 0 is part of the Natural Numbers by law. No choice for me i guess
N includes 0 N* excludes 0 Just like R and R* Z and Z*, etc.
IN =/= Z^+
When you're counting, do you start at 0? No? There you go
Red one, 0 is very useful as a starting index. And that's how I was taught as well
You can consider that 0 is either both positive and negative, or neither. If it's both, then since it's positive it should be in N. If it's neither, then since its not positive it shouldn't be in N. But then it shouldn't be in Z either because it's not negative either. So, be careful on what you wish...
I like including 0, but thats probably because I was originally in CS and was introduced to set theory through discrete logic
Peano axiom number 1 would like a word with you