Is 0 a natural number?

>>8342672
yes

No.

inb4 Maybe

>>8342729

no one is going to say it now retard

>>8342672
In my class that basically served as pre-number theory we constructed the natural numbers starting from 0.

So my answer is why not?

There's no answer to this, it's an arbitrary convention. Generally the phrases I choose to use in writing are positive integers (meaning 1,2,3...) and non-negative integers (meaning 0, 1, 2, 3).

If you want to use shorthand with a blackboard bold symbol, just write out (n=1,2,3...) instead or look into the common convention in your field.

As usual, the answer to this question is to clarify. Above all, it is important in math to convey your ideas without ambiguity.

Including zero makes [math]\mathbb{N}[/math] a monoid, which is nice.

>>8342748
>There's no answer to this, it's an arbitrary convention.

Except it's obviously better to include 0. Try defining the polynomial ring without 0, or anything else. Natural numbers are isomorphism classes of finite sets. There are very few cases in which you want to exclude 0.

>>8342767
>There's no answer to why -1 is not a natural number, it's an arbitrary convention.

Except it's obviously better to include -1. Try defining the solution for x + 1 = 0. -1 clearly solves this equation. There are very few cases in which you want to exclude -1.

>>8342672

Sometimes yes, sometimes no. It depends on who you are reading at the moment.

Often while working, a mathematician will write something like

"Consider [eqn] \mathbb{N} [/eqn] , where [eqn] \mathbb{N} = \{1,2,3,...\} [/eqn] ..."

or

"Consider [eqn] \mathbb{N} [/eqn] , where [eqn] \mathbb{N} = \{0,1,2,...\} [/eqn] ..."

in order to eliminate ambiguity. When we ask why this is necessary, we are sometimes told things along the lines of "both sets are bijective, have a least element", etc, and so 'there's no/little point' in specifying further. "They're the same", from the mathematician's point of view.

Like many of you, I still feel that such an answer is a disingenuous cop-out for multiple reasons. First, zero itself is just as much of a novelty and a cognitive leap as negative numbers, fractions, imaginary numbers, etc. The point being that it would seem to make sense, both for the above reason and also because zero is in fact a special number, to simply specify a further set to accompany our existing [eqn] \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} [/eqn] collection.

Second, it would be extremely easy to specify such a set. Ironically my high school education had a perfectly satisfactory solution: 1,2,3 etc are the /naturals/, while 0,1,2,3 etc are the /whole numbers/, as opposed to the /integers/, which then include the negatives. Now, you might instead specify that 0,1,2 etc are the naturals, while the subset 1,2,3 is called something different. That's the part that doesn't matter. What matters is that the two sets are unambiguously distinguished one from the other.

Third, declarations like the ones given at top are obvious wastes of ink and keystrokes.

If it were up to me I would recycle the above convention, "define" (informally) W = 0,1,2 etc, giving [eqn] \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} [/eqn] , and call it a day.

</autism>

>>8342750
Including -1 also makes it a semisymmetric comonad.

Now, here are my, ahem, irrational, unscientific, sentimental reasons for preferring such a scheme:

-the "W" fits the established tradition of using different consonants to denote key sets/number systems.

-Think about the word "natural" itself, in its more literary sense: comes naturally, it naturally occurs to me, clearly occurring in nature, etc. You will find example after example in the ancient world of working with 1,2,3 etc, but while some cultures did write zero-as-such, others had a slight confusion, used it (or some related symbol) only as a placeholder in number writing, etc, while not writing or conceiving of zero as a number as-such. The point being that zero-itself as a number required a little extra cognitive effort in antiquity. Thus zero did not "come naturally" to everyone as 1,2,3 always did to everyone with an IQ north of 60.

-I grew up with it and it was perfectly good until I went off to college and they fucked it all up for no good reason at all.

>>8342750
>>8342767
>>8342772

All this type of bitching just strengthens my central point that 1,2,3 is useful for certain things while 0,1,2 is useful for others. So the answer in terms of establishing a convention, jargon, etc, is to simply specify the two as being distinct sets, regardless of which of the two sets ends up being called the "natural numbers".

Concerning advantages of excluding zero, consider the following:

zero is the "first" natural number, one is the "second"... natural number(?), two is the "third"...???

Meanwhile, alternatively, one is the first natural number. Two is the second natural number... n is the nth natural number. ah!

Although in one sense pedantic, in another sense it really is a substantive observation. This is actually useful at times when indexing sums (say, starting from one), to know that your jth term is precisely j (or, involves plugging in precisely j), and not otherwise. The latter view is also a much "nicer" expression of ordinality.

----

Zero is a natural number. Every natural number can be expressed in a unique manner as a product of primes.* Please let me know the prime factorization of zero.

*To complain "durr that's not what FTA sez" misses the rhetorical point that is being made here. Which is that the confusion over whether to include/exclude zero is exactly what makes such sloppy troll-think as the above possible to begin with.

Again, it doesn't matter which-is-which. Call zero a natural, for all I care. But take care to specify some other set as being precisely 1,2,3, in that event. And then let's all speak the same language, is the point.

>>8342782
What's that? Explain it to me like I know almost no category theory

>>8342804
Let me know the prime factorisation of 1

>>8342672
0 does not exist in nature.

>>8342853
Sorry, I was just joking. A monad is a functor (along with some natural transformations and properties).

>>8342885
Man that's just wrong.
>>8342804
Let's just say that N is formed by numbers which have an integer prime divisor. So... No, 0 isn't part of N :)

>>8342772
Can you give an example where you would want a solution to x + 1 = 0 and not for x + 2 = 0 etc?

>>8342804
>Zero is a natural number. Every natural number can be expressed in a unique manner as a product of primes.* Please let me know the prime factorization of zero.

ok, this example is legit.

>>8342672
Is 0 a rational number?

Theorem: 0 is a natural number.

Proof: Let 0 be a natural number. Then we are done. Q.E.D.

Not that hard, r-tard.

>>8342918

It can be expressed as the quotient of two integers, so yes.

>>8342804

are you artistic

>>8342672
Generally, no 0 is not considered a natural number.

But realistically you will work with sets other than the canonical ones and as such you'll have to be more descriptive anyhow.

>>8342804
What's the prime factorization of 1?

>>8342672
Technically speaking, zero is the absence of numbers.

>>8343352
Technically speaking, you're an idiot.

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>>8342672
It seems only natural to consider it one.

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>>8342672
Kek, my memes are top qual

>>8342672
All numbers in possession of the qualities inherent in any natural numbers and not possessing any of the qualities not inherent to natural numbers are natural numbers. Which qualities do natural numbers possess, which qualities can they not posses.
Does the number zero possess the necessary qualities while not possessing the disqualifying qualities?
If so, zero is a natural number.

>>8342672
I felt that generally in Analysis and in related fields, 0 is not considered a natural number (mainly because indexing sequences is nicer starting with 1), while in abstract Algebra, 0 is considered natural (N is a monoid, defining polynomials as (sum a_j*x^j) for natural j etc.)

0 is a natural number. The whole numbers are the subset of positive integers, then natural, then integers, then their quotient field the rationals, then their completion the reals, then their ring of polynomials, then their quotient by x^2 + 1 and at least you've reached their algebraic closure.

>>8342804
>Every natural number can be expressed in a unique manner as a product of primes
[math]\mathbb{N} = \{2, 3, 4, ... \}[/math]
Now hold on just a second...

>>8343352
You must be at least 18 to post on 4chan.

>>8344126
1 = the empty product

>>8342915
I actually realized this isn't a good example. Unique factorization is usually formulated for RINGS (Z in this case) and you always have to exclude zero. So it's not a good argument.

>>8344126
Wait, couldn't you expand this to all rational numbers if you allow negative powers of primes?

>>8344488
1 = not a product of primes

>>8342672

Why does that cat not have a torso or back legs?

>>8342748

But 0 is also non positive right?

>>8344488

But this reply isn't even very good. Why speak of Z, or of rings to begin with (which by definition entail zero elements) if you're going to be forced to admit right at the top "well oh yeah, we exclude zero anyway in this case."?

You seem to believe that you are making a point because "one always excludes zero in such cases, so that example isn't special", when you're really missed the point: /zero is an extra-special number with certain extra-special properties and caveats that must always be made about it, and everyone knows this./ So depending on what one is doing it may be useful to consider a set that does not even have zero.

One may distinguish between zero and one vis a vis the prime-troll thusly: It may at least be said, for discussion (and not at all rigorously), that one can be represented as "the empty product of no primes" by taking the infinite product of every prime, each raised to the power of zero. this ordered list of infinite zeroes (that is zero-powers, or, terms evaluating to one) does uniquely correspond to one in the same wise as all other such ordered lists of exponents correspond to the primes and composites.

Such cannot even be done, even theoretically, for zero. Of course, to arrive at zero in such a way entails having actually /multiplied by zero/ at some point, implicating it as a prime in our list of primes. And of course zero is literally as far away from being prime as it is possible to get.

The point being that even in this pedantic example, zero and one are special numbers, special cases. So it makes perfectly good sense, depending on the situation, either to include zero, or to exclude it, depending on what one is doing. Which is why we ought to have one set {1,2,3,...} and another set {0,1,2,...}, agree as to their names (it doesn't actually matter which is called which, only that the two sets are described as distinct), and get on with our lives.

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we defined naturals in my programming languages class in terms of:

z, S(z), S(S(z))...

now z doesn't look like it stands for "one"

QED :)

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>>8345125
>>8345125
>Why speak of Z, or of rings to begin with (which by definition entail zero elements) if you're going to be forced to admit right at the top "well oh yeah, we exclude zero anyway in this case."?

In the VAST MAJORITY OF CASES you do not have to exclude zero. It is almost always preferable to include it when you can. Even in this case, one can define the prime factorization of 0 as having infinity in all components. Then you obtain the exact same results: where PF denotes the sequence of prime exponents in a number's prime factorization,

a | b => PF(a) <= PF(b) componentwise
PF(ab) = PF(a) + PF(b)
and where b != 0, PF(a/b) = PF(a) - PF(b)

It works perfectly.

>and not at all rigorously

It is rigorous you moran. The empty product is 1 by definition.

> Which is why we ought to have one set {1,2,3,...} and another set {0,1,2,...}, agree as to their names (it doesn't actually matter which is called which, only that the two sets are described as distinct), and get on with our lives.

It does matter what they're called.

>let's define R as R\{0}
>let's define Q as Q\{0}
>let's define C as C\{0}

Do you see how stupid this is? It is equally stupid for natural numbers. In all cases having an additive identity is preferable.

>>8345433

But you contradict your own rhetoric in your first paragraph of reply. You had earlier objected to my thing in your post of >>8344488 , where you pointed out that in such-and-such a situation (algebraic unique factorization with rings, Z, etc) "you always have to exclude zero". Your implicit point being that in a certain mathematical situation, one always excludes zero. Your words.

You then contradict your rhetoric in a way which undermines your own argument, when you say >>8345433 that "In the VAST MAJORITY OF CASES you do not have to exclude zero." At this point you will complain that I am confusing two different areas of math in your argument, but here's the thing. On the one hand, it is natural :^) to ask you what areas of math you have in mind in your second example. But it turns out not to matter, because you've just admitted that there are some situations (albeit a vast minority in your telling) where one wishes not to speak of zero, for whatever reason. And of course this is true.

Your latter point is also simply wrong. It does not matter by what name we call a particular set/object, as long as we understand what the object is, how it works, etc. We could historically have called the reals the complexes, and the complexes Parliament Funkadelic and come to no harm. You've also suggested that I demand that we "redefine" the reals etc to exclude zero, which I did not do. Since as I just said, the names don't matter, this point is moot. But as you know, we sometimes have occasion to speak of the punctured line plane etc, and so notations like [eqn] \mathbb{ R^{*} , C^{*} , R^{+} } [/eqn] are used. Because, for example, the simple notion of trichotomy from analysis.

My central point, which remains correct, is that we should have one name for {1,2,3,...}, another name for {0,1,2,...}, and be done with it. And as you've just conceded again, there are times when one wishes to exclude zero, and so I appreciate your concession of the point.

>>8342776
If anything, [math] \mathbb{W} [/math] should equal [math] \{ 1, 2, 3, ... \} [/math].

Nothing's an abstract concept and can't be quantified strictly by using measurements, meaning in the case of trying to measure something [extant], and not looking for something which exists solely as an idea, or as a concept e.g. We looked for something, but couldn't find it, therefore there are zero of those things. You still observed things in the search of that something, therefore never finding 'zero.'

0 does not naturally exist.

Further illustrating the lack of nothingness: https://en.wikipedia.org/wiki/Vacuum_energy

>>8345527
>My central point, which remains correct, is that we should have one name for {1,2,3,...}, another name for {0,1,2,...}, and be done with it.

Non-negative Integers. It's not a formal number set, but definitively describes the {0,1,2,...} concept.

>>8345584
>Further illustrating the lack of nothingness: https://en.wikipedia.org/wiki/Vacuum_energy

Okay, so [math]\mathbb{N} = \{\frac12, \frac32, \frac52, \frac72, \ldots\}[/math].

>>8342776
This is a good answer.

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>>8342672
where was cat

>>8344761
1*1

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if you're just using the naturals to construct some other object, including 0 just saves you an extra step down the line
your set is already a monoid, and you don't need to adjoin a 0 to whatever you build from the naturals, which you will almost certainly have to do

(if there is a case where N is required to only be a semigroup please tell me before i make a fool of myself in front of the other math nerds)


otherwise i use [math] \mathbb{Z}^+ [/math] to denote [math] \{1,2,3,..\} [/math] and
[math] \mathbb{N} [/math] to denote [math] \{0,1,2,..\} [/math]

fight me

>>8342745
I don't know

>>8346059
that literally means nothing in terms of it being a prime or not, retard

>>8345433

>thread is almost dead

>this guy never actually bothered to reply to my post >>8345527 where he got BTFO

>true, he might simply have never seen my post, but I much prefer to think that he actually read the thing, understood that he was cornered/wrong, and so didn't bother replying

feels so, so good.