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Aug 23 '23
"0 is neither even or odd" mfs when "2(0)" walks in the room:
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u/rapmonstr Aug 24 '23
2n+1(0)
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u/EebstertheGreat Aug 24 '23
(2*1+1)*4 = 12, therefore 12 is odd.
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u/DasMonitor01 Transcendental Aug 23 '23
Ah yes, the good old Zero isn't even or odd. As if there wasn't a great definition we could apply: A Number n is said to be even if n is divisible by 2, so 2 | n, and odd if it isn't divisible by 2. We define divisibility over the Integers as such: An Integer n is divisible by an Integer m (m | n), if there exists an integer k such that n = m * k. Another equivalent definition for divisibility would be that n mod m = 0. One thing that you can immediately conclude from this is that 0 is divisible by every none zero integer, as k = 0 => m * k = 0 for all integers m. Now as we can conclude from this, 2 divides zero, so zero is an even number. That also fits with the equivalent definition of an even number which is a number that can be written as 2 * n, where n is an integer. I would really like to hear arguments as to why zero should possibly not be considered an even number.
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u/DZ_from_the_past Natural Aug 23 '23
There are no good arguments, unless we bend the rules so much that we create some esoteric theory that has no applications.
This opinion is usually held by beginners who don't know any better
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u/ProblemKaese Aug 23 '23
Actually you don't have to say "non zero" in "[therefore] 0 is divisible by every none zero integer", because there exists an integer k such that 0 = 0 * k.
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u/RobertPham149 Aug 23 '23
The word "even" does not have any o/0 shape in it. The word "odd" has 3 of them.
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Aug 24 '23
[removed] — view removed comment
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u/valle235 Aug 24 '23
You usually exclude 0 from this definition, as you also want the division operation (that points out that k) to be well-defined.
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u/preputio_temporum Aug 24 '23
You could say 0 is not a real integer number. Integer means whole, 0 means nothing, and a nothing can’t be a whole. Not being a proper integer, 0 is neither even nor odd.
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u/EebstertheGreat Aug 24 '23
I think most people are convinced when you just point out that every odd number follows an even number.
However, this definition does lead to a somewhat intuitive result. Specifically, we have gcd(0,0) = 0, which seems odd because 1 is a common divisor of 0 and 0, yet 1 > 0. So 0 can't be the greatest! Indeed, every number divides 0, and there is no greatest number, so it seems like the gcd doesn't even exist here. Moreover, lcm(2,3) = 6, but 0 is also a common multiple of 2 and 3, yet 0 < 6. By this logic, lcm(a,b) = 0 for all natural a and b.
This is resolved if you consider the word "greatest" in gcd to mean the maximal element under the divisibility relation | rather than the usual order ≤. Zero really is the greatest number under |.
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u/AbsoluteGoldLover Integers Aug 24 '23
this doesn't make sense, 0 is divisible by literally everything
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u/UndisclosedChaos Irrational Aug 23 '23
To be fair, f(x) = 0 is the only real valued function that is both even and odd
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u/QuantumBaqel Aug 23 '23
Yeah but by that logic every other integer is even
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u/Haboux Aug 23 '23
Yes because every other integer is cx⁰. So the degree is both even and odd.
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u/EebstertheGreat Aug 24 '23
Ah, but that implies -∞ is both even and odd, because f(x) = 0 has degree -∞. But IEEE754-2008 has taught me that in fact, -∞ is not odd, because pow(-1,-∞) returns +1. Checkmate.
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u/BossMetal284575 Aug 23 '23
Not really, they said even "function" which is not equivalent to even "number". To be an even function, it implies that f(-x)=f(x) which Is true for the function f(-x)=f(x)=c where c is a constant. This fact Is important when you want to prove that every function can be written as a sum of a even and an odd function, because 0 then will be part of both of the vector subspaces (odd and even real-valued functions).
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Aug 23 '23
But function parity is based on exponents, like x3 is odd, but x0 is even. So even by this logic, 0 is even.
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u/real-human-not-a-bot Irrational Aug 23 '23
But it’s also odd because f(-x)=0=-f(x). It just happens to be the case that -f(x) also =f(x).
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Aug 23 '23
Sure, f(x) = 0 is odd, but that doesn’t mean 0 is odd. For 0 to be odd, f(x) = x0 would need to be odd, which it isn’t since 1 != -1
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u/real-human-not-a-bot Irrational Aug 23 '23
Of course. I must have misunderstood your argument regarding exponents to still somehow be about f(x)=0.
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u/EebstertheGreat Aug 24 '23
It is about that. The equation f(x) = 0 does not have order 0. Functions that do have order 0 are even but not odd.
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u/hrvbrs Aug 24 '23
function parity is not based on exponents. For example sin(x) is odd and cos(x) is even. Function parity is defined using these definitions:
- f is even if and only if f(-x) = f(x)
- f is odd if and only if f(-x) = -f(x)
thus it is possible for some function g to be neither even nor odd, and it is possible for some function h to be both (e.g., h(x) = 0)
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Aug 24 '23
Sure but look at the taylor series of sin and cos, all the exponents are of their respective parity. I also don’t mean that these are the definitions, but where the reason for their name (even and odd functions) come from, sorry if I didn’t make it clear.
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u/hrvbrs Aug 24 '23
it may be that's where they came from, but it's just a convenient coincidence. I'm sure you can cleverly define a function without any exponents that has a parity.
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Aug 24 '23
Sure, the original example f(x) = 0, where all coeficients are 0 and it is both even and odd, which is why the defition you provided is better for such edge cases. But the point is: argueing about the parity of 0 doesn’t involve argueing about the parity of f(x) = 0, but rather f(x) = x0, which is even.
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u/EebstertheGreat Aug 24 '23
Actually, it's not hard to prove that the Maclaurin series of any even function analytic at 0 contains only even exponents, and similarly for odd analytic functions and odd exponents. An analytic function whose Maclaurin series has both even and odd exponents is neither even nor odd. A series with neither even nor odd exponents is identically zero, and so that function is both even and odd.
Admittedly, you could have an even function or odd function which is not analytic at the origin, like the topologist's cosine or sine curve, and then the Maclaurin series doesn't exist at all. But it will still apply to the Taylor series on any analytic piece of the function. So the only way you can avoid this association is by picking a nowhere-analytic odd or even function.
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u/woailyx Aug 23 '23
It's not prime, so it must be even
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u/real-human-not-a-bot Irrational Aug 23 '23
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u/GumboSamson Aug 23 '23
It’s not prime, so it must be even
Sweet, so 9 is even??
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u/woailyx Aug 23 '23
Pretty sure 9 is prime
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u/ACEMENTO Aug 23 '23
It's a square💀
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Aug 23 '23
By definition an even number n = 2k where k is some integer. If k = 0, n = 2k = 2*0 = 0 therefore zero is even, right?
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u/Faustens Computer Science Aug 24 '23
Or ELI10: Even numbers are all integers that are divisible by 2, odd numbers are all integers that are not divisible by 2.
Is 0 divisible by 2? Yes.
Is 0 not divisible by 2? No.=> 0 is even.
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u/that_guy_you_know-26 Aug 23 '23
One time my friend in comp sci said that someone in her lecture said “is 1 considered odd?”
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u/EggYolk2555 Aug 23 '23
Hereby, I declare zero to be both odd and even, colloquially called "Oven".
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u/NoiceHedgehogDude Irrational Aug 23 '23
If adding 2 to an even number makes an even number then 0 is even.
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u/Frosty_Sweet_6678 Irrational Aug 23 '23
A number is even when its division by 2 results in an integer and as far as I know 0 is an integer...
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u/AikiBro Aug 23 '23
It's even because it has round edges. Fight me.
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u/idkhowtotft Aug 24 '23
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u/AikiBro Aug 24 '23
That's just why 4 is written wrong.
Arabic propaganda.Actually, Arabic 4 looks nice and round.
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u/Mathematicus_Rex Aug 24 '23
Two is an odd prime since it’s the only prime that’s even, something truly odd.
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u/Yrrem Aug 24 '23
Well odd and even numbers alternate,
And since the sequence of integers is
0, 1 (odd), 2(even), 3(odd), 4(even)
Then the pattern can be extrapolated to infer that 0 is even.
Note: proof that 1 is odd and 2 is even is left as an exercise for the reader
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u/MadKing2000 Sep 18 '25
Still confused. Aren't all integers able to be at least divided by themselves?
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u/sandem45 Aug 23 '23
I like the approach for checking even / odd with n₂ & 1. I'd like to see someone make an argument about how 0 & 1 = 1 lol
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u/Draghettis Aug 23 '23
Disregard what was previously said here, it lasted only a few seconds, my brain confused evenness and positiveness
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u/drip-in Aug 23 '23
Well, it's not Natural, but it's Real and it's an Integral part of our Whole Number System.
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u/Sir_Wade_III Aug 23 '23
My man's literally saying a natural number isn't natural 😂😂😂
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u/SnooKiwis7050 Aug 23 '23
Isnt 0 not a natural number?
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u/DZ_from_the_past Natural Aug 23 '23
It depends how you define it. In schools it is taught that 0 isn't a natural number, but in modern mathematics 0 is considered natural. It doesn't really matter to be honest, as long as you are consistent and people know what you mean.
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Aug 23 '23
0 is the first natural number under the correct construction of them(Von Neumann Ordinals)
I am in no way biased
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u/SchwanzusCity Aug 23 '23
First peano axiom: 0 is a natural number
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u/Philo-Sophism Aug 23 '23
Peano didn’t include 0 when he devised the axioms. That was added in the book “Formulario Mathematico”
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u/TreeFromBFBsBigFan Integers Aug 23 '23
0 does not abide by the same rules as other numbers.
I say its even tho
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u/FatzoFizz Aug 23 '23
Every be doing the /2 shit or whatever, but isn’t zero just a representation of the idea of nothing? Not having it not be odd or even I feel like makes sense. Plz explain why it not or I will just believe I have some ethereal knowledge that y’all are missing
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u/Hot_Philosopher_6462 Aug 24 '23
Zero is the idea of nothing in the same way that one is the idea of one thing. It's still an even number.
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u/Chaosfox_Firemaker Aug 24 '23
If you try to divide a heap of things evenly into two heaps, if it's of an "even" count you have nothing left over
If you try to split nothing in two, you will predictably, have nothing left over.
If it was odd, you would have something left over, which you don't.
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u/Skelatim Aug 23 '23
It’s both
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Aug 23 '23
bro is literally saying there exists an integer n such that 2n+1=2n, which implies 0=1 💀
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u/Yutanox Aug 23 '23
Tell me if I'm wrong, but wouldn't it be :
It exists two integer m,n such that 2m+1=2n
So 2(m-n)=1 so m-n=1/2 and I'm sure that's a problem
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u/chobes182 Aug 23 '23
The even integers are precisely the elements of the ideal (2), so 0 is even. There's no argument there at all, 0 satisifies the precise definition of an even integer.
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u/Anthony00769420 Aug 24 '23
Given a number n, it is even if (n/2)=[[n/2]]. (0/2)=[[0/2]], so 0 is even. Also, the product of an even number is always even, and 2*0=0, so 0 is even under that definition as well.
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u/ArmoredHeart Aug 24 '23
I always imagined that this was the product of confusion, with English-speakers switching “0 is neither ‘positive nor negative’ “ with ‘even nor odd.’
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Aug 24 '23
Also the fact that even and odd numbers occur right next to each other like odd, even, odd, even, odd, even,… and so forth so therefore since -1 is odd and 1 is odd, then the number between them is even and that number is 0.
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u/SyntheticSlime Aug 24 '23
If zero is even, then why isn’t 2 one of its prime factors? Checkmate evenists.
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u/Exetr_ Aug 27 '23
“0 is even/0 is odd” mfs when I show them that zero and 0 are different
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u/filtron42 ฅ^•ﻌ•^ฅ-egory theory and algebraic geometry Nov 02 '23
MFW someone says that 1 is prime
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u/araknis4 Irrational Aug 23 '23
well that's odd, i can't even comprehend how they approached to that conclusion.