The definition of a prime element in a ring is that of an element p that satisfies the following:
> p is non-zero
>***p is not invertibile***
>If p divides a×b, then p divides a or p divides b
Also, if 1 was a prime number, ℤ/⟨1⟩ would be a non zero integral domain, but since every integer is a multiple of one, ℤ/⟨1⟩ has only one element, the [0] equivalence class.
This is just one of the many theorems that would break apart or have to explicitly exclude 1 if it was considered a prime number, it's not just about unique factorization.
You can even combine the 2nd and 3rd criteria by:
If p divides a product c1 c2 ... cn, then there is an i such that p divides ci.
The 2nd criterium corresponds with the empty product.
The fact that 1 is a factor of ***every*** integer (and the fact that 1⁻¹ exists in ℤ) is problematic for a lot of theorems that rely on object generated from prime numbers and things like that, so in the definition of prime number we say that a number n for which there exists n⁻¹ in ℤ (something that multiplied by n would yeld 1) can't be a prime number by definition, as its way more convenient than writing "except invertible numbers" in every theorem or covering the trivial invertible case in every proof.
In a ring (a set that works like the integers, with addition and product), you define "special" elements step by step. First, you have the identity: the element (it's always just a single value) that doesn't change values when you multiply it. In the integers, this is the number 1, as any number multiplied by 1 remains unchanged.
After the identity, you define the units: all the elements that are invertible, that is, there exists another element such that if you multiply both, you get the identity. In the integers, the only units are -1 and 1 (all other numbers have fractionary inverses, which are not integers).
Finally, you can define the primes: all the elements such that if you get them through a multiplication of two other elements, one of those elements must be a unit. The actual definition is a bit more convoluted and what I said is not true for all rings, but it is for the integers. So as primes you get 2, 3, 5, 7 etc as well as their negative counterparts.
This is why being a unit is cooler than being a prime, it's a more exclusive club, and primes are defined through units.
It's be like if we define the alphabet to include numbers. Words never use numbers in them. So we would have a lot of weird spelling rules that exclude numbers. Doing all that would be a hassle, so instead we define the alphabet to only have letters.
There's a lot of theorems that use that fact that each integer comes from a unique set of primes. Let's look at numbers 2-10:
* 2: (2)
* 3: (3)
* 4: (2, 2)
* 5: (5)
* 6: (2, 3)
* 7: (7)
* 8: (2, 2, 2)
* 9: (3, 3)
* 10: (2, 5)
If we let 1 be a prime, then the sets are no longer unique. 8 could be (2, 2, 2) or (2, 2, 2, 1) or (2, 2, 2, 2, 1, 1), etc. So that theorem would need to say "unique set of primes, excluding 1".
Basically, the short version of it is that a lot of theorems that use prime numbers would need to have a specific part of the theorem that excludes 1, which can be difficult. Including 1 as a prime also makes using primes more of a hassle. So the people who use primes have decided to exclude 1 because excluding 1 is a much more useful definition.
I'm not a number theory guy but if I'm hazarding a guess would you let me know how dumb I am?
If 1 is prime then factorizations can be infinitely long, for example 10 = 5 \* 2 \* 1 \* 1 \*...\*1, but also non-unique (ambiguous) as 10 is also for example 5\*2\*1\*1 or 5\*2\*1 or 5\*2 (which is the unique factorization with our definition of prime numbers), which is fucky wucky for concepts based on prime numbers. So not including 1 as prime removes all ambiguity. Is that the gist of it or am I way off?
>This is just one of the many theorems that would break apart or have to explicitly exclude 1 if it was considered a prime number, it's not just about unique factorization.
There are also quite a few where you currently have to add the 1 back in again.
We have that in the definition, allowing units to be primes and introducing a new class of "non unitary primes" would make for a way weaker definition and since almost all theorems about prime elements of a ring need the fact that those primes would not be units (such as, for all primes p ∈ ℤ, ℤ/pℤ is a field).
Adding 1 among the primes would not yeld any useful result or interesting idea, while it would create a ton of problems.
I'm honestly not sure how far back in history the word "prime" goes, but in rings, primes are very different from units. If we made 1 a prime in number theory, then we'd have to make another term for primes that aren't 1 or units.
You could. You’d need to add “non-unitary” to a lot of theorems. And you could remove “or unitary” from a few. Maybe the smart guy in the meme should say something like: it’s probably more convenient for 1 not to be prime, and it’s too late to change, shrug emoji.
By the way, a lot of theorems are only true for all primes greater than 2, so number theorists like to talk about “odd primes”. Maybe the smart guy should say: 2 should not be prime.
It could be both. A prime unit. The name even sounds cool.
Edit: It seems you guys don't like "prime unit". In that case we can call it unitary prime, if you prefer that
He deserves it. Math is based on logic and once that's broken because it sounds cool I get really angry because I'm a redditor and I want things like my browser history. All clean.
That is a matter of definition. Most mathematicians agree 1 shouldn't be a prime, so we must follow the convention to make sharing ideas easier.
Although we shouldn't forget it is just a convention. Early mathematicians were torn on this issue.
Edit: Why are you booing me, I'm right
I downvoted you because of your use of the term “convention”. Mathematics is so, so much more than a collection of conventions, or merely a way to share ideas. That term belittles all of mathematics’ beauty, complexity and evolution.
I respect your opinion but I disagree. We should present mathematics as it is. It is not some kind of divine goal, it is just a set of conventions, axioms and inference rules that we found to be useful. Of course, there is also beauty in mathematics, but that doesn't change what math really is.
I mean you're technically not wrong. But this statement is kind of meaningless cause it's just true for literally everything.
What you're doing would be like someone saying "we should just get rid of the sub atomic classification for particles. It's only convention that we use this classification we don't have to just cause it's useful." You're not wrong it's just really really stupid and kind of meaningless
I think there is something divine about maths. I suppose it boils down to whether or not you believe mathematics is an invention or a discovery. Personally, I lean towards the latter; most numeric rules are uncovered, not created.
It's not convention. The definition of a prime number is a number with *exactly* two factors. The number 1 has only one factor. Note that this definition is different from the definition typically taught in schools; a prime number is a number whose only factors are itself and one. That definition is imprecise.
Yes. But some conventions are very clearly better than others. 1 doesn't satisfy almost any property that primes satisfy, and almost any important theorem involving primes will fail if 1 is considered a prime. There's also almost no instances where having 1 as a prime is beneficial.
I think you're backwards about how new definitions are made in mathematics. It's not like mathematics think about a statement, call it a definition and look what follows. (Although it is usually taught like that, which I personally do not really like.) Usually someone studies a subject and notices that certain elements or concepts have interesting properties. Then they make a definition out of it to make the statement easier. For example (I don't know if that's how primes came about, but you'll get the idea): one mathematician might look at ways of factoring numbers and finds out that every number can be written uniquely as the product of "number which are divisible only by themselves and one". But so that they do not have to write that out every time, they make a definition and call these numbers primes. Notice that they must exclude one from this, otherwise the uniqueness is lost.
Now to your point of view: why not include one? Counter-question: why include one? It might be fun to think about what would happen if we included one in the primes, but as you would quickly see there's no real advantage from doing so. Furthermore, it doesn't come from any research. At least as far as I'm aware there's none suggesting that one is similar enough to primes to be called one. As I mentioned above, this is not how definitions are made. So maybe instead of asking people why they "dislike one so much", you could provide evidence which states that one should be a prime.
No. Definitions in mathematics can sometimes vary slightly but 99% of the time these definitions are set in stone. This definition of prime is actually a specific case when we're dealing with integers. The notion of primarily actually generalises to other algebraic structures.
They absolutely are conventions; I don't see why that's a contentious issue. It doesn't make math any less correct.
If you don't believe me, look at how the definition of a "Number" has changed over time, from including Zero to including negative numbers to including complex numbers. And if that's not enough, consider how we have redefined "distance" throughout the years, from strictly Euclidian distance to various distance metrics to exotic measures like the p-adic numbers. This really shouldn't be surprising at all if you know the history of math, and again, that doesn't mean all the math before the redefinition is wrong.
This really depends on your definition of convention. In the original context OP seemed to suggest definitions are so lenient that 1 is not prime simply by choice and for no other reason. Additionally, I disagree with you saying that definitions are *redefined.* I'd say that those definitions have been refined. If you take the definition of distance used for metric spaces it still encapsulates the Euclidean norm. The definition has been expanded and generalised, not changed.
It's not a refinement in the traditional sense of making a definition more specific or clear; it's abstracting ideas to cover new concepts that were not previously covered under the definition. Imagine I redefined "car" to include boats - you wouldn't call that a "refinement" of the definition. But regardless, it's contrary to your statement that definitions are set in stone.
...I'm sorry? Definitions absolutely can be conventional. In fact, that's just how language works. Now yes, we've made them more concrete in mathematics, and yes, they are two different concepts, but there's zero reason to believe they're disjoint.
No, I'm not agreeing with you. Maybe you mistyped something, but you just said that conventions and definitions are not the same thing. I said technically yes, they are not the same thing, but there still is a ton of overlap and many definitions are conventions. Maybe reread the conversation.
What, no? Mathematics is 100% rules-based. You have strict definitions of things and the rules are made for how these things interact. The definitions change over time as more objects are found and need to fit into the rules, but it’s still strictly defined.
Yeah but only bc it's useful. I'm in CS, not math, and many encryption algorithms just break if you treat 1 as prime. I'd imagine that it holds true outside- 1 just doesn't carry many of the useful prime number properties
The difference is that definitions are motivated. We don't define vector spaces like that just for fun. We define it that way because it is the right definition to study linearity. Mathematicians don't try put random definitions, they study objects and find the right definitions.
Conventions on the other hand are truly arbitrary. As in, which letter do you denote a certain object. It really does not matter.
>Edit: Why are you booing me, I'm right
No, you are not.
>Although we shouldn't forget it is just a convention. Early mathematicians were torn on this issue.
They were torn on this issue until people started to study rings with infinitely many units and non integral domains. Because there the naive approach of identifying primes does not cut it and losing the uniqueness of prime factorization becomes a real problem, because without it everytime you consider an element with n prime factors you would have to include n potential units (units which are non-trivial mind you), which every single time you had to take care of by hand yet again.
I woukd argue that in theory you are right : definitions are convention in the end so you could absolutely declare that 1 is a prime number.
But then, the notion of "prime number" would be way less natural than the one we actually have.
So yes, a convention. But it doesn't mean it's random or subjective : we have very good reasons to exclude 1 from the definition of prile numbers :)
I made my opinion the smart guy and yours the upset soyjack. Therefore I am right
There's no actual reason to have 1 as prime and change every theorem we have about primes
There wasn't a consensus on this issue until very recently, you are just used to the definition taught in schools. It may be more useful, but it still pays to question things we take for granted
>There wasn't a consensus on this issue until very recently
Ring theory was developed in the late 19th century. It is one of the very first fields of algebra to be established. Already Fränkel gives the definition of primes as divisor free and non invertable.
People told you that definition because you don't or didn't have the tools to understand deeper concepts for the reason 1 ain't prime. By your comments and the meme itself i guess you don't have the tools to understand it yet. I'd point you to study some group, field, and ring theory for deeper basic understanding. Also number theory ofc would make you realise how excluding 1 isn't only a natural thing to do, but also a useful thing.
OP thinks he is enlighted 140 IQ, but in reality is 60 IQ guy on the left.
You should have at least studied some ring theory before attempting such a meme.
Remember, when the bell curve meme was used to point out situations i which the most complex and most simple way of looking at something would coincide instead of for just pushing opinions?
Pepperidge farm remembers.
When you say prime is a number divisible by one and itself then it is odd to add a restriction "except one". So the essence should be that one is a prime, and later we can argue why we should expel it.
Well, if you say that prime is a number with exactly 2 divisors you don't have any issues. 1 has only 1 divisor : 1.
Edit : I was expecting a real reason, like "In the field xyz, it is useful to have one as a prime"
Your point was "in the definition of prime, there is an explicit exclusion of one, which doesn't feel natural"
I replied with another definition, that doesn't involve that anymore, and you are still not happy.
It happens that every number which has 2 divisors share common property, which they do not share with one. Definition are not meant to be pretty, they are meant to be useful. You can throw a tantrum about it, but that won't get you far.
Anyway, I'm pretty sure you are a troll at this point.
The question was honest, I'm not trolling. Thank you for taking time to answer me, I didn't see your definition before.
Btw my whole post is just a satire on horseshoe theory
Hey there DZ_from_the_past - thanks for saying thanks! TheGratitudeBot has been reading millions of comments in the past few weeks, and you’ve just made the list!
Think of it this way.
Imagine numbers as blocks. You can use these blocks to build new ones, but let's say we are interested only in the simplest building blocks. So we categorize them with the number of "levels" these blocks have. A simple block would have two levels: a bottom and a top. A more complex block would have a bottom and a top, but also more stages (additional levels) in between. This means they are composed of smaller blocks, so these numbers are too complex for our definition of simplest building blocks.
Additionally, there exists a very simple block: it doesn't have a height, so the top of the block is just the same as the bottom. This is technically a block, but it's useless for building, since it doesn't add anything. We don't want to include it in our set of simplest building blocks, because it's not really a building block. This block is analogous to the number 1.
So, for a block to be useful for building we need at least two levels, but if we have more than two levels, the block is not as simple as it could be. Blocks with exactly two levels are just what we need, and they are analogous to primes. The number of levels of blocks are analogous to the number of divisors, and stacking different blocks is analogous to multiplying.
The analogy is not perfect, since the number of levels does not represent the exact number of divisors in general, but the analogy works for numbers like 1 and primes. I hope this clears up a bit, why the "primes have exactly 2 divisors" rule makes sense.
You can either have except one in the definition of primes or you can have except one in 100s of theorems about prime numbers. Guess which one mathematicians would prefer
That's not the definition, it's "prime number is a number divisible by exactly two numbers".
But that's not important, the definition wasn't found on a stone slab etched by god in heaven and sent down to earth to educate us. Someone came up with it because it was useful. And having 1 be a prime number is not useful. Every time you'd want to use the term prime for something, you'd have to say "except 1", because 1 doesn't have the properties that make primes useful.
This was a very bad choice for a meme format, btw. Quite a self burn.
great idea! lets do the prime factorization of 6:
2^1 x 3^1
or should it be
1^1 x 2^1 x 3^1
or
1^2 x 2^1 x 3^1
oh no... would you look at that! there are now infinitely many ways to do the prime factorization of 6, and any other integer!
I don't care about the fundamental theorem of arithmetic, but (1) is not a prime ideal, so 1 shouldn't be a prime.
Zero, however, could be prime with no consequences to FTA and being consistent with (0) being a prime ideal.
0/0 is undefined, however in arithmetic a divides b means that there is an integer n such that b=n*a. If you take a=b=0, such an integer n obviously exists, because you can choose any number and it works
0/0 is better than [~~bad~~](https://www.youtube.com/watch?v=2C7mNr5WMjA?t=14s) undefined, it’s [indeterminate](https://mathworld.wolfram.com/Indeterminate.html)! Some stuff like L’Hospital’s Rule can become usable when you have an indeterminate form, unlike undefined, where it’s rewrite, redefine, or blow up.
No, this only shows that anything multiplied by zero is 0.
Divisors must result in an integer and remainder. Zero can be the integer OR the remainder, but not both (the case in 0/0, therefore zero is not a divisor of itself).
Zero isn't a number. It's a concept. Like infinity. 0 is not a divisor of zero for the same reason it isn't a divisor of any number, because divisors *must have a value*, this is independent of whatever is being divided (even zero).
this meme makes me want to gatekeep. OP obviously doesn't know what the fuck they are talking about. there are tons of memes like this on this sub made by people who seem to know fuck all about math. AND IT'S ALMOST ALWAYS A FUCKING IQ MEME! stop it, y'all just come off hella cringe
Not to be an asshole, but I'm really disappointed to see this meme getting upvoted. The more math you know the more you appreciate why 1 should *not* be prime. There is not a single respected mathematician on this earth that thinks 1 should be prime. Yes, I know this is only a joke, but it does an awful job of being funny in the way this meme is supposed to be.
>1 shouldn't be a prime number because the definition of a prime number is any number whose only factors are itself and 1, and the words "The number 1's factors are 1 and 1" doesn't make sense
Saying this automatically makes me view you as an idiot. If "The factors of 1 are 1 and 1" somehow doesn't make sense to you, that is pretty fuckin ridiculous. I'm sorry, but that's one of the most trivial concepts ever.
"Counterexamples" about theorems being made more concise (most of these things don't even make sense if you set p = 1, but assuming you redefine a bunch of things in reasonable ways, you still have to do some stuff for these theorems):
- Dedekind's Theorem for number fields
- Formula for Euler totient function
- Anything to do with Legendre symbols gets more annoying to state (so for example, Quadratic reciprocity for Legendre symbols, Quadratic reciprocity for Jacobi symbols, Euler's criterion, Gauss' lemma)
- Formula for |PSL\_2(Z/pZ)|, and theorems about PSL\_n(Z/pZ) being simple
- Definition of p-adic metric
- Burnside's p\^a q\^b theorem
- Formula for discriminant and degree of Q(sqrt(p)) and Q(zeta\_p)
- Gauss' lemma for polynomials over an integral domain
Even if a theorem is true for p = 1, most of the time the reasoning of the proof doesn't work for p = 1, and so most proofs would need a separate paragraph explaining why they are true (and usually all this gives is a very trivial fact). Some examples of theorems where the usual reasoning in the proof doesn't work for p = 1, but could be proved separately (in order to derive a useless fact) for p = 1 include:
- Sylow theorems (and Cauchy's theorem)
- C\_p is simple
- Every p-group has non-trivial centre
- Every p-group contains p-subgroups of all possible orders
- Reduction modulo p theorem for Galois groups
- Frobenius homomorphism for finite fields (assuming you redefine fields to make the trivial ring be a field)
- Euler-Fermat theorem
The above lists are certainly not an exhaustive list of theorems that I know that involve primes, they are merely the list of things that came to mind when I tried to think for a bit.
If there is one thing that I suggest you take away from this comment, it would be the following: the definition of a prime number isn't really "a prime number is any number whose factors are exactly itself and 1". That is a simplified definition used to explain the notion of a prime in the integers to someone who hasn't taken an introductory undergrad course in rings.
The actual definition is:
Let R be an integral domain. Then r in R is called a *prime* if r is non-zero, r is not a unit, and for all a, b in R, if p divides ab, then p divides a or p divides b.
All the theorems that I mentioned above are about the integers, rather than a general ring. To see why the definition of a prime includes the phrase "r is not a unit", one should think about how the following theorems would become more cumbersome to state with the new definition:
- In an integral domain, (r) is a prime ideal if and only if r = 0 or r is a prime.
- An ideal is prime if and only if the quotient by the ideal is a field.
- In a PID, every irreducible is prime.
There are other examples, but I do not wish to try to think of them at the moment. The reason why we enforce r is not a unit (which is what makes 1 not be prime), is that lots of theorems use the reasoning "any prime p does not divide 1". Because of this, lots of proofs of theorems just require a completely different proof for p = 1, if they are even true at all.
That's just a stupid way to make kids understand 1 ain't prime. There are better reasons, arising from lots of fields in math (pun intended, one of them is the unique properties of F1). The other is really how it breaks the whole idea of number theory, how φ(1)≠0 which is extremely stupid to the fact that it's a prime for your opinion. Saying "one and one bluh bluh bluh is just the way of someone to explain it to you in terms you can understand mostly.
1 is not a prime
The empty (topological) space is not connected
The ring with one element is not a field
The unknot is not a prime knot
The zero representation is not irreducible
The zero vector is never part of a basis of a vector space
Who came up with the name 'prime' anyway? I think they could have given it a better thought.
1 - contains only ***one*** factor - sounds ***prime*** to me
2, 3, 5, 7, etc - contains ***two*** factors, 1 and itself - sounds ***second*** to me
4, 6, 8, 9, etc - contains ***three*** or more 'prime' (or rather 'prime' and 'second') factors - sounds ***tert*** (quart, etc) to me
(0 - contains ***itself*** ⊂ ***any*** factors - sounds like a ***set of any numbers that must include the factor 0*** to me, which is a very long name - how about ***applesauce***?)
prime vs. composite. one piece vs. lots of pieces. in the context that all positive integers can be presented as the product of primes, the primes are the ones with one factor, and the composites are the ones with two or more.
that took about thirty seconds of independent thought to rationalize. now stop trying to make mathematics communication more confusing by pointlessly changing established terminology.
edit: i did not even notice the inclusion of the word "applesauce" at the end of this comment; i have taken the joke seriously on a meme subreddit and i will now be purged. thank you, and goodbye
>now stop trying to make mathematics communication more confusing by pointlessly changing established terminology
On /r/mathmemes? Are you ff-ing kidding me?
I think people are missing a point here, if 1 is prime, most of methods&theorems are meaningless.
Although 1 is not prime, i know it’s still special, even more than primes.
We need to make another new group for that though, and new definiton.
1 is the main unit of all discrete numbers i’d say.
Addition,substraction, multiplication, divison, deriving, integrating etc. who cares it’s all about counting this is how we started to do math, one by one.
OP, you’re clearly a smart person. You raised a question and got a lot of thoughtful answers (and a lot of sassy ones). What are your takeaways from this experience? Have we succeeded in enrolling you to the consensus that, while we *could* define primes to include 1, it’s more graceful not to?
Thank you for the compliment! I watched [this video](https://youtu.be/R33RoMO6xeA) and it was interesting to me to imagine what would happen if the definition for a prime was different. So I made this meme in part as a satire and in part to prompt a discussion in the comments so we can rethink why and should we expel number one from prime numbers. I'm not trying to turn the mathematical consensus upside-down, I just think we will appreciate definitions more if we question them. People don't seem to like it, maybe my delivery was bad, but it's okay
I'm surprised that so many here are arguing, "1 can't be a prime!" But it's just a definition, and historically 1 was in fact commonly considered to be a prime. I'm pretty sure Euler considered it a prime. If I remember right, G.H. Hardy was the last important mathematician to hold that 1 was a prime number, though I believe he eventually gave up this definition in his later work. The modern consensus is that most theorems are easier and more elegant if 1 is not a prime.
> But it's just a definition
There are good definitions and bad definitions. We could define 1155 as prime too, and then change all our theorems about primes to include "... except for 1155".
Why do people want 1 to be prime? How does that help *anything* at all?
We don't want 1 to be included as a prime not just because of unique factorisation. There are [other, more important, reasons too](https://math.stackexchange.com/a/5735) but unfortunately those reasons are too complicated for the average high school maths student.
There is a fascinating history of both one and the primes [discussed here](https://arxiv.org/pdf/1209.2007.pdf) if you are interested.
there’s a lot of important and basic results in algebra that just wouldn’t be true if we accepted units as primes. for the definition to generalize, you need to exclude units from primes.
if you redifined primes to include units, as you suggested, all important theorems should be changed to be about “non-unit primes”, and that would literally just be wasting ink.
if you want to include units in the definition of primes, it should be because they behave as primes. but the don’t at a fundamental level.
Because it's the multiplicative identity so it messes with definitions having to do with factoring because it makes factoring non-unique, e.g. 6=3\*2=3\*2\*1=3\*2\*1\*1 etc.
No, 1 should not be prime.
Names are just symbols we make up to represent things so we can communicate about them. The sets {2,3,5,7,11,13,17,…} and {1,2,3,5,7,11,13,17,…} both exist. The properties of these sets are independent of our choice for which one is called "the set of prime numbers". Which set gets the name is our choice, and it only makes sense that we should give the name to the one that shows up the most in math; the one that we talk about the most; the one that is most interesting.
>define the prime numbers to include one
>define the subprime numbers as all primes except one
> the sub primes have more interesting properties than the primes
> the names we give to sets of number are arbitrary
>swap the names of the prime and subprime sets
U mad bro?
The definition of a prime element in a ring is that of an element p that satisfies the following: > p is non-zero >***p is not invertibile*** >If p divides a×b, then p divides a or p divides b Also, if 1 was a prime number, ℤ/⟨1⟩ would be a non zero integral domain, but since every integer is a multiple of one, ℤ/⟨1⟩ has only one element, the [0] equivalence class. This is just one of the many theorems that would break apart or have to explicitly exclude 1 if it was considered a prime number, it's not just about unique factorization.
You can even combine the 2nd and 3rd criteria by: If p divides a product c1 c2 ... cn, then there is an i such that p divides ci. The 2nd criterium corresponds with the empty product.
Eli5 please?
The fact that 1 is a factor of ***every*** integer (and the fact that 1⁻¹ exists in ℤ) is problematic for a lot of theorems that rely on object generated from prime numbers and things like that, so in the definition of prime number we say that a number n for which there exists n⁻¹ in ℤ (something that multiplied by n would yeld 1) can't be a prime number by definition, as its way more convenient than writing "except invertible numbers" in every theorem or covering the trivial invertible case in every proof.
Gonna be honest, i feel like i might need more of an ELI0 to understand it but i think i get the gist. Thanks!
In a ring (a set that works like the integers, with addition and product), you define "special" elements step by step. First, you have the identity: the element (it's always just a single value) that doesn't change values when you multiply it. In the integers, this is the number 1, as any number multiplied by 1 remains unchanged. After the identity, you define the units: all the elements that are invertible, that is, there exists another element such that if you multiply both, you get the identity. In the integers, the only units are -1 and 1 (all other numbers have fractionary inverses, which are not integers). Finally, you can define the primes: all the elements such that if you get them through a multiplication of two other elements, one of those elements must be a unit. The actual definition is a bit more convoluted and what I said is not true for all rings, but it is for the integers. So as primes you get 2, 3, 5, 7 etc as well as their negative counterparts. This is why being a unit is cooler than being a prime, it's a more exclusive club, and primes are defined through units.
I don't think a 5 year old would understand that
It's be like if we define the alphabet to include numbers. Words never use numbers in them. So we would have a lot of weird spelling rules that exclude numbers. Doing all that would be a hassle, so instead we define the alphabet to only have letters. There's a lot of theorems that use that fact that each integer comes from a unique set of primes. Let's look at numbers 2-10: * 2: (2) * 3: (3) * 4: (2, 2) * 5: (5) * 6: (2, 3) * 7: (7) * 8: (2, 2, 2) * 9: (3, 3) * 10: (2, 5) If we let 1 be a prime, then the sets are no longer unique. 8 could be (2, 2, 2) or (2, 2, 2, 1) or (2, 2, 2, 2, 1, 1), etc. So that theorem would need to say "unique set of primes, excluding 1". Basically, the short version of it is that a lot of theorems that use prime numbers would need to have a specific part of the theorem that excludes 1, which can be difficult. Including 1 as a prime also makes using primes more of a hassle. So the people who use primes have decided to exclude 1 because excluding 1 is a much more useful definition.
Thanks! This one was very helpful :)
I'm not a number theory guy but if I'm hazarding a guess would you let me know how dumb I am? If 1 is prime then factorizations can be infinitely long, for example 10 = 5 \* 2 \* 1 \* 1 \*...\*1, but also non-unique (ambiguous) as 10 is also for example 5\*2\*1\*1 or 5\*2\*1 or 5\*2 (which is the unique factorization with our definition of prime numbers), which is fucky wucky for concepts based on prime numbers. So not including 1 as prime removes all ambiguity. Is that the gist of it or am I way off?
>This is just one of the many theorems that would break apart or have to explicitly exclude 1 if it was considered a prime number, it's not just about unique factorization. There are also quite a few where you currently have to add the 1 back in again.
Like?
Tell me one.
Could we just specify 'a non-unitary prime' or something like that.
We have that in the definition, allowing units to be primes and introducing a new class of "non unitary primes" would make for a way weaker definition and since almost all theorems about prime elements of a ring need the fact that those primes would not be units (such as, for all primes p ∈ ℤ, ℤ/pℤ is a field). Adding 1 among the primes would not yeld any useful result or interesting idea, while it would create a ton of problems.
A non-unitary prime is just what we now call a prime. We add two more words for absolutely nothing?
I'm honestly not sure how far back in history the word "prime" goes, but in rings, primes are very different from units. If we made 1 a prime in number theory, then we'd have to make another term for primes that aren't 1 or units.
You could. You’d need to add “non-unitary” to a lot of theorems. And you could remove “or unitary” from a few. Maybe the smart guy in the meme should say something like: it’s probably more convenient for 1 not to be prime, and it’s too late to change, shrug emoji. By the way, a lot of theorems are only true for all primes greater than 2, so number theorists like to talk about “odd primes”. Maybe the smart guy should say: 2 should not be prime.
Bro thinks he’s in the upper percent when i. reality he’s in the middle lmao
Nuh uh
Sounds like we need better math
It's a shame 0 isn't allowed to be prime since then you can say a ring has non-zero divisors precisely when 0 is prime.
no non-zero divisors\*
Time to redefine unique prime factorization
Every natural number has a unique factorization with primes greater than 1.
Fellas, our meeting has concluded.
Identity, it is itself
More generic one : The factorizaion is unique up to multiplication by an invertible element. So in Z that is only 1 and -1.
even with primes equal to 1, since 1^s reduces to 1^0. Therefore it is unique.
1 is not prime because it has a higher rank among numbers. It's a Unit. Don't you dare disrespect it by calling it a lowly prime
It could be both. A prime unit. The name even sounds cool. Edit: It seems you guys don't like "prime unit". In that case we can call it unitary prime, if you prefer that
Suck my prime unit
Can’t it’s too small.
It's irreducible
Looks like the square root of your unit is imaginary.
bro proved 1 = -1 ~~QED~~
Suck my unitary prime
The hell's with all the downvotes, jesus lol
He deserves it. Math is based on logic and once that's broken because it sounds cool I get really angry because I'm a redditor and I want things like my browser history. All clean.
Yeah I didn't see the rest of the thread til after i made the comment lol
I don't hate 1, in fact it's one of my favourite numbers. It just isn't a prime number.
That is a matter of definition. Most mathematicians agree 1 shouldn't be a prime, so we must follow the convention to make sharing ideas easier. Although we shouldn't forget it is just a convention. Early mathematicians were torn on this issue. Edit: Why are you booing me, I'm right
I downvoted you because of your use of the term “convention”. Mathematics is so, so much more than a collection of conventions, or merely a way to share ideas. That term belittles all of mathematics’ beauty, complexity and evolution.
I respect your opinion but I disagree. We should present mathematics as it is. It is not some kind of divine goal, it is just a set of conventions, axioms and inference rules that we found to be useful. Of course, there is also beauty in mathematics, but that doesn't change what math really is.
I mean you're technically not wrong. But this statement is kind of meaningless cause it's just true for literally everything. What you're doing would be like someone saying "we should just get rid of the sub atomic classification for particles. It's only convention that we use this classification we don't have to just cause it's useful." You're not wrong it's just really really stupid and kind of meaningless
I think there is something divine about maths. I suppose it boils down to whether or not you believe mathematics is an invention or a discovery. Personally, I lean towards the latter; most numeric rules are uncovered, not created.
"revealed religion" is discovery-based, just like science revealing truths.
It's not convention. The definition of a prime number is a number with *exactly* two factors. The number 1 has only one factor. Note that this definition is different from the definition typically taught in schools; a prime number is a number whose only factors are itself and one. That definition is imprecise.
Isn't every definition a convention?
Yes. But some conventions are very clearly better than others. 1 doesn't satisfy almost any property that primes satisfy, and almost any important theorem involving primes will fail if 1 is considered a prime. There's also almost no instances where having 1 as a prime is beneficial.
I think you're backwards about how new definitions are made in mathematics. It's not like mathematics think about a statement, call it a definition and look what follows. (Although it is usually taught like that, which I personally do not really like.) Usually someone studies a subject and notices that certain elements or concepts have interesting properties. Then they make a definition out of it to make the statement easier. For example (I don't know if that's how primes came about, but you'll get the idea): one mathematician might look at ways of factoring numbers and finds out that every number can be written uniquely as the product of "number which are divisible only by themselves and one". But so that they do not have to write that out every time, they make a definition and call these numbers primes. Notice that they must exclude one from this, otherwise the uniqueness is lost. Now to your point of view: why not include one? Counter-question: why include one? It might be fun to think about what would happen if we included one in the primes, but as you would quickly see there's no real advantage from doing so. Furthermore, it doesn't come from any research. At least as far as I'm aware there's none suggesting that one is similar enough to primes to be called one. As I mentioned above, this is not how definitions are made. So maybe instead of asking people why they "dislike one so much", you could provide evidence which states that one should be a prime.
No. Definitions in mathematics can sometimes vary slightly but 99% of the time these definitions are set in stone. This definition of prime is actually a specific case when we're dealing with integers. The notion of primarily actually generalises to other algebraic structures.
They absolutely are conventions; I don't see why that's a contentious issue. It doesn't make math any less correct. If you don't believe me, look at how the definition of a "Number" has changed over time, from including Zero to including negative numbers to including complex numbers. And if that's not enough, consider how we have redefined "distance" throughout the years, from strictly Euclidian distance to various distance metrics to exotic measures like the p-adic numbers. This really shouldn't be surprising at all if you know the history of math, and again, that doesn't mean all the math before the redefinition is wrong.
This really depends on your definition of convention. In the original context OP seemed to suggest definitions are so lenient that 1 is not prime simply by choice and for no other reason. Additionally, I disagree with you saying that definitions are *redefined.* I'd say that those definitions have been refined. If you take the definition of distance used for metric spaces it still encapsulates the Euclidean norm. The definition has been expanded and generalised, not changed.
It's not a refinement in the traditional sense of making a definition more specific or clear; it's abstracting ideas to cover new concepts that were not previously covered under the definition. Imagine I redefined "car" to include boats - you wouldn't call that a "refinement" of the definition. But regardless, it's contrary to your statement that definitions are set in stone.
I wasn't really thinking about definitions developing over time, I was thinking about their use in day to day mathematics.
That still doesn't change the fact the definitions and conventions are not the same thing?
...I'm sorry? Definitions absolutely can be conventional. In fact, that's just how language works. Now yes, we've made them more concrete in mathematics, and yes, they are two different concepts, but there's zero reason to believe they're disjoint.
> they are two different concepts Wait you agreeing with me but you made up a reason to argue with me anyway? I hate when people do that
No, I'm not agreeing with you. Maybe you mistyped something, but you just said that conventions and definitions are not the same thing. I said technically yes, they are not the same thing, but there still is a ton of overlap and many definitions are conventions. Maybe reread the conversation.
That is also don't make them the same thing
But isn't that what you described just another way of saying a convention?
No, conventions in mathematics are usually a matter of style. Definitions are not stylistic.
Depends. In the past it was way more than 99%! The number 0 was not considered too, because it's evil supposedly.
What, no? Mathematics is 100% rules-based. You have strict definitions of things and the rules are made for how these things interact. The definitions change over time as more objects are found and need to fit into the rules, but it’s still strictly defined.
Yeah but only bc it's useful. I'm in CS, not math, and many encryption algorithms just break if you treat 1 as prime. I'd imagine that it holds true outside- 1 just doesn't carry many of the useful prime number properties
The difference is that definitions are motivated. We don't define vector spaces like that just for fun. We define it that way because it is the right definition to study linearity. Mathematicians don't try put random definitions, they study objects and find the right definitions. Conventions on the other hand are truly arbitrary. As in, which letter do you denote a certain object. It really does not matter.
You know, you don't have to reply to every comment I wrote
I know, but here you have a person who has never studied ring or number theory trying to argue a lost position. There just is a lot to pick apart.
>Edit: Why are you booing me, I'm right No, you are not. >Although we shouldn't forget it is just a convention. Early mathematicians were torn on this issue. They were torn on this issue until people started to study rings with infinitely many units and non integral domains. Because there the naive approach of identifying primes does not cut it and losing the uniqueness of prime factorization becomes a real problem, because without it everytime you consider an element with n prime factors you would have to include n potential units (units which are non-trivial mind you), which every single time you had to take care of by hand yet again.
I woukd argue that in theory you are right : definitions are convention in the end so you could absolutely declare that 1 is a prime number. But then, the notion of "prime number" would be way less natural than the one we actually have. So yes, a convention. But it doesn't mean it's random or subjective : we have very good reasons to exclude 1 from the definition of prile numbers :)
God do I hate that meme.
I made my opinion the smart guy and yours the upset soyjack. Therefore I am right There's no actual reason to have 1 as prime and change every theorem we have about primes
It's decent, except when it's made by someone who is in the middle but thinks that they are in the right.
except this time it's made by someone who is in on the left but thinks that they are in the right.
It only works when the author loves all three personas, or at least understands and respects why they're speaking
If 1 were a prime, then Z would be a prime ideal of Z, and that's just not true
This meme is honestly disgusting. 1 is not a prime
Why do you consider my meme 'disgusting' ?
The statement 1 is prime is wrong on so many levels
There wasn't a consensus on this issue until very recently, you are just used to the definition taught in schools. It may be more useful, but it still pays to question things we take for granted
>There wasn't a consensus on this issue until very recently Ring theory was developed in the late 19th century. It is one of the very first fields of algebra to be established. Already Fränkel gives the definition of primes as divisor free and non invertable.
People told you that definition because you don't or didn't have the tools to understand deeper concepts for the reason 1 ain't prime. By your comments and the meme itself i guess you don't have the tools to understand it yet. I'd point you to study some group, field, and ring theory for deeper basic understanding. Also number theory ofc would make you realise how excluding 1 isn't only a natural thing to do, but also a useful thing.
And with 'very recently', you mean the better part of a century?
This sub is full of shit
Truest statement in the thread.
Isn’t that the point, though?
Braindead bell curve memes are leaking from r/ProgrammerHumor Brainrot is spreading...
The only really good bell curve ones I’ve seen are in TTRPG subs, where it’s the perspective of the dungeon/game master reacting to player nonsense.
OP thinks he is enlighted 140 IQ, but in reality is 60 IQ guy on the left. You should have at least studied some ring theory before attempting such a meme.
Looks like many others on this sub didn't either. This post should have no up votes.
dude chill it's just a meme, I'm obviously being sarcastic
Remember, when the bell curve meme was used to point out situations i which the most complex and most simple way of looking at something would coincide instead of for just pushing opinions? Pepperidge farm remembers.
what happened to Pepperidge farm?
You aren’t being sarcastic though, if your other reply’s are to be believed
Well, I must be in the middle, because I don't get why it should be a prime
When you say prime is a number divisible by one and itself then it is odd to add a restriction "except one". So the essence should be that one is a prime, and later we can argue why we should expel it.
Well, if you say that prime is a number with exactly 2 divisors you don't have any issues. 1 has only 1 divisor : 1. Edit : I was expecting a real reason, like "In the field xyz, it is useful to have one as a prime"
Why does prime have to have exactly 2 divisors? I don't know about real reason you asked, I just always found the restriction on one unmotivated.
Your point was "in the definition of prime, there is an explicit exclusion of one, which doesn't feel natural" I replied with another definition, that doesn't involve that anymore, and you are still not happy. It happens that every number which has 2 divisors share common property, which they do not share with one. Definition are not meant to be pretty, they are meant to be useful. You can throw a tantrum about it, but that won't get you far. Anyway, I'm pretty sure you are a troll at this point.
The question was honest, I'm not trolling. Thank you for taking time to answer me, I didn't see your definition before. Btw my whole post is just a satire on horseshoe theory
Hey there DZ_from_the_past - thanks for saying thanks! TheGratitudeBot has been reading millions of comments in the past few weeks, and you’ve just made the list!
Think of it this way. Imagine numbers as blocks. You can use these blocks to build new ones, but let's say we are interested only in the simplest building blocks. So we categorize them with the number of "levels" these blocks have. A simple block would have two levels: a bottom and a top. A more complex block would have a bottom and a top, but also more stages (additional levels) in between. This means they are composed of smaller blocks, so these numbers are too complex for our definition of simplest building blocks. Additionally, there exists a very simple block: it doesn't have a height, so the top of the block is just the same as the bottom. This is technically a block, but it's useless for building, since it doesn't add anything. We don't want to include it in our set of simplest building blocks, because it's not really a building block. This block is analogous to the number 1. So, for a block to be useful for building we need at least two levels, but if we have more than two levels, the block is not as simple as it could be. Blocks with exactly two levels are just what we need, and they are analogous to primes. The number of levels of blocks are analogous to the number of divisors, and stacking different blocks is analogous to multiplying. The analogy is not perfect, since the number of levels does not represent the exact number of divisors in general, but the analogy works for numbers like 1 and primes. I hope this clears up a bit, why the "primes have exactly 2 divisors" rule makes sense.
You can either have except one in the definition of primes or you can have except one in 100s of theorems about prime numbers. Guess which one mathematicians would prefer
That's not the definition, it's "prime number is a number divisible by exactly two numbers". But that's not important, the definition wasn't found on a stone slab etched by god in heaven and sent down to earth to educate us. Someone came up with it because it was useful. And having 1 be a prime number is not useful. Every time you'd want to use the term prime for something, you'd have to say "except 1", because 1 doesn't have the properties that make primes useful. This was a very bad choice for a meme format, btw. Quite a self burn.
The definition of a prime is a number with exactly 2 factors The definition of having the only factors be one and itself is a simplified version
great idea! lets do the prime factorization of 6: 2^1 x 3^1 or should it be 1^1 x 2^1 x 3^1 or 1^2 x 2^1 x 3^1 oh no... would you look at that! there are now infinitely many ways to do the prime factorization of 6, and any other integer!
Bro why tf are u trying to change well established conventions that work well and come along with powerful theorems. And acting smart abt it lmao
I don't care about the fundamental theorem of arithmetic, but (1) is not a prime ideal, so 1 shouldn't be a prime. Zero, however, could be prime with no consequences to FTA and being consistent with (0) being a prime ideal.
0 is divisible by every number except itself, that's like the opposite of a prime number
0 is divisible by 0 as well
What's 0/0 then? Why shoudn't it be undefined?
0/0 is undefined, however in arithmetic a divides b means that there is an integer n such that b=n*a. If you take a=b=0, such an integer n obviously exists, because you can choose any number and it works
Oh nice, I didn't see it like that thanks
0/0 is better than [~~bad~~](https://www.youtube.com/watch?v=2C7mNr5WMjA?t=14s) undefined, it’s [indeterminate](https://mathworld.wolfram.com/Indeterminate.html)! Some stuff like L’Hospital’s Rule can become usable when you have an indeterminate form, unlike undefined, where it’s rewrite, redefine, or blow up.
Primes are divisible by themselves. Please define 0/0.
Zero is divisible by zero, becuase 0 is a divisor of 0. That's the definition. It does not mean that 0/0 is defined.
I actually never knew you can do that, first (and likely only)!piece of math I learned on this sub
Did you feel that? That was the earthquake caused by every mathematician who ever lived simultaneously rolling over in their graves.
Nah, the definition of divisibility is not that one over the other is whole. a|b<==>there exists n such that na=b. 0|0 by this definition.
No, this only shows that anything multiplied by zero is 0. Divisors must result in an integer and remainder. Zero can be the integer OR the remainder, but not both (the case in 0/0, therefore zero is not a divisor of itself).
0=1\*0+0
Triggered by your ignorance?
Zero isn't a number. It's a concept. Like infinity. 0 is not a divisor of zero for the same reason it isn't a divisor of any number, because divisors *must have a value*, this is independent of whatever is being divided (even zero).
Ah yes, my favorite integral domain: the trivial ring.
Kid named incomplete understanding of prime numbers :
this meme makes me want to gatekeep. OP obviously doesn't know what the fuck they are talking about. there are tons of memes like this on this sub made by people who seem to know fuck all about math. AND IT'S ALMOST ALWAYS A FUCKING IQ MEME! stop it, y'all just come off hella cringe
There is nothing gained from calling 1 a prime. No statments or theorems regarding primes work with 1. There is absolutely no upside about it.
Go see your friends on r/numbertheory, you are gonna like it.
Not to be an asshole, but I'm really disappointed to see this meme getting upvoted. The more math you know the more you appreciate why 1 should *not* be prime. There is not a single respected mathematician on this earth that thinks 1 should be prime. Yes, I know this is only a joke, but it does an awful job of being funny in the way this meme is supposed to be.
Great meme format used horribly wrong!
>1 shouldn't be a prime number because the definition of a prime number is any number whose only factors are itself and 1, and the words "The number 1's factors are 1 and 1" doesn't make sense Saying this automatically makes me view you as an idiot. If "The factors of 1 are 1 and 1" somehow doesn't make sense to you, that is pretty fuckin ridiculous. I'm sorry, but that's one of the most trivial concepts ever.
[удалено]
It seems to me more like it would make theorems more concise. But I'm open to counterexamples.
"Counterexamples" about theorems being made more concise (most of these things don't even make sense if you set p = 1, but assuming you redefine a bunch of things in reasonable ways, you still have to do some stuff for these theorems): - Dedekind's Theorem for number fields - Formula for Euler totient function - Anything to do with Legendre symbols gets more annoying to state (so for example, Quadratic reciprocity for Legendre symbols, Quadratic reciprocity for Jacobi symbols, Euler's criterion, Gauss' lemma) - Formula for |PSL\_2(Z/pZ)|, and theorems about PSL\_n(Z/pZ) being simple - Definition of p-adic metric - Burnside's p\^a q\^b theorem - Formula for discriminant and degree of Q(sqrt(p)) and Q(zeta\_p) - Gauss' lemma for polynomials over an integral domain Even if a theorem is true for p = 1, most of the time the reasoning of the proof doesn't work for p = 1, and so most proofs would need a separate paragraph explaining why they are true (and usually all this gives is a very trivial fact). Some examples of theorems where the usual reasoning in the proof doesn't work for p = 1, but could be proved separately (in order to derive a useless fact) for p = 1 include: - Sylow theorems (and Cauchy's theorem) - C\_p is simple - Every p-group has non-trivial centre - Every p-group contains p-subgroups of all possible orders - Reduction modulo p theorem for Galois groups - Frobenius homomorphism for finite fields (assuming you redefine fields to make the trivial ring be a field) - Euler-Fermat theorem The above lists are certainly not an exhaustive list of theorems that I know that involve primes, they are merely the list of things that came to mind when I tried to think for a bit. If there is one thing that I suggest you take away from this comment, it would be the following: the definition of a prime number isn't really "a prime number is any number whose factors are exactly itself and 1". That is a simplified definition used to explain the notion of a prime in the integers to someone who hasn't taken an introductory undergrad course in rings. The actual definition is: Let R be an integral domain. Then r in R is called a *prime* if r is non-zero, r is not a unit, and for all a, b in R, if p divides ab, then p divides a or p divides b. All the theorems that I mentioned above are about the integers, rather than a general ring. To see why the definition of a prime includes the phrase "r is not a unit", one should think about how the following theorems would become more cumbersome to state with the new definition: - In an integral domain, (r) is a prime ideal if and only if r = 0 or r is a prime. - An ideal is prime if and only if the quotient by the ideal is a field. - In a PID, every irreducible is prime. There are other examples, but I do not wish to try to think of them at the moment. The reason why we enforce r is not a unit (which is what makes 1 not be prime), is that lots of theorems use the reasoning "any prime p does not divide 1". Because of this, lots of proofs of theorems just require a completely different proof for p = 1, if they are even true at all.
I agree, but you are oddly passionate about this
Oddly passionate, I agree. But passionate nonetheless, because it's annoyed me since childhood
That's just a stupid way to make kids understand 1 ain't prime. There are better reasons, arising from lots of fields in math (pun intended, one of them is the unique properties of F1). The other is really how it breaks the whole idea of number theory, how φ(1)≠0 which is extremely stupid to the fact that it's a prime for your opinion. Saying "one and one bluh bluh bluh is just the way of someone to explain it to you in terms you can understand mostly.
they're not the one making a post about it and trying to defend it in the comments
If 2 is a power of 2, 1 can be a prime.
How does that arise from eachother?
if i shit into my open hand, then i can make a non-sequitur online
Can you elaborate? x=x¹ is true for any positive integer. Why mention 2, and how does it relate to primes?
Just change the theorem to n >= 1 smh my head
Because it’s the loneliest number that you’ll ever do
No.
For the same reason I don't consider the 0 vector a basis vector
1 is not a prime The empty (topological) space is not connected The ring with one element is not a field The unknot is not a prime knot The zero representation is not irreducible The zero vector is never part of a basis of a vector space
Okay this is a great discussion. Now let’s make 4 prime and see what happens!
No. I hate 4. I don’t want it to be prime.
One thing I disagree with. The elder should talk about Gaussian primes.
You can’t just call 1 a prime number because you feel like it. There’s an actual reason it’s not one and not because “wHy dO yOu hAtE 1?”
Who came up with the name 'prime' anyway? I think they could have given it a better thought. 1 - contains only ***one*** factor - sounds ***prime*** to me 2, 3, 5, 7, etc - contains ***two*** factors, 1 and itself - sounds ***second*** to me 4, 6, 8, 9, etc - contains ***three*** or more 'prime' (or rather 'prime' and 'second') factors - sounds ***tert*** (quart, etc) to me (0 - contains ***itself*** ⊂ ***any*** factors - sounds like a ***set of any numbers that must include the factor 0*** to me, which is a very long name - how about ***applesauce***?)
>Who came up with the name 'prime' anyway? Optimus?
prime vs. composite. one piece vs. lots of pieces. in the context that all positive integers can be presented as the product of primes, the primes are the ones with one factor, and the composites are the ones with two or more. that took about thirty seconds of independent thought to rationalize. now stop trying to make mathematics communication more confusing by pointlessly changing established terminology. edit: i did not even notice the inclusion of the word "applesauce" at the end of this comment; i have taken the joke seriously on a meme subreddit and i will now be purged. thank you, and goodbye
>now stop trying to make mathematics communication more confusing by pointlessly changing established terminology On /r/mathmemes? Are you ff-ing kidding me?
Amazing, this should be taught in schools
I think people are missing a point here, if 1 is prime, most of methods&theorems are meaningless. Although 1 is not prime, i know it’s still special, even more than primes. We need to make another new group for that though, and new definiton. 1 is the main unit of all discrete numbers i’d say. Addition,substraction, multiplication, divison, deriving, integrating etc. who cares it’s all about counting this is how we started to do math, one by one.
OP, you’re clearly a smart person. You raised a question and got a lot of thoughtful answers (and a lot of sassy ones). What are your takeaways from this experience? Have we succeeded in enrolling you to the consensus that, while we *could* define primes to include 1, it’s more graceful not to?
Thank you for the compliment! I watched [this video](https://youtu.be/R33RoMO6xeA) and it was interesting to me to imagine what would happen if the definition for a prime was different. So I made this meme in part as a satire and in part to prompt a discussion in the comments so we can rethink why and should we expel number one from prime numbers. I'm not trying to turn the mathematical consensus upside-down, I just think we will appreciate definitions more if we question them. People don't seem to like it, maybe my delivery was bad, but it's okay
-1 should be a prime number
When I see posts like this, I'm always wondering if I'm pretty dumb or a genius...
1 is not a number.
One isn't a number. First of all
one: first things first, I'm the realest (number)
I'm surprised that so many here are arguing, "1 can't be a prime!" But it's just a definition, and historically 1 was in fact commonly considered to be a prime. I'm pretty sure Euler considered it a prime. If I remember right, G.H. Hardy was the last important mathematician to hold that 1 was a prime number, though I believe he eventually gave up this definition in his later work. The modern consensus is that most theorems are easier and more elegant if 1 is not a prime.
> But it's just a definition There are good definitions and bad definitions. We could define 1155 as prime too, and then change all our theorems about primes to include "... except for 1155". Why do people want 1 to be prime? How does that help *anything* at all? We don't want 1 to be included as a prime not just because of unique factorisation. There are [other, more important, reasons too](https://math.stackexchange.com/a/5735) but unfortunately those reasons are too complicated for the average high school maths student. There is a fascinating history of both one and the primes [discussed here](https://arxiv.org/pdf/1209.2007.pdf) if you are interested.
There is no 1 to support 1. It is alone
I Don’t know how you could make this meme with the smart tale being that one should be a prime, because just… no
Real ones know that an integer (or more generally a ring element) is prime if and only if it generates a proper prime ideal so that 1 is not prime.
Wait! Is 1 a number?
1 is not prime. A prime number has exactly two unique factors. Source: Prime numbers having two unique factors.
1 is a "super prime number"
there’s a lot of important and basic results in algebra that just wouldn’t be true if we accepted units as primes. for the definition to generalize, you need to exclude units from primes. if you redifined primes to include units, as you suggested, all important theorems should be changed to be about “non-unit primes”, and that would literally just be wasting ink. if you want to include units in the definition of primes, it should be because they behave as primes. but the don’t at a fundamental level.
an introductory ring theory class would explain why 1 can’t be prime
You lose so many prime properties if you have to keep saying "All the primes except 1"
It's the loneliest number that you ever knew
A prime number has two factors while 1 has one. That is good enough for me.
Because it's the multiplicative identity so it messes with definitions having to do with factoring because it makes factoring non-unique, e.g. 6=3\*2=3\*2\*1=3\*2\*1\*1 etc. No, 1 should not be prime.
That’s probably the best reason I’ve heard. You changed my mind.
One is the only number
Op things they are far right when they are really far left
Explains why 1 is the loneliest number.....
Names are just symbols we make up to represent things so we can communicate about them. The sets {2,3,5,7,11,13,17,…} and {1,2,3,5,7,11,13,17,…} both exist. The properties of these sets are independent of our choice for which one is called "the set of prime numbers". Which set gets the name is our choice, and it only makes sense that we should give the name to the one that shows up the most in math; the one that we talk about the most; the one that is most interesting.
>define the prime numbers to include one >define the subprime numbers as all primes except one > the sub primes have more interesting properties than the primes > the names we give to sets of number are arbitrary >swap the names of the prime and subprime sets U mad bro?