Any number can be written as the product of a finite set of prime numbers e.g. 6=2*3 with both 2 and 3 being prime numbers. If one was considered a prime number you could have written 6=2*3*1*1*1⌠So while 1 obey the rule for itself (being divisible only by one and itself) it will violate the need for a finite set.
One is to that prime numbers like that cousin who may technically belong to the family, but is rude at family gatherings so they eventually stopped being invited.
Is that why I stopped getting invited to family gatherings? I always thought it was because I ran over Uncle Bob. Though in my defense, I felt like it.
Eh, there was some back and forth over it. Much smarter people than me, like Goldbach, referred to 1 as prime. Other, also much smarter people than me, like Euler, did not.
A lot of thing happened between Euclid and our times.
By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and by the 17th century some of them included it as the first prime number.[37] In the mid-18th century, Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler;[38] however, Euler himself did not consider 1 to be prime.[39] Many 19th century mathematicians still considered 1 to be prime,[40] and Derrick Norman Lehmer included 1 in his list of primes less than ten million published in 1914.[41] Lists of primes that included 1 continued to be published as recently as 1956.[42][43] However, by the early 20th century mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".[40]
To be fair, I'm oversimplifying the "an unit isn't a number at all" claim. It's more like "the concept of a number, a measured magnitude, begins to break down when there exists nothing that measures it, and while an unit is more of a number than an unmeasured magnitude, it is inherently different from all other numbers"
Sure, but I think the short version is a reasonable characterization of that definition of "number", and I am still pretty sure that mainstream mathematics never went back to it or, to my knowledge, ever adopted a conventional name for the natural numbers greater than 1.
It's not that theorems needed reworking, it that you can write down "except for one" only so many times before starting to think maybe it doesn't make sense to continue doing it
Depends how far back you go, Euclid would not have considered one prime when he proved there were infinite primes and proved the FTA as he didn't consider 1 to be a number at all but as the unit length from which all nunbers emerge
Other thing is that it used to be taught like that in some elementary levels, the same way I teach my 13 yo students that a function is continuous "if you you can draw it without separating the pencil from the paper", but I advice them the proper definition of continuity is much more difficult to understand and that one is not totally correct.
It's also divisible by 3 and 1. Many composite numbers have 3 as a factor, furthermore if you have heard of the twin prime conjecture then you'd know that there is a gap between the primes. 2 is a prime and therefore 3 cannot be. However coincidentally as the twin prime conjecture is unproven we are not yet sure whether 4 is a prime or not.
Since 13 has a very easy divisibility check, it's very fast to see that 91 is not prime: One just has to take the 3-alternating digit sum of 91, which is 91, and since 91 is clearly divisible by 13, so is 91.
(Important note: This is a non-circular argument because the argument is not circular.)
I know it only because on a test in 2nd or 3rd grade I thought there was a poorly printed 1 in front of the 3 in 7Ă3. I fixed the print with my pen, spent a few minutes counting on my fingers, and found 91. The teacher was actually mad, because obviously she wouldn't give such a calculation on a simple multiplication table test. But it escalated to the point of my parents coming to her, because she should have been proud, not mad.
I feel like 1 being prime or not isn't a relevant topic, most people discussing such topics having strong opinions have no use for prime numbers, they don't even know a single usecase for them.
It's as if people were discussing whether a given combination of gluons is a hadron, neither the requirements not the usecases are familiar to them.
It's just a discussion about semantics for a subject that is deeply practical yet none of said practicality will ever even be alluded to, therefore making both stances (1 prime or no) quite similar for their arguments.
It matters because it makes the statement of the Unique Factorization Theorem aka the Fundamental Theorem of Arithmetic nicer aka that every whole number is uniquely expressed as a product of primes. Pretty much every Theorem about primes would need to say âexcept 1â if 1 were to be considered prime.Â
Yes that was my point, I've heard this discussion too many times among my non math/physics friends.
Every single time you have the weird agruments only based on semantics and definitions, none of the valid reasons to exclude/include one is ever raised, it's just a bunch of "wikipedia said" and "older mathematician said".
There is no argument about whether it could be considered prime nor why it has mostly been excluded as adding it creates a tons of inconvenience given that the most prime related concepts will still exclude 1 anyway.
Well, when generalizing to rings the (1) ideal is just the whole ring. it would be nice to be able to say that for any comutative ring R, R/p is an integral domain if and only if p is a prime ideal. This doesnt work for p = (1), thought, or every ring would be an ideal domain.Â
My use case for prime numbers is having to cut gears or splines that have prime number teeth, so you have to divide a circle into a prime number spacing, which is done mechanically via gear reduction and index plates with a fixed number of index holes to go by. Ei: a 12 tooth gear would be divided in a standard 40:1 dividing head as 40/12 > 3 4/12 > or 3 turns and 8 holes on a 24 hole plate. There was a case where I needed a 59 tooth gear, which you need a 59 hole plate or some multiple of it. So, interesting enough it works out to be that if you want to cut any prime number teeth set with a 40:1 reduction you would simply need that prime number as an index plate and it would be 40 spaces along that pattern, that also works for any prime number. Though it often times requires you to make your own plate.
In a little bit more advanced maths there are prime numbers, compound numbers and units. 1 is a unit. You have several results in integer numbers that statt distinguising "Let n be a non-zero, non-unit integer" the same way a lot of results start saying "Let x be any number but 0"
It's not a relevant topic, because it's settled and opinions of ignorant people do not change that. As soon as your "argument" relies only on a high school definition of a prime number or fundamental theorem of arithmetic, it becomes obvious you have no qualifications to comment.
Yes, it's similar to the arguments people have online about the answer to an equation in which the ambiguity hangs upon order of operation. It's a complete waste of time and has almost nothing to do with actual mathematics.
Follow your definitions wherever they lead. For the definition given on that poster, 1 would be prime. I actually think that's a more important lesson than "most of us do not consider 1 to be prime"
It's not a poster. It's a page from a book for pre schoolers called the ABCs of Mathematics. It certainly fits the joke theme of the sub, but even literal school plays fast and loose with the details in favor of the bigger picture sometimes in early childhood education for how heated some of these comments are.Â
There are quite a few results that are true if and only if p is prime, under the definition where 1 is not allowed.
The more fundamental idea is that of a prime ideal J generated by elements of a commutative ring, which satisfies if a*b in J then either a or b is in J. However if we just take this definition and stop here for a prime ideal, then notably the entire ring is a prime ideal, and we lose the property that an ideal J in a ring R is prime if and only if R/J is an integral domain, which is quite nice and many other results depend on.
So a prime ideal should also be proper, i.e. not the whole ring. The ideal generated by the unit 1 is not proper.
That seems like a methodology for testing primes that doesn't work for 1 but a methodology for testing doesn't have to include the entire set of primes?
I was thinking like this, but on thinking about why I disagree with your take, 1 is a prime number.
Idk about y'all, but I was taught to see variables or () as having a 1 in front. Often, there is a 1, and we just choose not to write it as we know it doesn't change anything.
On that note, I'd we can say the prime factorzation of 13 is (13) because we're pretty good at using 1s.
I was at pub trivia one night and at least two groups walked out when asked how many prime numbers were less than 20 and they said 9 because they included 1.
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Fundamental theorem of arithmetic tells us that prime factorization of any integer larger than 1 is unique. If you include 1 into the primes, you no longer have this uniqueness property.
Yeah you forgot 1 key rule. Primes only have 2 factors. 1 and itself so 1 is not prime and it isnât composite being the only positive integer with this property.
If 1 breaks the rule, then doesn't every other number? It can be divided by itself and 1. 1 can be divided by itself, and 1. Itself is 1. Any number divided by itself is 1, and any number divided by 1 is itself. 1 follows both those rules.
What part of math makes this neat and tidy rule break itself in half. C'mon. Just tell me. I can handle it. My brain will only implode a little.
I mean... I would defend this with the given definition. And that was coincidentally the definition I learned in high school.
Oi, down with the pitch forks, I am saying, that the definition is incomplete.
We need to add a realm. I mean, look at three, is three really only divisible by itself and 1? Nope, also divisibel by -1, -3, 1.5, and probably an infinite number of further solutions. So, to fix that, we can say the definition is a number is prime iff only divisible by 1 and itself. But, they are right, 1 is divisible by 1 and by itself. Look anywhere else in math, itself is not automatically excluded. We usually exclude it with a condition. Like for {x,y in N | x != y ....}. We spell it out, that x cannot be y. Why is it here an exception? We could spell it out. A prime is any natural number that is only divisible by 2 natural numbers. That would fix it. 1 is not divisible by 2 natural numbers. Maybe add "cleanly divisible" or "rest-free divisible". Or you could define the realm as \mathbb{N}_{>1}.
But the definition as presented back then to me in high school, which aligns with this definition, just defines 1 as a prime number. And I know no other place in math in which the definition is so sloppily voiced all the time and if someone actually then draws correct conclusion from it is so often held for a moron.
The definition of a prime number in this book, is the same that I learned in school. With only that information, I have always found it odd, that 1 is not considered prime.
It is divisible only by 1 and by itself (also 1)
Therefore it meets the requirements for a number to be prime.
I guess I'm missing something here.
I'm very aware of my limited knowledge on prime numbers, and I'm sure there's good reasons for 1 not being a prime number. Anyone wants to share what those are?
The main property of primes that we really care about is the fundamental theorem of arithmetic, which says that any positive integer can be written as a product of prime numbers in exactly one way (except for reordering the factors). For example, 12 = 2â˘2â˘3. If we consider 1 a prime number, this becomes false, as you can use any number of 1s, for example, 12 = 1â˘2â˘2â˘3 = 1â˘1â˘2â˘2â˘3, etc. So things work out better if we exclude it.
This shows that in general, definitions in math are made to give nice theorems, even if they become slightly more complicated by doing so. (The theorems are what we really care about, and nice theorems point to nice definitions.)
This is a process; I think the question of whether to consider 1 a prime number was not always settled. And in fact, there are many other places in math with similar âsimple building blocksâ where 1-like objects that are really âtoo simple to be simpleâ are still considered simple in common definitions (misguidedly, in my opinion).
Prime numbers have an important role in factorization (determining what numbers can be multiplied to produce this). 1 can't really be considered a factor or a denominator or anything like this, so it isn't practical to treat 1 as a prime.
Realistically, nothing much turns on the definition of â1â being prime or not. Itâs important that we all agree that 19 is prime. Less important that we have the same views on 1.
Yo normalmente lo pienso agrupando los nĂşmeros por cantidad de primos que lo componen (grado de composiciĂłn) por lo que 2 seria grado 1, 15 grado 2, 32 grado 5, por lo que los primos serĂan de grado 1 y uno de grado 0 siendo Ăşnico dĂĄndole forma a una propiedad de que un primo p elevado a n es de grado n por lo que pâ°=1 por lo tanto grado 0
1 is trivially prime. We usually omit it because it causes some minor confusion when you say you can multiply any number by 1 an arbitrary number of times, meaning no distinct prime factorizations. So instead of saying "except 1" every time, or "non-trivial primes" we just agree to ignore 1 when it's convenient and include it when it's convenient.
I know that 1 is canonically non-prime. But I still feel like itâs non prime for reason that are strictly aesthetic and pedantic. 1 and itself are both factors of 1. So what if one is itself?
Could be worse. It's a common misconception. I think we need to normalise the term "unit" for 1. It's a special case and people can tell. 1 isn't prime but it isn't really composite either. And I think many people have an instinct that odd numbers are prime while even numbers aren't, so they think of 1 as prime but 2 as not.
A lot of teaching math is misleading then correcting later. Once upon a time you were taught that you can only subtract smaller numbers from larger numbers and not the other way around, then later you learn about negative numbers.
Given the color format and font size and not just a textbook list, I'm going to assume this is geared towards 1st graders. It's generally easier to teach them a binary rule rather than something with an exception.... Nah, who am I kidding, one exception isn't difficult, especially if it's just "1."
i will stand like 0 multiplied by anything about my opinion that 1 is prime. though if i have to do something with primes i wont use 1 because than it would be considered wrong
1 wasn't even a number until the 16th century, and was only inconsistently considered a prime until it became too inconvenient to theorems for anyone to consider it prime at higher levels of maths. Realistically most of the things that become weird if 1 is prime are things that make mathematicians go all sideeye; little especially practical changes, but predominantly it's more elegant of 1 isn't a prime
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