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I understood prime as the minimal set of numbers with which you can generate all natural numbers by multiplying them. There are infinite natural numbers and in
by antaviana 7y ago
I understood prime as the minimal set of numbers with which you can generate all natural numbers by multiplying them.
There are infinite natural numbers and infinite prime numbers, but you need each and every prime number to generate any other natural number.
So I'm puzzled to learn that 1 is not a prime number because I do not know how to generate 1 if we exclude 1 from the prime set.
- ummwhat 7y agoFor that matter, even with 1, how do you generate 0?
- roywiggins 7y agoZero being a member of the natural numbers or not depends on convention.
- traderjane 7y agoI would argue that Zero is generally considered to be a natural. The naturals are often constructed via an initial object, and you can call that initial object whatever you want (some people write Z or Zed), but it's metaphorically a zero.
- rocqua 7y agoThe pushback I have gotten om my tongue-in-cheek rants defending 0 as a natural number would suggest not everyone agrees. Some people really care about 1/n being possible for every natural number N.
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- yakubin 7y agoAnd other people care about natural numbers being the set of cardinal numbers for finite sets. Others want the set of natural numbers with addition to be a monoid. And actually most-known series in analysis are usually indexed starting from zero, because that's when the formulas are usually simpler.
- traderjane 7y agoBut if you asked the same people to build the naturals, they'd likely include an initial object that behaves like 0.
- rocqua 7y agoI fully agree. However, there is also pushback on ordinals vs cardinals, and monoids vs semigroups.
- OscarCunningham 7y agoYou could just use 1 as the initial element though. Peano's axioms would become 1 != Sn Sn = Sm implies n=m n + 1 = Sn n + Sm = S(n + m) n × 1 = n n × Sm = (n × m) + n Induction That's not any more complex than the usual set of axioms.
- OscarCunningham 7y agoThere are actually many circumstances in which we do want to treat 0 as a prime. For example it generates a prime ideal in the integers. One way to state unique prime factorisation including 0 is to treat the set of primes as a pointed set with 0 as the point. The smash product makes the category of pointed sets into a symmetric monoidal category. Then unique prime factorisation can be stated by saying that the multiplicative monoid of natural numbers (with 0) is isomorphic to the free commutative monoid object on the set of primes.
- yakubin 7y agoPrime numbers are defined as elements of rings, so we need two operations, the other being addition (understood as a group operation). So you generate 0 by 1-1.
- OscarCunningham 7y agoThe number 1 is what you get if you multiply no numbers together. Just like if you add no numbers together you get 0.
- davnicwil 7y agoCan't you divide any number by itself (multiply by its inverse) to get 1? Or does that not count?
- codethief 7y agoWith the exception of the number 1, the inverse of any natural number with respect to multiplication is not a natural number.
- davnicwil 7y agoBy inverse I was meaning 1/x, i.e. x * 1/x = 1
- ColinWright 7y agoBut we're working in the integers, so 1/2, for example, doesn't exist.
- davnicwil 7y agoI'm just saying to get to 1 through multiplication you can always multiply any number by its own inverse (which is another way of saying divide any number by itself) I.e. X * 1/X = X/X = 1, always. I guess, technically, it's a division needed to get the inverse, so it's perhaps it doesn't count :-) Edit: Ah, yeah, now with help from the reply below I see my mistake - you're saying the inverse itself cannot exist since we're only dealing with natural numbers, that makes sense. I guess that's pretty much the reason that only multiplication in allowed in the OP's original question, allowing division lets us get all kinds of fractional numbers which wouldn't be allowed.
- jfarmer 7y agoIt's important to separate the structure from the objects. There's 2-the-natural-number, 2-the-integer, 2-the-rational, and 2-the-real. In each context, it's the "same" 2, but the structure we put around it is different. The structure is what makes it interesting. 2-the-rational is not prime. There are no primes when every non-zero element has a multiplicative inverse. An element with a multiplicative inverse is called a unit and for the same reason we exclude 1 from being prime we exclude units from being primes in other structures. Every non-zero real number is a unit. Every non-zero rational is a unit. Only -1 and 1 are units in the integers. Only 1 is a unit in the naturals.
- tobiasSoftware 7y agoThe problem is that the usefulness of primes comes from the ability to break down a number into a unique set of primes, and including 1 as a prime defeats that usefulness. While your understanding of primes is an elegant idea, in sciences, usefulness trumps elegance. From a programmer's point of view, a similar thing happens with Lisp. Lisp is a very elegant language due to its minimal structure, but not a very useful one with respect to how often it is used.
- kemitche 7y agoIt almost seems to me that there are two very similar, overlapping mental models* of "primeness". The first is: "numbers that can be multiplied together to get all other numbers". This mental model by necessity excludes 1. The second is: "numbers that cannot be 'made' by multiplying other, smaller numbers together". This mental model is "looser," and ends up including 1, and maybe 0 and -1. The first mental model tends to be more mathematically useful, so becomes the official definition of prime. \* Note that I explicitly use "mental model" here instead of "definition," because I am discussing different ways that different humans try and _understand_ different sets of numbers.
- Sniffnoy 7y ago1 is the empty product. You generate it by multiplying nothing together. (Or to put it differently, the set of primes forms a generating set for the monoid of natural numbers under multiplication, not the semigroup. Semigroups are kind of dumb. :P )