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It only makes sense to define factorization on numbers that don't have an inverse, otherwise you can make infinite factorizations of a number by multiplying by

It only makes sense to define factorization on numbers that don't have an inverse, otherwise you can make infinite factorizations of a number by multiplying by a number that has an inverse and then by its inverse. Therefore it makes sense to classify numbers as either primes (they are their own unique factorization), or composites (they are the result of the multiplication of other numbers, and the factorization may or may not be unique), or units (numbers that have an inverse). 1 is the only unit of the natural numbers. If you add negative integers, this results into all negated primes being also primes, and there is no unique factorization (for example 6 is 2*3 but also -2*-3).

bmacho 3y ago
Yeah, I don't buy this argument. Or anything similar for analogous questions. They obviously have zero proving strength. Especially if it is only 1 example. To the question "why professionals use the tool they use instead of a slightly different version of that tool", the answer is either: - they like it, or - they don't like it, but they still use it, say for historical reasons Now you can list various examples to illustrate your point why would they prefer the tool as it is, instead of a hypothetical, slightly different version of said tool. But you can't just point at 1 example and say that this is the reason. That is just a wrong answer. Logically wrong. Lying. Annoying and hurting the readers head. Don't do that.
- bonzini 3y ago
  Ok, then the answer is "they do use a slightly different version of the tool than what you learnt at school. That version is more rigorous (which is why they like it) and in that version 1 is neither a prime nor a composite. In the version you learnt at school you're told that 1 is not a prime and you can't really say that it's composite either, but no clear explanation of why it is not a prime; in the version that mathematicians use there's a name for what 1 is, and a rule to find things like it in other number systems."

bmacho 3y ago

Yeah, I don't buy this argument. Or anything similar for analogous questions. They obviously have zero proving strength. Especially if it is only 1 example. To the question "why professionals use the tool they use instead of a slightly different version of that tool", the answer is either: - they like it, or - they don't like it, but they still use it, say for historical reasons Now you can list various examples to illustrate your point why would they prefer the tool as it is, instead of a hypothetical, slightly different version of said tool. But you can't just point at 1 example and say that this is the reason. That is just a wrong answer. Logically wrong. Lying. Annoying and hurting the readers head. Don't do that.

bonzini 3y ago
Ok, then the answer is "they do use a slightly different version of the tool than what you learnt at school. That version is more rigorous (which is why they like it) and in that version 1 is neither a prime nor a composite. In the version you learnt at school you're told that 1 is not a prime and you can't really say that it's composite either, but no clear explanation of why it is not a prime; in the version that mathematicians use there's a name for what 1 is, and a rule to find things like it in other number systems."