Quadratic Reciprocity: The connection that changed number theory
31–33 of 33 posts
Re: Quadratic Reciprocity: The connection that changed number theory
#32Earlier quoted context omitted.
If it conforms the definition, then 1 is a prime. This is evident. Why would they choose a definition so that 1 is not a prime? They found it the best and most useful definition that they knew at that time. While there are many bad definitions, concepts and names in math, I don't think prime number is an example. Also please notice that sometimes the more useful concepts have longer definitions, especially if one lim…
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…
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
Re: Quadratic Reciprocity: The connection that changed number theory
#33Earlier quoted context omitted.
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…
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 i…
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."