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I am not a mathematician. I reached solace regarding this itchy issue (not even joking) by considering the following: Primes are the minimal set of numbers to

I am not a mathematician. I reached solace regarding this itchy issue (not even joking) by considering the following: Primes are the minimal set of numbers to generate another set of numbers, using multiplication (with repetition, yada yada) For the positive numbers, [2, 3, 5, 7, ...] is all we need, considering that, as many others have said, 1 = Product([]). In that sense, to generate all integers, the integers you need are: [-1, 0, 2, 3, 5, 7..] Similarly, to generate all gaussian integers (complex numbers with integer components,) you can follow the following link: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim... however, am not sure why 0 is not considered a prime number in this context. Gaussian primes are symmetric over the real and imaginary axes, and to me it's quite curious why 0 is not considered prime there, but I guess uniqueness of the multiplication matters too (0 can be repeated multiple times, which could make some theorems about uniqueness of factorization a bit cumbersome as well.)

rocqua 7y ago
> I am not sure why 0 is not considered a prime number in this context. So, if you have a set with multiplication and addition, that is called a ring. A very nice class of rings is called fields. This is where division is possible. Another way to state that 'division is possible' is to say that multiplication is a group operator on the entire set except zero.
contravariant 7y ago
You might be happy to know that 0 is considered a prime element in typical cases (barring those cases where there are 0 divisors, i.e. where two elements can be multiplied to create 0, obviously this is not an issue with regular numbers). Edit: for historical reasons prime elements are slightly different from the usual prime numbers, and 0 remains a bit of a special case even now.
yakubin 7y ago
> I am not sure why 0 is not considered a prime number in this context. The greatest common divisor of any two primes is one. To say that two numbers are coprime, is to say their greatest common divisor is 1. If we considered 0 a prime, it would be a bit weird (and messy) that two prime (0 and any other prime number) numbers aren't coprime.

rocqua 7y ago

> I am not sure why 0 is not considered a prime number in this context. So, if you have a set with multiplication and addition, that is called a ring. A very nice class of rings is called fields. This is where division is possible. Another way to state that 'division is possible' is to say that multiplication is a group operator on the entire set except zero.

contravariant 7y ago

You might be happy to know that 0 is considered a prime element in typical cases (barring those cases where there are 0 divisors, i.e. where two elements can be multiplied to create 0, obviously this is not an issue with regular numbers). Edit: for historical reasons prime elements are slightly different from the usual prime numbers, and 0 remains a bit of a special case even now.

yakubin 7y ago

> I am not sure why 0 is not considered a prime number in this context. The greatest common divisor of any two primes is one. To say that two numbers are coprime, is to say their greatest common divisor is 1. If we considered 0 a prime, it would be a bit weird (and messy) that two prime (0 and any other prime number) numbers aren't coprime.