In my discrete math my textbook is now covering prime factorization. It states that the Fundamental Theorem of Arithmetic states:
Every positive integer other than 1 can be expressed uniquely as a product of prime numbers where the prime factors are written in non-decreasing order.
My confusion is how can you write a prime number as a factor of prime numbers? Take 7 for instance. You cannot write 7 as the product of prime numbers, right? Since 1 is not a prime number, you cannot express 7 as 7 * 1.
Is my misunderstanding something with the definition of "product"? Can only one number be considered a product?
Example: 7 can be written as the product of 7. (This doesn't sound right to me)
Hopefully someone can help.
Best Answer
You can define products with any number of factors in $\mathbb{Z}_{\ge 0}$. The only product with $0$ factors is, by definition, $1$. This kind of makes sense, since $1$ is the neutral element for multiplication. Moreover, this definition is handy whenever you define a product of $n$ factors but $n$ could be $0$. And then of course the product of one factor is the factor itself. From $2$ on, we know how it works. Hence it is just a matter of defining what a product is.