One of the surprising things about mathematics is its insistence
that every assertion needs a justification. Non-mathematicians are
often surprised by the extent to which mathematicians enforce this
dictum. For example, consider the following result, which is usually
called the Fundamental Theorem of Arithmetic.
Theorem:Every integer N > 1 can be written uniquely as a product of finitely many prime numbers.
Most people not only can't figure out where to start looking for a proof of this result, but also don't understand why it needs a proof. Of course it's true; it has been drilled into them since grade school. The whole issue of reducing a fraction to lowest terms doesn't make sense unless you can factor both numerator and denominator as a product of primes, and cancel the common factors.
To a mathematician, however, it is precisely because this result is so basic that it needs to be questioned. Is it so fundamental that it needs to be made an axiom? Or is it a consequence of other, simpler statements?
Another thing to keep in mind when confronted with the statement of a mathematical result is that mathematicians are laconic. They omit needless words. In particular, every word in the statement of a theorem is there for a reason. You won't really understand the statement until you ferret out the reason for the inclusion of each word.
In the case of the present theorem, there are three critical phrases, and you should try to explain why each one is there.
1 = 1.1 = 1.1.1 = 126.96.36.199 = ....
Now, having excluded 1 from the list of primes, you certainly can't write 1 as a product of other primes (all of which are bigger than 1). Well, you could if you allowed a product of zero primes, but that seems too much like cheating.
Minus one has a similar problem, but you can handle other negative
numbers by factoring out the -1 and then using this theorem. It would
be possible to state the theorem to include this case, but it would
distort the main meaning so much that it's not worthwhile.
The theorem talks about finite products because it can.
Mathematicians are usually careful to distinguish things that can be
accomplished in finitely many steps from things that truly require an
infinite process. Arithmetic (and, more generally, algebra) are
essentially finite mathematics; analysis is essentially the
mathematics of the infinite.
The main point of the Fundamental Theorem of Mathematics is the uniqueness of factorizations. It is relatively easy to show that some kind of factorization into primes exists; it takes considerably more care to show that there is only one way to factor an integer.
This idea has played an important role in the history of mathematics. In attempting to prove Fermat's Last Theorem, mathematicians of the nineteenth century generalized the notion of numbers to collections of things called algebraic integers. They soon realized that if you could factor algebraic integers uniquely into a product of prime algebraic integers, then you could prove Fermat's Last Theorem. Unfortunately, that approach failed. Unique factorization is a special property of the usual integers, and it rarely holds for algebraic integers.
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