Any positive integer has a unique representation as a composite of prime numbers.

First, it must be understood that an integer is a whole number with no fractional parts and a prime number is any number which can only be represented, through multiplication of positive integers, as the number one multiplied by itself, the number one itself being excluded. (The number one is not considered a prime number simply by convention.)

2, 3, 5, 7, 11, 13, and 17 are examples of prime numbers.

According to the Fundamental Theorem of Arithmetic, if we were to take a random number, say 84,598,206, we would be able to factor it and turn it into a composite of primes and this set of primes would uniquely identify the number. (That is to say, no other set of primes could be found to multiply together to form the same number.)

As an example 84,598,234 is an even number and thus, divisible by 2 therefore 84,598,206=42,299,103 * 2. We can keep expanding this and get:

Based on this factorization, we can say a good number of things. First, the square root of a prime number is irrational, and so the square root of 84,598,206 will be irrational (or 7 time an irrational number). if the factorization were 101*101*37*37*7*7*3*3, then we would very easily see that the square root was the rational number codified by 101*37*7*3.

When multiplying and dividing it is very easy to see that the primes are added to the composite and removed accordingly. 5*3*2 multiplied by 7*2 equals 7*5*3*2, while 7*5*3*2 divided by 5*3 equals 7*2. This is all obvious enough, but then again, it's only the Fundamental Theorem of Arithmetic.

Another interesting consideration is that the Fundamental Theorem of Arithmetic sets up numbers as existing in an infinitely dimensional space, which we might call prime space. We could then say that each prime number serves as the basis for a dimension in prime space. (ie. A 3-vector (1,2,3) might be considered 2*3*3*5*5*5=2,250 and vice versa.)