Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic asserts that every natural number greater than 1 is either a prime number or can be uniquely expressed as a product of prime numbers.
This theorem is one of the foundations of mathematics—a truth that feels almost intuitive once grasped, yet holds immense significance.
This uniqueness of prime factorization is what allows mathematics to stay organized and predictable: every number has its own distinct “numerical DNA.”
For example, the number 12 can be broken down into the prime numbers 2 and 3 as follows:
$$ 12 = 2 \times 2 \times 3 = 2^2 \times 3 $$
In this prime factorization, you can rearrange the order of the factors, like writing $ 12 = 3 × 2 × 2 $ or $ 12 = 2 \times 3 \times 2 $, but the product remains unchanged.
Thus, the set of prime factors is the only one that produces the number 12.
According to the Fundamental Theorem of Arithmetic, the uniqueness of prime factorization holds for every natural number greater than 1.
The heart of this theorem lies in understanding prime numbers. These are the essential “building blocks” of arithmetic, indivisible elements that combine in various ways to form all other numbers. Imagine each number as a structure—prime numbers are the bricks it’s built with. Without them, the entire framework of arithmetic would crumble, as there would be no consistent way to break down numbers.
Likewise, you can write 30 as a unique product of the prime numbers 2, 3, and 5, no matter the order of the factors:
$$ 30 = 2 \cdot 3 \cdot 5 $$
In general, any number can be expressed as a product of prime numbers.
The Unique Case of the Number 1
An interesting detail that often raises questions is why the number 1 is not considered a prime number.
Why isn’t it considered a prime?
At first glance, it might seem like it is, since it’s divisible by 1 and itself. However, it’s not classified as prime due to the Fundamental Theorem of Arithmetic.
If 1 were counted as a prime, the uniqueness of prime factorization would break down.
For instance, consider the number 24, which can be factored as \( 2 \cdot 2 \cdot 2 \cdot 3 \).
$$ 24 = 2 \cdot 2 \cdot 2 \cdot 3 $$
If 1 were prime, you could also write 24 as \( 1 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \), or \( 1 \cdot 1 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \), and so on, infinitely.
$$ 24 = 1 \cdot 2 \cdot 2 \cdot 2 \cdot 3 $$
$$ 24 = 1 \cdot 1 \cdot 2 \cdot 2 \cdot 2 \cdot 3 $$
$$ 24 = 1 \cdot 1 \cdot 1 \cdot 2 \cdot 2 \cdot 2 \cdot 3 $$
$$ \vdots $$
This endless repetition would disrupt the uniqueness of factorization, making it impossible to represent numbers in a single, consistent way.
This is why the number 1 is not considered a prime number!
