Why isn't 1 a prime number?

1 isn't prime because mathematicians decided it shouldn't be, and they had a very good reason. A prime is a whole number greater than 1 that can't be written as a product of two smaller whole numbers, and the phrase "greater than 1" is there on purpose: 1 is specifically excluded. Another way to say it: a prime has exactly two divisors, 1 and itself. The number 1 has only one divisor, itself, so it doesn't qualify.

The real reason is what primes are for. Every whole number above 1 can be written as a product of primes in exactly one way, apart from the order: 12 is 2 × 2 × 3 and nothing else. Primes are the building blocks of all the other numbers. If 1 counted as a building block, that uniqueness would collapse, because you could tack on as many copies of 1 as you like: 2 × 3, 1 × 2 × 3, 1 × 1 × 2 × 3, and so on forever.

This wasn't always obvious. Christian Goldbach listed 1 as a prime in his letters to Leonhard Euler in the 18th century, and printed tables of primes that began with 1 were still being published as recently as 1956. By then, most mathematicians had agreed to give 1 its own special category: a "unit".