indivisibility

Indivisibility is the quality of being unable to be divided into smaller, meaningful parts within a given framework. In mathematics, it most often refers to numbers that cannot be factored into smaller positive integers.

In number theory, a prime number is an integer greater than 1 that has no positive divisors other than 1 and itself. Equivalently, it is indivisible by any smaller positive integer other than 1. The fundamental theorem of arithmetic states that every integer greater than 1 is either prime or a product of primes in a unique way, underscoring the role of indivisibility as the building blocks of the integers.

The historical idea of indivisibility extends beyond arithmetic. In ancient Greek philosophy, the concept of atoms described indivisible fundamental constituents of matter. In modern mathematics and logic, there are notions of indivisibility in combinatorics and model theory, where a structure may be called indivisible if it cannot be partitioned into smaller parts without preserving a copy of the whole.

Examples help illustrate the distinction. The numbers 2, 3, 5, and 7 are primes and thus indivisible by smaller numbers. The numbers 4, 6, and 9 are composite and can be divided by smaller integers. The number 1 is typically treated as a unit, not a prime, and is not described as indivisible in the same sense.

Indivisibility remains a central idea across disciplines, illustrating how complex systems can be built from simple, indivisible components.