Nicomachus writes almost entirely without diagrams or worked calculations, which signals from the outset that Introduction to Arithmetic is not a manual for practical reckoning. Its actual subject is the nature of number as a category, in the Neopythagorean conviction, inherited from Pythagoras through several centuries of transmission, that number is not merely a tool for counting objects but the underlying structural principle of reality itself, prior to and more fundamental than the physical world number is used to measure. Working from that premise, the text spends most of its length classifying whole numbers by type rather than teaching operations on them: odd against even, prime against composite, and a set of ranked qualitative categories, deficient numbers whose divisors sum to less than the number itself, abundant numbers whose divisors sum to more, and perfect numbers, rare cases where the divisors sum exactly to the number, treated as a kind of numerical ideal.
Nicomachus also introduces figurate numbers, quantities that can be arranged into a regular geometric shape, triangular numbers forming a triangle of dots, square numbers forming a square, and so on, treating the fact that number and geometric form align so neatly as further evidence that number sits beneath and generates the visible structure of the physical world rather than merely describing it after the fact. That framing, arithmetic as a doorway into metaphysics rather than a practical skill, is what separates this text from a simple mathematics primer and situates it squarely within the Pythagorean tradition's larger project of finding divine order encoded in number.