There is a philosophy of mathematics and a Science on the theory of Numbers that has discussed this question and de-constructed our notions of Number, and lain bare the deficiencies of our Understanding on this matter better than we can hope to do so here. We remind the Student however, that in Mathematics itself, even a thing so simple as Number is defined according to its scope and needs. Thus in Geometry, a number represents a Length or a Ratio or a Quantity. In Arithmetic, it is always an identity; in Calculus, the limit of a function. In mathematical and philosophic theory, every number is infinite in itself and contains within it the possibility for engendering all other Identities by combining with other Numbers. Even the transcendental numbers which cannot be expressed finitely, can be written down as the operation between two numbers to some degree.^{1} Thus ∏, though incomprehensible as a string of numbers, is simply the circumference of the Circle divided by its diameter and expressible as a simple ratio of two quantities. Though the result is mathematically, a fiction; we allow it into our calculations, for we witness Circles in Nature and know this ratio to exist in Reality, though it does not in our current Field of integral Number.^{2}

...Thus in Geometry, a number represents a Length or a Ratio or a Quantity. In Arithmetic, it is always an identity; in Calculus, the limit of a function. In mathematical and philosophic theory, every number is infinite in itself...