their exponents. Any number can be conceived and expressed as the power of any other number (logarithms, y =ax). And this transformation of one form into the opposite one is no idle trifling, it is one of the most powerful levers of mathematical science, without which today hardly any of the more difficult calculations are carried out. If negative and fractional powers alone were abolished from mathematics, how far could one get?
(—.— =+, —/— =+, √-1, etc., to be expounded earlier.)
The turning point in mathematics was Descartes’ variable magnitude. With that came motion and hence dialectics in mathematics, and at once, too, of necessity the differential and integral calculus, which moreover immediately begins, and which on the whole was completed by Newton and Leibniz, not discovered by them.
Quantity and quality. Number is the purest quantitative determination that we know. But it is chock-full of qualitative differences. 1. Hegel, number and unity, multiplication, division, raising to a higher power, extraction of roots. Thereby, and this is not shown in Hegel, qualitative differences already make their appearance: prime numbers and products, simple roots and powers. 16 is not merely the sum of 16 ones, it is also the square of 4, the fourth power of 2. Still more. Prime numbers communicate new, definitely determined qualities to numbers derived from them by multiplication with other numbers; only even numbers are divisible by 2, and there is a similar determination in the case of 4 and 8. For 3 there is the rule of the sum of the figures, and the same thing for 9 and also for 6, in the last case in combination with the even number. For 7 there is a special rule. These form the basis for tricks with numbers which seem incomprehensible to the uninitiated. Hence what Hegel says (Quantity, p. 237) on the absence of thought in arithmetic is incorrect. Compare, however, Measure.
When mathematics speaks of the infinitely large and infinitely small, it introduces a qualitative difference which even takes the form of an unbridgeable qualitative opposition: quantities so enormously different from one another that every rational relation, every comparison, between them ceases, that they become quantitatively incommensurable. Ordinary incommensurability, for instance of the circle and the straight line, is also a dialectical qualitative difference; but here it is the difference in quantity of similar magnitudes that increases the difference of quality to the point of incommensurability.
Number. The individual number becomes endowed with quality already in the numerical system itself, and the quality depends on the system used. 9 is not only 1 added together 9 times, but also the basis for 90, 99, 900,000, etc. All numerical laws depend upon and are determined by the system adopted. In dyadic and triadic systems 2 multiplied by 2 does not equal 4, but=100 or=11. In all systems with an odd basic number, the difference between odd and even numbers falls to the ground, e.g., in the system based on 5, 5=10, 10=20, 15=30. Likewise in the same system the sums of digits 3n of products of 3 or 9 (6=11, 9=14). Hence the basic number determines not only its own quality but also that of all the other numbers.
With powers of numbers, the matter goes still further: any number can be conceived as the power of any other number-there are as many logarithmic systems as there are whole and fractional numbers.
One. Nothing looks simpler than quantitative unity, and nothing is more manifold than it, as soon as we investigate it in connection with the corresponding plurality and according to its various modes of origin from plurality. First of all, one is the basic number of the whole positive and negative system of numbers, all other numbers arising by the successive addition of one to itself.
One is the expression of all positive, negative, and fractional powers of one: 12, √1, 1-2 are all equal to one.
It is the content of all fractions in which the numerator and denominator prove to be equal. It is the expression of every number that is raised to