ero has a very definite content. As the border-line between all positive and negative magnitudes, as the sole really neutral number, which can be neither positive nor negative, it is not only a very definite number, but also in itself more important than all other numbers bounded by it. In fact, zero is richer in content than any other number. Put on the right of any other number, it gives to the latter, in our system of numbers, the tenfold value. Instead of zero one could use here any other sign, but only on the condition that this sign taken by itself signifies zero, =0. Hence it is part of the nature of zero itself that it finds this application and that it alone can be applied in this way. Zero annihilates every other number with which it is multiplied; united with any other number as divisor or dividend, in the former case it makes this infinitely large, in the latter infinitely small; it is the only number that stands in a relation of infinity to every other number. 0/0 can express every number between – ∞ and +∞, and in each case represents a real magnitude.
The real content of an equation first clearly emerges when all its members have been brought to one side, and the equation is thus reduced to zero value, as already happens for quadratic equations, and is almost the general rule in higher algebra. The function F (x, y)=O can then also be put equal to z, and this z, although it is =0, differentiated like an ordinary dependent variable and its partial derivative determined.
The nothing of every quantity, however, is itself quantitatively determined, and only on that account is it possible to calculate with zero. The very same mathematicians who are quite unembarrassed in reckoning with zero in the above manner, i.e., in operating with it as a definite quantitative concept, bringing it into quantitative relation to other quantitative concepts, clutch their heads in desperation when they read this in Hegel generalised as: the nothing of a something is a determinate nothing.
But now for (analytical) geometry. Here zero is a definite point from which measurements are taken along a line, in one direction positively, in the other negatively. Here, therefore, the zero point has not only just as much significance as any point denoted by a positive or negative magnitude, but a much greater significance than all of them: it is the point on which they are all dependent, to which they are all related, and by which they are all determined. In many cases it can even be taken quite arbitrarily. But once adopted, it remains the central point of the whole operation, often determining even the direction of the line along which the other points-the end points of the abscissae – are to be inserted. If, for example, in order to arrive at the equation of the circle, we choose any point of the periphery as the zero point, then the line of the abscissae must go through the centre of the circle. All this finds just as much application in mechanics, where likewise in the calculation of the motions the point taken as zero in each case forms the main point and pivot for the entire operation. The zero point of the thermometer is the very definite lower limit of the temperature section that is divided into any desired number of degrees, thereby serving as a measure both for temperature stages within the section as also for higher or lower temperatures. Hence in this case also it is a very essential point. And even the absolute zero of the thermometer in no way represents pure abstract negation, but a very definite state of matter: the limit at which the last trace of independent molecular motion vanishes and matter acts only as mass. Wherever we come upon zero, it represents something very definite, and its practical application in geometry, mechanics, etc., proves that - as limit - it is more important than all the real magnitudes bounded by it.
Zero powers. Of importance in the logarithmic series:
√—1. The negative magnitudes of algebra are real only in so far as they are connected with positive magnitudes and only within the relation to 