An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

4

Rational numbers.

Bk. [. ch. I.

besides the series of positive ones; zero separates the two. The
remaining operations with negative numbers are given by the equations
(I) (II) (III), since they are required to comply with the rules for
differences in general. A special consequence is, that the product of
two negative numbers is positive. The rules of signs for division by
positive and negative numbers are also determined by the inversion of
multiplication.
4. In order that the operation of division may be always possible,
the positive or negative unit must be broken up into subordinate
units; it is sufficient to introduce the numerical conception + b,
where b can be any integer. For if the symbol - be employed to denote that number which when multiplied by b produces 1, then a
times this number will express the value of. Here again it is to
be remarked that in nature there exists no fractional number, but
that this conception also has a sense only in reference to numerical
combinations.
Any fraction can be replaced by another whose denominator is a
multiple of the original denominator:
b ba,
a a a,
Employing this transformation we derive the rules
b bt _ ba + bla
a - ai aat
We have further from the conception of multiplication
b bc
-- C =
a a
Multiplication of a fraction - by another -, in analogy with this last
a aj
equation, is understood to be division of the fraction by a, and multiplication of this part by b,; whence
b b bb
a al aa~
This definition complies with the fundamental proposition of multiplication. By inversion we get the equation of division
b b _ b ac
a -ai ab,
Thus we can now complete the equations for sums and differences
(IV) a a b a b a -b C b
C C -- C c d d c+ d c +cl
But there is one very important exception to these equations: the
difference which occurs in the denominator must not be zero; a division