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.

2

Rational numbers.

Bk.. ch. T.

has alway the same value in whatever order the summands be reckoned
up. The truth of this proposition cannot be deduced from simpler
conceptions; it is immediately perceived from the intuition of an
arbitrary but finite number of units, which form the sum. Addition
is always possible, the sum is continually greater than any of the
summands.
The problem of Sub traction follows from inversion of
addition. What this requires is, given the sum and one summand to
calculate the other summand (difference); that is, to reckon off from
a given number (minuend) as many units as the other number (subtrahend) contains. This is only possible when the minuend is greater
than the subtrahend. In case they are both equal, i.e. contain the same
number of units, we denote the result by 0. The numerical conception
0 is therefore defined by the equation a - a 0=. Hence we obtain
for calculating with 0 the equations: a -- 0 = a, a - 0 =- a.
20 To form a sum in which the several summands are equal to
the same number a and the number of summands is b, is, to Multiply
the number a by b. The result is called a multiple of a. But without
distinguishing a and b they may be called factors and the result simply
the product, since for multiplication of two or more numbers we have
the fundamental proposition - which can be proved from the conception
of summation: The product of given numbers has always the same
value, in whatever way the factors may be interchanged or combined
in groups.*) Multiplication is always possible.
The problem of Division follows from inversion of multiplication, when, given the product and one factor it is required to
calculate the other factor; that is, to find that number (quotient) which
multiplied by one of the given numbers (divisor) yields a product equal
to the other (dividend). This is impossible unless the dividend is a
multiple of the divisor; calling a the divisor, b the dividend, then if
a < b there is always an equation of the form: b= a. q + r, where
r (the remainder) must be a number of the series 0, 1, 2,... a- 1.
Two numbers can be multiples of a third number, this is then
a common divisor of both. Two numbers, which have no common
divisor besides unity, are relatively prime. A number is called
absolutely prime which is not divisible by any other except unity.
This distinction of divisible and prime numbers leads to the important
theorem: Any number can be expressed only in a single manner as a
product of absolutely prime numbers; but the investigation of the
divisibility of a number rests on the following rule (employed by
*) The proofs of the theorems for rational numbers cannot be presented
here: they are found in Baltzer, die Elemente der Mathematik, Vol. I.