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.

8

Radicals and irrational numbers in general.

Bk. I. ch. II.

than each f3. Further, we can make the constantly positive differences /ft+ - ac+,r a,~+v - an,.P, - Pt+V for each value of v, smaller
than any assigned value however small, merely by selecting only
those terms in the series for which #u has a correspondingly great
value. For if these differences are to be smaller than 6, we have only
to determine the first value of p for which /,, - a, < 6; then we
shall also have by the foregoing inequalities for every v:
SLtYV - a&+~v < An, aur - ~U < 4, jt -- IP+ <y 4*
The numbers of the unlimited series of the a and likewise of the
/ have the property that their Mth powers are always coming nearer to
the value -, so that they ultimately differ from it as little as we please.
Hence it appears, that the numbers a and / themselves also approach
more and more to one definite quantity, which, even though it does not
exist among rational numbers, is yet called a numerical value, because
it is connected with a rational number by a perfectly determinate arithmetical operation. We denote the quantity as the limiting value
of the series a and /3; it is written in the form y -/
When the quantity is a rational number, its exposition as a
limiting value depends solely on the choice of the denominators
a, a', 6" etc. All periodic decimals belong to this case; for instance,
the value of
limiting 0e3:; 0o 3; 0333; 0 3333 etc.
is a. Similarly, by the geometric progression, the limiting value of
1; 1 + + + (); v 1 + + ()2 + (.)3.
is 2. By suitably choosing the denominator, therefore, a rational form
can also be discovered for such values.
When on the other hand the quantity is not a rational number,
- and of this we can make sure at the outset in the case of a
radical by means of our opening proposition, - the only possible
way of expressing the number, which is called an Irrational
number, is as limiting value of a series. This exposition and definition
embraces therefore besides rational quantities a new class, namely irrational radicals; but it will be subsequently seen to embrace still more
than the roots of fractions. At the same time this process of evolution fixes our attention on what is properly to be understood henceforth by the calculation of a number which is to possess a given
property. It cannot in general be required that this number shall
be assigned in finite closed form, but rather we see that: To calculate a number which shall have a definite property relatively to other given
numbers, mecans, to find a series of rational numbers that can be