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.

10

Radicals and irrational numbers in general.

Bk. I. ch. II.

A series whose terms sink numerically below any assignable value
is said to have the limiting value zero; thus we have the important
fact, that zero also is included in this general conception of number.
A series whose limiting value is not zero, if it is to have both
the properties we have stated, must have its terms after some definite
place either only positive or only negative. For if a,, and,,,, differ
in sign, the absolute amount of a,,+, --,, is not less than the greater
of the two numbers, and so it is only possible for it to become smaller
than 6 when the series of the a has zero for limit.
7. We calculate with the numbers thus defined, by applying the
operations of arithmetic to the terms of the series expressing thenm,
since the limiting value can always be replaced with arbitrary approximation by a term of the series and thus the requirement of calculation
just laid down is complied with. Suppose the given series are
a, cq, 2r... c a,... and /,, P 2,.....j *;
each representing a number, we can embody this in the symbolical
expressions
A = Lim (cc,,), B = Lim (/i,,).
Applying addition or subtraction it follows that in either case
Ct + At ~ t, 2 +) *** +at * *
form a new series possessing both the properties necessary for expressing a number. For when the absolute amount of the difference
of cc,+, and c,, which we write briefly abs [ca_ — - J,], is < 6 and
abs [ /3,+t --,t] < 4, the absolute amount of the difference of two terms
of the new series is less than 26, for ac,+^ and,,t+ have at the most
increased or diminished by the quantity 6 in comparison with c,
and /,,. But the limiting value of this new series differs from the
algebraic sum of the two given limiting values by less than an arbitrarily small assigned number, since the quantities a, and /,, differ
arbitrarily little from these limiting values; i. e. the limiting value
of the new series is equal to the algebraic sum of the given numbers
A and B. We formulate this in the equation
(I) Lim (o,) + Lim (,S) == Lim (,c +~ i,).
Subtraction here furnishes the special theorem: Two series of
numbers which express the same limiting value yield when subtracted
a series of numbers whose limit is zero. But also conversely we define: Two series of numbers whose difference has the limiting value
zero express the same number, or, two numbers are called equal when
the difference of corresponding terms of their defining series has the
limiting value zero. This is to be regarded as the definition of
equality of two numbers. The original definition: Two rational
numbers are equal if they contain the same number of units or