P. HISTORY OF ' AATHEMATICAL - School of Mathematics

124 A HISTORY OF MATHEMATICAL NOTATIONS
alle diejenigen Werthe beilegt, fur welche Ix I= r ist, eine obere
Grenze, die mit g bezeichnet werde: und es gilt der Satz: IAI' I'$.gr-I'
fur jeden ganzzahligen Werth von JJ.." This article was not printed
at the time. Weierstrass used I I again in an article read before the
Berlin Academy of Sciences on December 12, 1859. 1 Weierstrass also
employed the notation I I for determinants," and avoided confusion
by the use of such phrases as "Die Determinante 1Wall I."
Andre 3 states that the mark z has also been used to designate
absolute value. The abbreviation' "mod z" from Argand's and
Cauchy's word! "module" is not very convenient; nor is it desirable
in view of the fact that "mod" has been widely adopted, since the
time of Gauss, in the theory of numbers .
. 493. Zeroes of different origin.-Riabouchinski 6 gives a new extension
of the notion of number leading to transfinite numbers differing
from those of G. Cantor; he represents the ordinary symbolism
lim . ~ = 0 by the symbol J~ = +o, As a is taken arbitrarily, he assumes
n-7oon ­
the existence of an infinity of zeroes of different origin; if a is positive,
the corresponding zero is marked +0 or 0, if negative the zero is
marked - o, Let ~ be 0, and J the inverse of passage to the limit,
then J . ~ =J . 0 =5. When the origin of a zero is not defined, he
writes J. ±o=a, where a is arbitrary. He has a . 0= ±o, J =~ ,
J • a= ~,a. 0= ~ =ab .0= ±o, I+J =J+l. He designates by ± I
the operation of the return to relative values or the inverse of the passage
to absolute values. Accordingly, he lets F1 or j represent the
result of an impossible return to a relative value. Then, IiI = -1,
j=±F!.
1 K. Weiel'lltr8BB, op, cit., Vol. I, p. 252.
2 K. Weiel'lltr88s, op. cit., Vol. IV, p. 524.
a D. Andre, op. cit., p. 86.
'See G. Peano, Formulaire mathbnatique, Vol. IV (1903), p. 93.
I A. L. Cauchy, Exercices de mathbnatiques (Paris, 1829), Tome IV, p. 47;
(Euvres (2d ser.), Vol. IX, p. 95.
I D. Riabouchinski, "Sur Ie caloul des valeurs absolues," Comptes rendw du
congres international des mathbnaticien8 (Strasbourg, 22-30 Septenibre 1920;
Toulouse, 1921), p. 231-42.