(1465–1526), N
ICCOLÒ
T
ARTAGLIA
(1499–1557), and
to his assistant L
UDOVICO
F
ERRARI
(1522–65), Car-
dano unified general methods. He was an outstanding
mathematician of the time in the fields of
ALGEBRA
,
TRIGONOMETRY
, and
PROBABILITY
.
At the age of 19, Cardano entered Pavia University
to study medicine but quickly transferred to the Uni-
versity of Padua to complete his degree. He excelled at
his studies and earned a reputation as a top debater. He
graduated with a doctorate in medicine in 1526.
Cardano set up a small medical practice in the vil-
lage of Sacco, but it was not at all successful. He
obtained a post as a lecturer in mathematics at the
Piatti Foundation in Milan, where he pursued interests
in mathematics while continuing to treat a small clien-
tele of patients.
In 1537 Cardano published two mathematics texts
on the topics of arithmetic and mensuration, marking
the start of a prolific literary career. He wrote on such
diverse topics as theology, philosophy, and medicine in
addition to mathematics.
In 1539 Cardano learned that an Italian mathe-
matician by the name of Tartaglia knew how to solve
cubic equations, a topic of interest to Cardano since he
and Ferrari, his assistant, had discovered methods for
solving quartics, if the method for cubics was clear.
Tartaglia revealed his methods to Cardano under the
strict promise that the details be kept secret. (At the
time, Renaissance scholars, such as Tartaglia, were
often supported by rich patrons and had to prove their
worth in public challenges and debates. Keeping meth-
ods secret was thus of key importance.) When Cardano
later learned that another scholar by the name of del
Ferro had discovered methods identical to those of
Tartaglia decades earlier, Cardano no longer felt
obliged to keep the solution secret and published full
details in his famous 1545 piece Ars magna (The great
art). Tartaglia was outraged by this act, and a bitter
dispute between the two men ensued.
Although Cardano properly credits Ferrari and
Tartaglia as the first scholars to solve the cubic equa-
tion, it should be noted that Cardano properly identi-
fied general approaches that unified previous methods.
Cardano also recognized that solutions would often
involve
COMPLEX NUMBERS
and was the first scholar to
make steps toward understanding these quantities. He
died on September 21, 1576, in Rome.
See also R
AFAEL
B
OMBELLI
.
Cardano’s formula (Cardano-Tartaglia formula) See
CUBIC EQUATION
.
cardinality In common usage, the cardinal numbers
are the counting numbers 1, 2, 3, … These numbers
represent the sizes of
FINITE
sets of objects. (Unlike the
ORDINAL NUMBERS
, however, the cardinal numbers do
not take into account the order in which elements
appear in a given set.)
In the late 1800s German mathematician G
EORG
C
ANTOR
(1845–1918) extended the notion of cardinal-
ity to include meaningful examination of the size of
infinite sets. He defined two sets to be of the same car-
dinality if their members can be matched precisely in a
one-to-one correspondence. That is, each element of
the first set can be matched with one element of the
second set, and vice versa. For example, the set of peo-
ple {Jane, Lashana, Kabeer} is of the same cardinality
as the set of dogs {Rover, Fido, Spot}, since one can
draw leashes between owners and dogs so that each
owner is assigned just one dog, and each dog is leashed
to one owner. Each of these sets is said to have cardi-
nality 3. (Both sets are of the same cardinality as the set
{1, 2, 3}.) Two sets of the same cardinality are said to
be equipotent (equipollent, equinumerable, or, simply,
equivalent).
The set of all counting numbers {1, 2, 3, …} is
equipotent with the set of all integers {…, –2, –1, 0, 1,
2, …}. This can be seen by arranging each set of num-
bers in a list and matching elements according to their
positions in the list:
123456…
0–11–22–3…
This procedure shows that any two sets whose elements
can be listed are equipotent. Such a set is said to be
DENUMERABLE
, and Cantor denoted the cardinality of
any denumerable set ℵ0, pronounced “aleph null.” Can-
tor’s first
DIAGONAL ARGUMENT
shows that the set of all
counting numbers, the set of integers, and the set of all
rational numbers are each denumerable sets and so each
have cardinality ℵ0. Mathematicians have shown that
the set of all
ALGEBRAIC NUMBER
s is also denumerable.
Not all infinite sets, however, are denumerable, as
Cantor’s second diagonal argument shows. For
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60 Cardano’s formula