when exploring the theoretical subtleties of area and
volume in greater detail.
See also
SCALE
.
Argand, Jean Robert (1768–1822) Swiss Complex
number theory Born on July 8, 1768, in Geneva,
Switzerland, amateur mathematician Jean Argand is
remembered today for his famous geometrical interpre-
tation for
COMPLEX NUMBERS
. An A
RGAND DIAGRAM
uses two perpendicular axes, one representing a real
number line, the second a line of purely complex num-
bers, to represent complex numbers as points in a plane.
It is not well known that Argand, in fact, was not
the first to consider and publish this geometric
approach to complex numbers. The surveyor Casper
Wessel (1745–1818) submitted the same idea to the
Royal Danish Academy in 1797, but his work went
unnoticed by the mathematics community. At the turn
of the century, Argand independently began to interpret
the complex number igeometrically as a rotation
through 90°. He expounded on the convenience and
fruitfulness of this idea in a small book, Essai sur une
manière de représenter les quantités imaginaires dans
les constructions géometriques (Essay on a method for
representing imaginary quantities through a geometric
construction), which he published privately, at his own
expense, in 1806. He never wrote his name in the
piece, and so it was impossible to identify the author.
By chance, French mathematician Jacques Français
came upon the small publication and wrote about the
details of the work in an 1813 article, “A Memoir on
the Geometric Representation of Imaginary Numbers,”
published in the Annales de Mathématiques. He
requested that the unknown originator of the ideas
come forward and receive credit for the work. Argand
made himself known by submitting his own article to
the same journal, presenting a slightly modified and
improved approach to his methods. Although histori-
ans have since discovered that the mathematicians
J
OHN
W
ALLIS
(1616–1703) and C
ARL
F
RIEDRICH
G
AUSS
(1777–1855) each considered their own geomet-
ric interpretations of complex numbers, Argand is usu-
ally credited as the discoverer of this approach.
Argand was the first to develop the notion of the
MODULUS
of a complex number. It should also be noted
that Argand also presented an essentially complete
proof of the
FUNDAMENTAL THEOREM OF ALGEBRA
in
his 1806 piece, but has received little credit for this
accomplishment. Argand was the first to state, and
prove, the theorem in full generality, allowing all num-
bers involved, including the coefficients of the polyno-
mial, to be complex numbers.
Argand died on August 13, 1822, in Paris, France.
Although not noted as one of the most outstanding
mathematicians of his time, Argand’s work certainly
shaped our understanding of complex number theory.
The Argand diagram is a construct familiar to all
advanced high-school mathematics students.
Argand diagram (complex plane) See
COMPLEX
NUMBERS
.
argument In the fourth century
B
.
C
.
E
., Greek
philosopher
ARISTOTLE
made careful study of the struc-
ture of reasoning. He concluded that any argument,
i.e., a reasoned line of thought, consists, essentially, of
two basic parts: a series of
PREMISE
s followed by a con-
clusion. For example:
If today is Tuesday, then I must be in Belgium.
I am not in Belgium.
Therefore today is not Tuesday.
is an argument containing two premises (the first two
lines) and a conclusion. An argument is valid if the con-
clusion is true when the premises are assumed to be true.
Any argument has the general form:
If [premise 1 AND premise 2 AND premise 3
AND…], then [Conclusion]
Using the symbolic logic of
FORMAL LOGIC
and
TRUTH
TABLE
s, the above example has the general form:
p→q
¬
q
Therefore
¬
p
The argument can thus be summarized: ((p→q) (
¬
q))
→(
¬
p).
One can check with the aid of a truth table that
this statement is a tautology, that is, it is a true state-
ment irrespective of the truth-values of the component
statements pand q. (In particular, it is true when both
premises have truth-value T.) Thus the argument pre-
sented above is indeed a valid argument.
∨
argument 25