180 Algebra of Complex Numbers
point of the construction.
The set N is naturally equipped with two binary operations, addition
(a,b) I—> a + b, and multiplication (a,b) >—> ab. These operations are
associative:
(a + b) + c = a + (b + c), (ab)c = a(bc),and distributive:
a(b + c) = ab + ac.To solve the linear equation x + a = b for every a, b G N the set of
positive integers is extended to the set Z = {0, ±1, ±2,...} of all integers
by introducing the special number 0 so that a+0 = a for all a and adjoining
to every non-zero a G N the additive inverse —a so that a + (—a) = 0.
To solve the linear equation ax = b for every a, b G Z, the set of integers
is extended to the set Q of rational numbers, that is, expressions of the
form p/q = pq_1, where p, q G Z and q ^ 0. For every non-zero p/q G Q,
the element q/p satisfies {p/q)(q/p) — 1 and is called the multiplicative
inverse of p/q.
To solve the quadratic equation x^2 — 2 = 0 and other general polynomial
equations with coefficients in Q, the set of rational numbers is extended
by creating and including the algebraic irrational numbers, that is, the
numbers such as \/2 or v^4, that can be roots of polynomial equations with
coefficients in Q. The result is the algebraic closure of Q.
It turns out there are other irrational real numbers that are not alge-
braic, that is, are not roots of any polynomial with rational coefficients.
These irrational numbers are called transcendental. To put it differently,
the ordered set of algebraic numbers contains many holes, and these holes
are filled with transcendental numbers. It was only in 1844 that the French
mathematician JOSEPH LIOUVILLE (1809-1882) showed the existence of
such numbers by explicitly constructing a few. Later, it was proved that
the two familiar numbers, n and e, are also transcendental. Together, the
algebraic and transcendental numbers make up the real numbers. The
rigorous definition of a real number is necessary to put calculus on a sound
basis.
Cantor defined a real number as the limit of a sequence of rational num-
bers. The operations of addition, subtraction, multiplication, and division
are introduced on the set of real numbers in an obvious way as the limits
of the corresponding sequences; the non-trivial part is the proof that the