Basic Definitions 181definitions do not depend on the approximating sequences. The set of real
numbers is denoted by R. Geometrically, we represent E as points on a
one-dimensional continuum, known as the real line. The construction of
Cantor ensures that the set of real numbers, also known as the real line,
does not contain any holes.
Finally, to solve equation x^2 + 1 = 0 and other similar polynomial
equations with real coefficients but without real solutions, the set of real
numbers is extended by adjoining to M an element i (also denoted by j
in some engineering books) such that i^2 = -1. This element is called
the imaginary unit. The resulting extension is denoted by C and is the
collection of the expressions x + iy, where x, y € M.. The elements of C
are called complex numbers. As far as solving polynomial equations, we
are all set now: the Fundamental Theorem of Algebra states that every
polynomial with coefficients in C has at least one root in C; in other words,
the set C is algebraically closed.
If z = x + iy G C, and i,i/£l, then x, denoted by 3?z, is called the
real part of z. Also, y = 9z is called the imaginary part of z, and
~z = a — hi, the complex conjugate of z. By definition,
(a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c+di) = (ac — bd) + (ad + bc)i,where the multiplication rule follows naturally from the distributive law
and the equality i^2 = —1.
EXERCISE 4.1.l.c (a) Verify that "Siz = (z + z)/2, Sz = (z - z)/(2i).
(b) Verify that the complex conjugate of the sum, difference, product, and
ratio of two complex numbers is equal to the sum, difference, product, and
ratio, respectively, of the corresponding complex conjugates: for example,
z\Z2 = z\Z2- (c) Verify that if P = P(z) is a polynomial with real coeffi-
cients, then P(z) = P(z), and therefore the non-real roots of this polynomial
come in complex conjugate pairs, (d) Conclude that a polynomial of odd
degree and with real coefficients has at least one real root.
Complex numbers appear naturally in computations involving square
roots of negative numbers. Some records indicate that such computa-
tions can be traced to the Greek mathematician and inventor HERON OF
ALEXANDRIA (c.10 - c.70 AD). In 1545, the Italian mathematician GERO-
LAMO CARDANO (1501-1576) published the general solutions to the cubic
(degree three) and quartic (degree four) equations. The publication boosted
the interest in the complex numbers, because the corresponding formulas
required manipulations with square roots of negative numbers, even when