Page 15 - Schaum's Outline of Theory and Problems of Advanced Calculus

6 NUMBERS [CHAP. 1

A set containing all its limit points is called a closed set. The set of rational numbers is not a closed
ffiffiffi
p
set since, for example, the limit point 2 is not a member of the set (Problem 1.5). However, the set of
all real numbers x such that 0 @ x @ 1isa closed set.

BOUNDS
If for all numbers x of a set there is a number M such that x @ M, the set is bounded above and M is
called an upper bound. Similarly if x A m, the set is bounded below and m is called a lower bound.If for
all x we have m @ x @ M, the set is called bounded.
If M is a number such that no member of the set is greater than M but there is at least one member
which exceeds M for every > 0, then M is called the least upper bound (l.u.b.) of the set. Similarly
if no member of the set is smaller than m but at least one member is smaller than m þ for every > 0,
m
m
then m is called the greatest lower bound (g.l.b.) of the set.
m
BOLZANO–WEIERSTRASS THEOREM
The Bolzano–Weierstrass theorem states that every bounded infinite set has at least one limit point.
A proof of this is given in Problem 2.23, Chapter 2.

ALGEBRAIC AND TRANSCENDENTAL NUMBERS
A number x which is a solution to the polynomial equation
n
a 0 x þ a 1 x n 1 þ a 2 x n 2 þ þ a n 1 x þ a n ¼ 0 ð1Þ
where a 0 6¼ 0, a 1 ; a 2 ; ... ; a n are integers and n is a positive integer, called the degree of the equation, is
called an algebraic number. A number which cannot be expressed as a solution of any polynomial
equation with integer coefficients is called a transcendental number.

2
EXAMPLES. 2 and p ffiffiffi 2 which are solutions of 3x 2 ¼ 0 and x 2 ¼ 0, respectively, are algebraic numbers.
3
The numbers and e can be shown to be transcendental numbers. Mathematicians have yet to
determine whether some numbers such as e or e þ are algebraic or not.
The set of algebraic numbers is a countably infinite set (see Problem 1.23), but the set of transcen-
dental numbers is non-countably infinite.

THE COMPLEX NUMBER SYSTEM
2
Equations such as x þ 1 ¼ 0 have no solution within the real number system. Because these
equations were found to have a meaningful place in the mathematical structures being built, various
mathematicians of the late nineteenth and early twentieth centuries developed an extended system of
numbers in which there were solutions. The new system became known as the complex number system.
It includes the real number system as a subset.
We can consider a complex number as having the form a þ bi, where a and b are real numbers called
p ffiffiffiffiffiffiffi
1 is called the imaginary unit. Two complex numbers a þ bi
the real and imaginary parts, and i ¼
and c þ di are equal if and only if a ¼ c and b ¼ d.We can consider real numbers as a subset of the set
of complex numbers with b ¼ 0. The complex number 0 þ 0i corresponds to the real number 0.
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
a þ b . The complex conjugate of
The absolute value or modulus of a þ bi is defined as ja þ bij¼
a þ bi is defined as a bi. The complex conjugate of the complex number z is often indicated by z or z .
z

The set of complex numbers obeys rules 1 through 9 of Page 2, and thus constitutes a field. In
performing operations with complex numbers, we can operate as in the algebra of real numbers, replac-
2
ing i by 1 when it occurs. Inequalities for complex numbers are not defined.