Page 10 - Schaum's Outline of Theory and Problems of Advanced Calculus

Numbers

Mathematics has its own language with numbers as the alphabet. The language is given structure
with the aid of connective symbols, rules of operation, and a rigorous mode of thought (logic). These
concepts, which previously were explored in elementary mathematics courses such as geometry, algebra,
and calculus, are reviewed in the following paragraphs.

SETS
Fundamental in mathematics is the concept of a set, class,or collection of objects having specified
characteristics. For example, we speak of the set of all university professors, the set of all letters
A; B; C; D; ... ; Z of the English alphabet, and so on. The individual objects of the set are called
members or elements. Any part of a set is called a subset of the given set, e.g., A, B, C is a subset of
A; B; C; D; ... ; Z. The set consisting of no elements is called the empty set or null set.

REAL NUMBERS
The following types of numbers are already familiar to the student:
1. Natural numbers 1; 2; 3; 4; ... ; also called positive integers, are used in counting members of a
set. The symbols varied with the times, e.g., the Romans used I, II, III, IV, . . . The sum a þ b
and product a b or ab of any two natural numbers a and b is also a natural number. This is
often expressed by saying that the set of natural numbers is closed under the operations of
addition and multiplication,or satisfies the closure property with respect to these operations.
2. Negative integers and zero denoted by 1; 2; 3; ... and 0, respectively, arose to permit solu-
tions of equations such as x þ b ¼ a, where a and b are any natural numbers. This leads to the
operation of subtraction,or inverse of addition,and we write x ¼ a b.
The set of positive and negative integers and zero is called the set of integers.
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3. Rational numbers or fractions such as , ,. .. arose to permit solutions of equations such as
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bx ¼ a for all integers a and b, where b 6¼ 0. This leads to the operation of division,or inverse of
multiplication,and we write x ¼ a=b or a b where a is the numerator and b the denominator.
The set of integers is a subset of the rational numbers, since integers correspond to rational
numbers where b ¼ 1.
p ffiffiffi
4. Irrational numbers such as 2 and are numbers which are not rational, i.e., they cannot be
expressed as a=b (called the quotient of a and b), where a and b are integers and b 6¼ 0.
The set of rational and irrational numbers is called the set of real numbers.
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