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Chapter 2
Real Numbers and
Complex Numbers
What is a number?
• What qualifies a mathematical object to be
identified as some type of number?
Exactly what basic properties objects called
‘numbers’ should possess can be a subject of
debate: is a telephone number a number?
• One answer comes by introducing the idea of
a number system .
What is a number system?
• A number system is a set of objects, together
with operations (+, x, others?) and relations (=
and perhaps order) that satisfy some
predetermined properties (commutativity,
associativity, etc.)
• Chapter 2 examines the numbers that up the
rational, real and complex numbers systems,
starting from their most familiar geometric
representations: the real number line and the
complex plane.
2.1.1 Rational numbers
and Irrational numbers
• Defn: A number is rational if and only if (iff) it
can be written as the indicated quotient of
two integers: a/b, a ÷ b, a
b
• Note: A rational number is not the same as a
fraction! π/3 or 0.25
What makes rational numbers so nice?
Theorem 2.1
a. The set Q of rational numbers is closed under
addition, subtraction and multiplication.
b. The set Q – {0} of non-zero rational numbers
is closed under division.
Also, the algorithms we have for operations with
fractions make rational numbers easy to add ,
subtract, multiply and divide.
Estimating rational numbers
• It is easy to estimate the value of a positive
rational number a/b if we write it as a mixed
number (the sum of an integer and a fraction
between 0 and 1 written with no space
2
between them). 14
3
• The integer part of a positive rational number
t is denoted by  t 
 
This is the greatest integer less than or equal to t
Division Algorithm
• When we divide one integer by another, what
guarantees that our quotient and remainder
are unique? The Division Algorithm.
• Theorem 5.3 If a and b are integers with b > 0,
then there exist unique integers q and r such
that a = bq + r, and 0 ≤ r < b.
• (or a/b = q + r/b, with 0 ≤ r < b)
Irrational Numbers
• Defn. An irrational number is a real number
which is not a rational number.
• They show up everywhere—in roots,
logarithms, and trig functions to name a few.
In fact we will show later that there are more
irrational numbers than rational ones!
A classic example of indirect proof
Theorem 2.2 Let n be a positive integer. Then
the square root of n is either an integer or it is
irrational.
This theorem is equivalent to asserting that if p
is not a perfect square, then x² - p = 0 has no
rational solutions. (A special case of the
Rational Root Theorem.)
Generating Irrational Numbers
Theorem 2.3 Let s be any non-zero rational number
and v any irrational number. Then s+v, s-v, sv and
s/v are irrational numbers.
s
What about the following power?
v
• Sums, differences, products and quotients of
irrational numbers may be either rational or
irrational, so the set I of irrational numbers is not
closed under any of the arithmetic operations.