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 numbersand 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,

• Note: A rational number is not the same as a fraction! π/3 or 0.25

a

b

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 between them).

• The integer part of a positive rational number t is denoted by

This is the greatest integer less than or equal to t

2143

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.

What about the following power?

• 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.

sv