Numbers and Functions

1. What is a Number?

This section introduces the foundational concept of numbers, which is essential for understanding calculus. It explores different types of numbers, their properties, and the motivation for extending the number system beyond rational numbers.

1.1 Different Kinds of Numbers

Numbers can be classified into several categories based on their properties and how they are constructed.

• Positive Integers: The simplest numbers, also called natural numbers, are 1, 2, 3, 4, …

• Zero: The integer 0 is included in the set of whole numbers.

• Negative Integers: These are …, -4, -3, -2, -1.

• Integers: The set of all positive integers, zero, and negative integers forms the set of integers (or "whole numbers").

1.2 Rational Numbers

Rational numbers are formed by dividing one integer by another nonzero integer. These are also known as fractions.

• Definition: A rational number is any number that can be written as , where and are integers and .

• Examples:

  • Any integer can be written as , so all integers are rational numbers.

• Properties:

  • You can add, subtract, multiply, and divide (except by zero) any pair of rational numbers, and the result is always a rational number.

1.3 The Existence of Irrational Numbers

Not all numbers are rational. The discovery of irrational numbers (numbers that cannot be written as a ratio of integers) was a significant development in mathematics.

• Example: The square root of 2 () is not a rational number. There is no fraction such that , or equivalently, .

• Numerical Investigation: By checking values between 1 and 2, we see:

  x
  1.2	1.44
  1.3	1.69
  1.4	1.96 < 2
  1.5	2.25 > 2
  1.6	2.56

  This suggests there is a number between 1.4 and 1.5 whose square is exactly 2, which we call .

• Algebraic Properties: We assume these new numbers (irrationals) obey the same algebraic rules as rationals, such as .

1.4 Decimal Expansions

To handle both rational and irrational numbers, we often represent numbers as infinite decimal expansions.

• Example: Certain fractions can be written as decimal fractions:

• Rational Numbers: Their decimal expansions either terminate or repeat.

• Irrational Numbers: Their decimal expansions are non-terminating and non-repeating.

Summary Table: Types of Numbers

Key Point: The real number system includes both rational and irrational numbers, and is often represented using infinite decimal expansions. This system forms the foundation for calculus and the study of functions of a real variable.