Numbers and Functions

1. What is a Number?

Understanding the types and properties of numbers is fundamental to the study of calculus. This section introduces the main classes of numbers used in mathematics, especially those relevant to functions of a real variable.

1.1 Different Kinds of Numbers

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

• Zero: The integer 0 is included as a distinct number.

• 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").

• Rational Numbers: Numbers that can be expressed as the ratio of two integers (with a nonzero denominator) are called rational numbers. Examples include:

  Note: Every integer is also a rational number (e.g., , ).

• Arithmetic Operations: The set of rational numbers is closed under addition, subtraction, multiplication, and (except for division by zero) division. That is, performing these operations on rational numbers yields another rational number.

1.2 Beyond Rational Numbers: Irrational Numbers

Not all numbers are rational. The first well-known example is the square root of 2. It is impossible to find rational numbers such that:

• , or equivalently,

This means is not rational; it is an irrational number.

Example: Approximating

To estimate , consider values of between 1 and 2 and compute :

This shows that lies between 1.4 and 1.5.

• Existence and Properties: We assume the existence of such numbers (irrational numbers) and that they obey the same algebraic rules as rational numbers (e.g., ).

• Infinite Decimal Expansions: In calculus, numbers are often represented as infinite decimal expansions, which allows us to work with both rational and irrational numbers.

1.3 Decimal Representation of Rational Numbers

Rational numbers can be written as decimal fractions. For example:

Summary Table: Types of Numbers

Additional info: The text introduces the concept of real numbers as those that can be represented by infinite decimal expansions, which includes both rational and irrational numbers. This foundational understanding is essential for the study of functions and calculus.