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").

1.2 Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. These include:

• Fractions: Examples include , , , , , , , …

• Properties:

  • Any integer is a rational number (e.g., , ).

  • The set of rational numbers is closed under addition, subtraction, multiplication, and (except by zero) division.

1.3 Irrational Numbers

Not all numbers are rational. Some numbers, such as the square root of 2, cannot be written as a ratio of two integers. These are called irrational numbers.

• Example: is irrational because there are no integers and (with ) such that , or equivalently, .

• Historical Note: The existence of irrational numbers was known since the time of the ancient Greeks.

1.4 Existence and Properties of Irrational Numbers

To understand the existence of numbers like , we can examine the squares of numbers between 1 and 2:

This table shows that there is a number between 1.4 and 1.5 whose square is exactly 2. We call this number , even though it cannot be written as a rational number.

1.5 Decimal Expansions

One way to represent numbers, especially rational numbers, is through their decimal expansions. For example:

Decimal expansions provide a practical way to work with both rational and irrational numbers in calculations.

Additional info: In calculus, understanding the distinction between rational and irrational numbers, as well as the concept of decimal expansions, is essential for grasping the real number system and the behavior of functions defined on real numbers.