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

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 as a whole 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. Some numbers, such as the square root of 2, cannot be written as a ratio of two integers.

• Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers are called irrational numbers.

• Example: is irrational. There is no fraction (with ) such that , or equivalently, .

1.3 Existence and Properties of Irrational Numbers

Although cannot be written as a fraction, we can approximate it by checking values between 1 and 2:

This table shows that must be a number between 1.4 and 1.5. Such numbers are assumed to exist and are called irrational numbers. They obey the same algebraic rules as rational numbers (e.g., commutativity, associativity).

1.4 Decimal Expansions

To work with both rational and irrational numbers, we often use their decimal expansions. Rational numbers can be represented as terminating or repeating decimals.

• Example:

In calculus, we often think of numbers as infinite decimal expansions, which allows us to handle both rational and irrational numbers in a unified way.

Additional info: In later sections, the concept of real numbers will be formalized further, and the properties of functions defined on these numbers will be explored.