Numbers and Functions

1. What is a Number?

Understanding the concept of a number is fundamental to calculus and higher mathematics. This section introduces the different types of numbers and their properties, laying the groundwork for the study of 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 known as the natural numbers, are 1, 2, 3, 4, …

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

• Negative Integers: These are the additive inverses of the positive integers: …, -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 quotient of two integers (with a nonzero denominator) are called rational numbers. Examples include:

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

• Rational numbers are closed under addition, subtraction, multiplication, and (except by zero) division.

1.2. Beyond Rational Numbers: Irrational Numbers

Not all numbers are rational. The discovery of irrational numbers was a significant development in mathematics. The classic example is the square root of 2.

• Definition: An irrational number is a number that cannot be expressed as a ratio of two integers.

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

Table: Approximating

By squaring numbers between 1 and 2, we can see that lies between 1.4 and 1.5:

This suggests that is a number between 1.4 and 1.5 whose square is exactly 2.

• Algebraic Properties: A natural question is whether irrational numbers obey the same algebraic rules as rational numbers (e.g., commutativity, associativity).

• Existence: The existence of such numbers is assumed, and they are essential for a complete understanding of the real number system.

1.3. Decimal Expansions

One practical way to represent numbers, especially rational numbers, is through their decimal expansions.

• Decimal Representation: Any rational number can be written as a decimal fraction. For example:

• Some decimals terminate (like 0.5 or 11.16), while others repeat (like 0.333... for ).

• Irrational numbers have non-terminating, non-repeating decimal expansions (e.g., ).

Summary Table: Types of Numbers

Example: (repeating decimal), (non-repeating decimal)

Additional info: The real numbers, which include both rational and irrational numbers, form the basis for calculus and analysis. They can be thought of as all possible infinite decimal expansions, providing a complete and continuous number line.