Numbers and Functions

1. What is a Number?

This section introduces the foundational concept of numbers, which is essential for understanding calculus and functions of a real variable. We explore different types of numbers, their properties, and how they are represented.

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 natural numbers, are 1, 2, 3, 4, ...

• Zero: The number 0 is included as a whole number.

• Negative Integers: These are ..., -4, -3, -2, -1. Together with positive integers and zero, they form 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. They include all integers and fractions.

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

• Examples: , , , , , , , ...

• Negative Rational Numbers: , , , , , , , ...

• Properties:

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

  • All integers are rational numbers (e.g., , ).

1.3. Irrational Numbers

Not all numbers are rational. Some numbers cannot be written as a ratio of two integers. The first well-known example is the square root of 2.

• Definition: An irrational number is a number that cannot be expressed as for any integers and (with ).

• Example: is irrational because there is no fraction such that , i.e., has no integer solutions.

• Approximation: By checking values between 1 and 2:

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

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

• Properties: Irrational numbers, like , are assumed to exist and are essential for a complete understanding of the real number system.

1.4. Infinite Decimal Expansions

To address the challenge of defining numbers precisely, we often represent numbers as infinite decimal expansions. This approach is especially useful for irrational numbers, which cannot be written as terminating or repeating decimals.

• Rational Numbers as Decimals: Some fractions can be written as terminating or repeating decimals.

• Examples:

• Irrational Numbers: Their decimal expansions are non-terminating and non-repeating (e.g., ).

Summary Table: Types of Numbers

Key Points:

• Numbers can be classified as integers, rational, or irrational.

• Rational numbers can be written as fractions and have terminating or repeating decimals.

• Irrational numbers cannot be written as fractions and have non-terminating, non-repeating decimals.

• All these numbers are essential for understanding the real number system, which is the foundation for calculus.