Numbers and Functions

1. What is a Number?

Understanding the concept of a number is fundamental to calculus and all of mathematics. This section introduces the different types of numbers and explores their properties and representations.

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 number 0 is included as a special integer.

• 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. These include fractions such as:

  By definition, any integer is also a rational number (e.g., ), and zero is a rational number ().

• Arithmetic Operations: The set of rational numbers is closed under addition, subtraction, multiplication, and division (except division by zero). 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 expressed 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 ancient Greek times.

1.3 Approximating Irrational Numbers

Although irrational numbers cannot be written as exact fractions, they can be approximated by rational numbers. For example, to estimate , we can check the squares of numbers between 1 and 2:

This table shows that is between 1.4 and 1.5.

• Key Question: How do we know there really is a number whose square is exactly 2? How many such numbers exist? Do these numbers follow the same algebraic rules as rational numbers?

• Mathematical Assumption: We assume the existence of such numbers (irrational numbers) and denote them using symbols like .

1.4 Decimal Expansions

To better understand numbers, especially rational numbers, we often represent them as decimal expansions.

• Example: The fraction can be written as a decimal:

• Rational Numbers as Decimals: Every rational number can be written as a terminating or repeating decimal.

Summary Table: Types of Numbers

Additional info: In calculus, understanding the real number system (including both rational and irrational numbers) is essential, as it forms the foundation for defining limits, continuity, and functions of a real variable.