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 known as the 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:

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

• 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 first well-known example is the square root of 2. The ancient Greeks proved that there is no rational number whose square is exactly 2. In other words, there are no integers and (with ) such that:

• , i.e.,

Numbers like are called irrational numbers because they cannot be written as a ratio of two integers.

Example: Approximating

To estimate , consider the following table:

This shows that is between 1.4 and 1.5.

• Key Point: There are infinitely many irrational numbers, and they fill the gaps between rational numbers on the number line.

• Algebraic Properties: A natural question is whether irrational numbers obey the same algebraic rules as rational numbers (e.g., commutativity, associativity). In fact, they do, as part of the real numbers.

1.3 Decimal Expansions

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

Rational numbers have decimal expansions that either terminate or repeat. Irrational numbers have non-terminating, non-repeating decimal expansions.

Summary Table: Types of Numbers

Additional info: The real numbers consist of all rational and irrational numbers. In calculus, we primarily work with real numbers and their properties, especially as they relate to functions and limits.