Numbers and Functions

1. What is a Number?

This section introduces the foundational concept of numbers, which is essential for understanding functions of a real variable in calculus. We explore different types of numbers, their properties, and the motivation for extending the number system beyond rational numbers.

1.1. Different Kinds of Numbers

Numbers can be classified into several types, each with unique properties and uses in mathematics.

• 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, positive integers, zero, and negative integers form the set of integers (or "whole numbers").

• Rational Numbers: Numbers that can be expressed as the ratio of two integers (with a nonzero denominator). Examples include:

  • Negative rational numbers:

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

• Closure Properties: The set of rational numbers is closed under addition, subtraction, multiplication, and division (except division by zero).

1.2. Beyond Rational Numbers: Irrational Numbers

Not all numbers can be written as a ratio of integers. The first well-known example is the square root of 2.

• Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. For example, is irrational because there are no integers and (with ) such that: , i.e.,

• Historically, the discovery of irrational numbers dates back to the ancient Greeks.

Example: Approximating

To estimate , consider squaring numbers between 1 and 2:

This table shows that is between 1.4 and 1.5.

• We assume the existence of such numbers (like ) to fill the "gaps" between rational numbers.

• These numbers, together with rational numbers, form the set of real numbers.

• It is important to ask whether these new numbers obey the same algebraic rules as rational numbers (e.g., ).

1.3. Decimal Expansions

To avoid the complexities of defining numbers rigorously, we often represent numbers as infinite decimal expansions.

• Decimal Representation: Many rational numbers can be written as terminating or repeating decimals.

• For example:

• Some numbers (irrational numbers) have non-terminating, non-repeating decimal expansions.

Summary Table: Types of Numbers

Key Point: The real number system includes both rational and irrational numbers, and is fundamental for calculus.

Additional info: In calculus, understanding the properties of real numbers is essential for defining limits, continuity, and functions of a real variable.