Numbers and Functions

1. What is a Number?

Understanding the different types 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

• Positive Integers: The simplest numbers, also called 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). These include fractions such as:

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

• Properties of Rational Numbers:

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

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

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 cannot be written as a ratio of two integers.

• Key Fact: There is no rational number such that , or equivalently, .

• Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. Examples include , , and .

1.3 Approximating Irrational Numbers

Although is not rational, it can be approximated by rational numbers. By squaring numbers between 1 and 2, we can find values that get closer to 2:

This table shows that is between 1.4 and 1.5.

• Assumption: We assume the existence of such numbers (like ) even though they are not rational.

• Algebraic Properties: These new numbers (irrationals) are assumed to obey the same algebraic rules as rational numbers (e.g., commutativity, associativity).

1.4 Decimal Expansions

To handle both rational and irrational numbers, we often represent numbers as infinite decimal expansions.

• Example: The rational number can be written as a 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 domain for most functions studied in the course.