Numbers and Functions

1. What is a Number?

Understanding the concept of a number is fundamental to calculus and higher mathematics. Numbers are classified into various types based on their properties and how they are constructed.

1.1 Different Kinds of Numbers

Numbers can be grouped into several categories, each with distinct characteristics and uses in mathematics.

• Positive Integers: These are the counting numbers: 1, 2, 3, 4, …

• Zero: The integer 0 serves as the additive identity.

• Negative Integers: These are numbers less than zero: …, -4, -3, -2, -1.

• Integers: The set of all positive integers, zero, and negative integers. Integers are sometimes called "whole numbers."

1.2 Rational Numbers

Rational numbers are numbers that can be expressed as the quotient 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: , , , , ,

• 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., ).

• Decimal Representation: Rational numbers can be represented as terminating or repeating decimals. For example:

1.3 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.

• Definition: An irrational number is a number that cannot be written as , where and are integers.

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

• Properties:

  • Irrational numbers have non-terminating, non-repeating decimal expansions.

  • They cannot be written as exact fractions.

1.4 Infinite Decimal Expansions

To address the existence and properties of irrational numbers, mathematicians often define real numbers as infinite decimal expansions. This approach allows for a precise description of both rational and irrational numbers.

• Rational numbers: Have decimal expansions that either terminate or repeat.

• Irrational numbers: Have decimal expansions that neither terminate nor repeat.

1.5 Example: Approximating

To find a number whose square is 2, we can test values between 1 and 2:

This table shows that is between 1.4 and 1.5, but it cannot be written exactly as a fraction.

Additional info: The distinction between rational and irrational numbers is foundational for calculus, as the real number system includes both types and is essential for defining limits, continuity, and functions.