Numbers and Functions

1. What is a Number?

Understanding the concept of a number is fundamental to calculus and mathematical analysis. 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 starting from 1 and increasing without bound: 1, 2, 3, 4, …

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

• Negative Integers: These are the opposites of positive integers: …, -4, -3, -2, -1.

• Integers: The set of all positive integers, zero, and negative integers together form the set of integers or "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 in the form , where and are integers and .

• Examples: , , , , etc.

• Properties:

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

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

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.

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

• Table: Approximating

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

• Algebraic Properties: We assume irrational numbers like exist and that they obey the same algebraic rules as rational numbers (e.g., ).

1.4 Decimal Expansions

Numbers can be represented as decimal expansions. Rational numbers have either terminating or repeating decimals, while irrational numbers have non-repeating, non-terminating decimals.

• Example:

• Rational numbers: Their decimal expansions either terminate or repeat.

• Irrational numbers: Their decimal expansions neither terminate nor repeat.

Additional info: The concept of real numbers includes both rational and irrational numbers. In calculus, we work with real numbers, which can be thought of as points on an infinite number line, each corresponding to a unique decimal expansion.