Numbers and Functions

1. What is a Number?

1.1. Different Kinds of Numbers

Understanding the types of numbers is fundamental to calculus and higher mathematics. Numbers can be classified into several categories based on their properties and how they are constructed.

• Positive Integers: The simplest numbers, also known as natural numbers, are 1, 2, 3, 4, ...

• Zero: The number 0 is a unique integer that represents the absence of quantity.

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

• Integers: The set of all positive integers, zero, and negative integers forms the set of integers (or "whole numbers").

1.2. Rational Numbers

Rational numbers are numbers that can be expressed as the ratio 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: , , , , , , , ...

• Negative Rational Numbers: , , , , , , , ...

• 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., , ).

1.3. Irrational Numbers

Not all numbers are rational. Some numbers cannot be written as a ratio of two integers. The first well-known example is the square root of 2.

• Definition: An irrational number is a number that cannot be expressed as for any integers and (with ).

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

• Historical Note: The Greeks proved that is irrational.

1.4. Approximating Irrational Numbers

Although irrational numbers cannot be written as exact fractions, they can be approximated by rational numbers. For example, to estimate , we can check the squares of numbers between 1 and 2:

From the table, we see that and , so is between 1.4 and 1.5.

1.5. Infinite Decimal Expansions

To handle both rational and irrational numbers, we often represent numbers as infinite decimal expansions. Rational numbers have repeating or terminating decimals, while irrational numbers have non-repeating, non-terminating decimals.

• Example (Terminating Decimal):

• Example (Non-Terminating Decimal): (decimal expansion continues without repeating)

Key Point: In calculus, we often work with real numbers, which include both rational and irrational numbers, and are typically represented as infinite decimals.

Additional info: The real numbers form the foundation for calculus, as they allow us to define limits, continuity, and other key concepts.