Numbers and Functions

1. What is a Number?

This section introduces the concept of numbers, which are fundamental to calculus and all of mathematics. Understanding the different types of numbers and their properties is essential for studying functions and calculus.

1.1 Different Kinds of Numbers

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

• Zero: The number 0 is included as a special integer.

• 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").

1.2 Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. These include fractions and all integers (since any integer n can be written as n/1).

• Definition: A rational number is any number that can be written as , where m and n are integers and n ≠ 0.

• Examples: , , , , etc.

• 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.

  • Zero is a rational number (e.g., ).

1.3 Irrational Numbers

Not all numbers are rational. Some numbers, such as the square root of 2, cannot be written as a ratio of two integers. These are called irrational numbers.

• Example: is irrational because there are no integers m and n such that , or equivalently, .

• Historical Note: The Greeks discovered the existence of irrational numbers when they proved that cannot be written as a fraction.

• Approximation: By checking values between 1 and 2, we see that and , so is between 1.4 and 1.5.

Summary: The table above shows that is between 1.4 and 1.5, since .

1.4 Infinite Decimal Expansions

To address the question of what numbers like "really" are, we often think of numbers as infinite decimal expansions. This approach allows us to represent both rational and irrational numbers as decimals, some of which terminate or repeat (for rationals), and some of which do not (for irrationals).

• Example:

• Rational numbers have decimal expansions that either terminate or repeat.

• Irrational numbers have decimal expansions that neither terminate nor repeat.

Additional info: The real numbers consist of both rational and irrational numbers, and they form the basis for calculus. Understanding their properties is essential for defining limits, continuity, and functions of a real variable.