BackNumbers and Functions: Foundations for Calculus
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Numbers and Functions
1. What is a Number?
This section introduces the foundational concept of numbers, which is essential for understanding calculus and functions of a real variable. We explore the different types of numbers, their properties, and how they are represented.
1.1. Different Kinds of Numbers
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 included as a unique integer.
Negative Integers: These are ..., -4, -3, -2, -1.
Together, the positive integers, zero, and negative integers form the set of integers (or "whole numbers").
Rational Numbers: Numbers that can be expressed as the ratio of two integers (with a nonzero denominator) are called rational numbers. Examples include:
Positive Rational Numbers | Negative Rational Numbers |
|---|---|
By definition, any integer is also a rational number (e.g., ).
Closure Properties: The set of rational numbers is closed under addition, subtraction, multiplication, and division (except division by zero). This means that performing these operations on rational numbers always yields another rational number.
1.2. Beyond Rational Numbers: 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., has no integer solutions for and (with ).
1.2 | 1.44 |
1.3 | 1.69 |
1.4 | 1.96 < 2 |
1.5 | 2.25 > 2 |
1.6 | 2.56 |
This table shows that increases as increases, and there is a value between 1.4 and 1.5 where . This value is , which is not rational.
Infinite Decimal Expansions: Irrational numbers cannot be written as terminating or repeating decimals. Instead, they have non-repeating, non-terminating decimal expansions.
1.3. Decimal Representation of Numbers
Many numbers, especially rational numbers, can be represented as decimal fractions. For example:
This shows that different fractions can have the same decimal representation.
Summary Table: Types of Numbers
Type | Examples | Decimal Representation |
|---|---|---|
Integer | -3, 0, 4 | Terminating |
Rational | , | Terminating or repeating |
Irrational | , | Non-terminating, non-repeating |
Key Points:
Numbers can be classified as integers, rational numbers, or irrational numbers.
Rational numbers can be written as fractions and have terminating or repeating decimals.
Irrational numbers cannot be written as fractions and have non-repeating, non-terminating decimals.
Understanding these types of numbers is essential for studying functions and calculus.
Additional info: The text alludes to the importance of understanding the real number system for calculus, as it forms the basis for defining limits, continuity, and differentiability in later chapters.
