BackNumbers and Functions: Foundations for Calculus
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Numbers and Functions
1. What is a Number?
This section introduces the foundational concept of numbers, which is essential for understanding calculus. It explores different types of numbers, their properties, and the motivation for expanding the number system beyond rational numbers.
1.1. Different Kinds of Numbers
Numbers can be classified into several types, each with unique properties and uses in mathematics.
Positive Integers: These are the counting numbers: 1, 2, 3, 4, …
Zero: The integer 0 is included in the set of whole numbers.
Negative Integers: These are the opposites of positive integers: …, -4, -3, -2, -1.
Together, the positive integers, zero, and negative integers form the set of integers (also called whole numbers).
Rational Numbers: Numbers that can be expressed as the ratio of two integers (with a nonzero denominator). These are also called fractions. Examples include:
Fraction | Decimal Representation |
|---|---|
0.5 | |
0.666… | |
0.75 | |
11.16 |
By definition, any integer is also a rational number (for example, and ).
Operations on Rational Numbers: You can add, subtract, multiply, and divide any pair of rational numbers (except division by zero), and the result will always be a rational number.
1.2. Beyond Rational Numbers: Irrational Numbers
Not all numbers are rational. Some numbers cannot be written as the ratio of two integers. The classic example is the square root of 2.
Irrational Numbers: Numbers that cannot be expressed as , where and are integers and . For example, is irrational because there are no integers and such that .
Historical Note: The ancient Greeks proved that is irrational.
Example: Approximating
To estimate , consider values of between 1 and 2 and compute :
1.2 | 1.44 |
1.3 | 1.69 |
1.4 | 1.96 |
1.5 | 2.25 |
1.6 | 2.56 |
Since and , must be between 1.4 and 1.5.
Infinite Decimal Expansions: Many numbers, including irrationals, are best represented as infinite decimal expansions. For example,
1.3. Summary Table: Types of Numbers
Type | Definition | Examples |
|---|---|---|
Integer | Whole numbers, positive, negative, or zero | -3, 0, 4 |
Rational Number | Can be written as , | , , |
Irrational Number | Cannot be written as | , , |
Key Point: The real number system includes both rational and irrational numbers, and is often represented using infinite decimal expansions.
Additional info: In calculus, understanding the different types of numbers is crucial because many concepts (such as limits, continuity, and differentiability) rely on the properties of real numbers, including both rationals and irrationals.
