BackNumbers and Functions: Foundations of 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 and functions of a real variable. We explore different types of numbers, their properties, and the motivation for extending the number system beyond rational numbers.
1.1. Different Kinds of Numbers
Numbers can be classified into several types, each with distinct properties and uses in mathematics.
Positive Integers: The simplest numbers, also known as natural numbers, are 1, 2, 3, 4, ...
Zero: The number 0 is included as a whole number.
Negative Integers: These are ..., -4, -3, -2, -1. Together with positive integers and zero, they form the set of integers (or "whole numbers").
Rational Numbers
Rational numbers are formed by dividing one integer by another nonzero integer. They can be written as fractions:
Examples:
Negative examples:
By definition, any integer (including zero) is also a rational number (e.g., ).
Properties of Rational Numbers:
You can add, subtract, multiply, and divide any pair of rational numbers (except division by zero), and the result is always another rational number.
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 is no fraction such that , i.e., .
Approximating Irrational Numbers
Although is not rational, we can approximate it by checking squares of numbers between 1 and 2:
x | |
|---|---|
1.2 | 1.44 |
1.3 | 1.69 |
1.4 | 1.96 < 2 |
1.5 | 2.25 > 2 |
1.6 | 2.56 |
This suggests that is a number between 1.4 and 1.5. We assume such numbers exist and call them irrational numbers.
Infinite Decimal Expansions
To address the existence and properties of irrational numbers, we often represent numbers as infinite decimal expansions. This approach allows us to work with both rational and irrational numbers in a unified way.
Example: Certain fractions can be represented as decimal fractions:
Fraction | Decimal Expansion |
|---|---|
11.16 | |
11.16 |
Key Takeaways:
Numbers can be classified as integers, rational numbers, and irrational numbers.
Rational numbers can be expressed as fractions; irrational numbers cannot.
Infinite decimal expansions provide a way to represent all real numbers, both rational and irrational.
Example Application: Approximating using decimal expansions helps us understand the concept of irrational numbers and their place in the real number system.
Additional info: In advanced mathematics, the set of real numbers is rigorously defined using concepts such as Dedekind cuts or Cauchy sequences, but for calculus, thinking of real numbers as infinite decimals is sufficient.
