BackNumbers and Functions: Foundations for Calculus
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Numbers and Functions
1. What is a Number?
Understanding the types and properties of numbers is fundamental to calculus. This section introduces the main categories of numbers used in mathematics and explores their properties and representations.
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 integer 0 is included as a unique number representing 'nothing.'
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").
Rational Numbers: Numbers that can be expressed as the ratio of two integers (with a nonzero denominator). Examples include:
Fraction | Decimal Representation |
|---|---|
$\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{2}{4}$, $\frac{3}{4}$, $\frac{4}{3}$, … | 0.5, 0.333..., 0.666..., 0.25, 0.5, 0.75, 1.333..., … |
By definition, any integer is also a rational number (e.g., $3 = \frac{3}{1}$, $0 = \frac{0}{1}$).
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 a 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 written as a ratio of two integers. These are called irrational numbers.
Example: $\sqrt{2}$ is irrational because there are no integers $m$ and $n$ (with $n \neq 0$) such that $\left(\frac{m}{n}\right)^2 = 2$, or equivalently, $m^2 = 2n^2$.
Historical Note: The existence of irrational numbers was known since ancient Greek times.
1.3 Approximating Irrational Numbers
Although irrational numbers cannot be written as exact fractions, they can be approximated by decimals or by checking values between rational numbers.
$x$ | $x^2$ |
|---|---|
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 $\sqrt{2}$ is between 1.4 and 1.5, since $1.4^2 = 1.96 < 2$ and $1.5^2 = 2.25 > 2$.
1.4 Infinite Decimal Expansions
To handle both rational and irrational numbers, we often represent numbers as infinite decimal expansions. For example, rational numbers can be written as terminating or repeating decimals:
Example: $\frac{279}{25} = \frac{1116}{100} = 11.16$
For irrational numbers, the decimal expansion neither terminates nor repeats (e.g., $\sqrt{2} \approx 1.4142135\ldots$).
Summary Table: Types of Numbers
Type | Examples | Decimal Expansion |
|---|---|---|
Integer | -3, 0, 7 | Terminating |
Rational | $\frac{1}{2}$, $\frac{3}{4}$ | Terminating or repeating |
Irrational | $\sqrt{2}$, $\pi$ | Non-terminating, non-repeating |
Key Points:
Numbers can be classified as integers, rational, or irrational.
Rational numbers can be written as fractions and have terminating or repeating decimals.
Irrational numbers cannot be written as fractions and have non-terminating, non-repeating decimals.
All these numbers are used in calculus, especially when dealing with functions of a real variable.
