Explanation of Mathematics

$\infty$ Explanation of Mathematics $\infty$

Greek "Apeiron":
a- without
perion- end/limit/boundary

Latin: "Infinitas":
in-"not"
finis-"end"
tas - "state of being"

'Infinite' means 'having no limit or end', 'boundless', 'unlimited', 'endless'.

(Oppy, Graham, et al, 2021)

The Size of Infinity

Infinite sets of numbers are often categorized as "countably infinite" or "uncountably infinite".

Sets are considered to be countably infinite if they can be "put into one-to-one correspondence with the natural numbers". Countable sets also have an ordering, where the elements can be counted. The set of all natural numbers is an example of a countable set.

Sets are considered to be uncountably infinite if they "are too vast to be put into a one-to-one correspondence with the natural numbers". Uncountably infinite sets have more elements than the natural numbers, and cannot be ordered in a specific way. An example of an uncountably infinite set is the set of numbers between $0$ and $1$ ($\{0,1\}$).

(GeeksforGeeks 2024)

The "Hilbert Hotel Paradox" provides an excellent visualization of the differences between countable and uncountable infinities.

Counting the Infinite | The Notation of Infinity

Set notation was created to formally define different infinite sets of numbers. Some common sets of numbers are:

$\mathbb{Z}$: The set of Integers
The set of nonnegative integers together with the negatives of the positive integers.
$\{...-3,-2,-1,0,1,2,3 ...\}$

$\mathbb{Q}$: The set of Rational Numbers
Numbers which are ratios of integers
$\{p/q : p,q \in \mathbb{Z}\}$

$\mathbb{I}$: The set of Irrational Numbers
Real numbers that cannot be represented as simple fractions
Irrational numbers cannot be represented as a ratio of two integers

$\mathbb{R}$: The set of Real Numbers
The union of rational and irrational numbers
$\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$

$\mathbb{C}$: The set of Complex Numbers
This set contains real numbers and imaginary numbers
$\mathbb{C}= \{a+bi : a,b \in \mathbb{R}\}$ where $i^2 = -1$

(Libretexts 2021)

Theorems, Laws and Properties

Addition with Infinity:
$x + \infty = \infty$
$\infty + \infty = \infty$

Subtraction with Infinity:
$\infty - x = \infty$
$\infty - \infty =$ indeterminate

Multiplication with Infinity:
$x \cdot \infty = \infty$ (where $x \neq 0$)
$\infty \cdot \infty = \infty$

Powers of Infinity:
$\infty^0 =$ indeterminate
$x^\infty = \infty$ (for $x>1$)
$x^\infty = 0$ (for $0 < x < 1$)
$0^\infty = 0$
$\infty^\infty = \infty$
$1^\infty =$ indeterminate

Division with Infinity:
$\infty / \infty$ = indeterminate
$x/ \infty = 0$
$0/ \infty = 0$
$\infty/ \infty =$ indeterminate

(GeeksforGeeks 2024)

The Limits of Infinity

Evaluating limits as $x$ approaches infinity, we will observe the function behavior. This is as follows: $$\lim_{x \to \infty} f(x)$$ What is the function value doing as $x$ approaches infinity? Or in other words, what is the function value doing as $x$ gets really really big? $$\lim_{x \to -\infty} f(x)$$ What is the function value doing as $x$ approaches negative infinity? Or in other words, what is the function value doing as $x$ gets really really small? Consider the following limit, $$\lim_{x \to \infty} \frac{f(x)}{g(x)}$$ Where $f(x)$ and $g(x)$ are both polynomials. We can look at the order (the highest exponents) of the two polynomials to help us evaluate the limit. There are three possible cases:

Case 1. The order of $f(x) >$ the order of $g(x)$
- $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)} = \pm \infty$
- The numerator is approaching a larger number faster than the denominator, thus our fraction gets bigger and bigger in either the positive or negative direction as x approaches infinity.
- To determine if the limit is $+\infty$ or $-\infty$, observe the signs of the leading coefficients.
Case 2: The order of $f(x) <$ the order of $g(x)$
- $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 $
- The denominator is approaching a larger number faster than the numerator, thus our fraction gets smaller and smaller as x approaches infinity.
Case 3: The order of $f(x) =$ the order of $g(x)$
- $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{\text{The leading coefficient of } f(x)}{\text{The leading coefficient of } g(x)}$

(Strang, et al., 2018)(Strang, et al., 2022)