Some Mathematics

Greetings, wimps. I hope you have been practicing your logic. As every barbarian knows, practice and exercise are the key to building strong muscles, and strong muscles are big!

AND BIGGER IS BETTER!

Quite a few of you are so pathetically weak as to be frightened of numbers, the stepping stone to great mathematics. In this blog, I will slay your fears, or they will slay you. Either way, my Valkyrie is keeping my bed warm, and I do not intend to keep her waiting.

I shall introduce the runes required to properly understand what I will say. If you think this is somehow a sign of weakness, it is because you yourself are weak and I could crush you with just a FART!

ODIN IMPALED HIMSELF ON HIS SPEAR AND TOOK OUT ONE OF HIS EYES SO THAT WE MAY BE ABLE TO USE RUNES!

Behold $\mathbb{R}$, this rune, or symbol, represents the collection of all the so called "real numbers". Do not bother yourself with what other kinds of numbers there may be just yet, for the strain will cause your slow and painful death. Content yourself that any number you can concieve of at this moment is a real number, or as true barbarians would say: $x \in \mathbb{R}$, where $x$ is any number you can think of.

Recalling your basic logic (YES, LOGIC IS IMPORTANT IN MATHEMATICS YOU WHELP!), we shall start with statements assumed to be true. In mathematics, such statements are known as axioms. You may disagree with the choice of axioms, and you are free to select new ones. Indeed, mathematicians do this on a daily basis, else they starve. And in doing so, you are no longer discussing real numbers.

But we are barbarians, we please no-one but ourselves, such considerations are beneath us!

Even the most lowly weakling can comprehend the following:

In all that follows $a,b,c \in \mathbb{R}$

1. There exists an operation called addition, represented by $+$, such that for any $a,b$; $a+b = c$ for some $c$

2. $a+b = b+a$. This property means that the operation, in this case additon, is commutative.

3. $(a + b) + c$ = $a + (b +c)$. Such a property means that the operation is associative.

4. There exists a number zero, whose symbol is $0$ such that $a + 0 = a$ for all $a$ there exists $b$ such that $a + b = 0$. $b$ is known as the additive inverse of $a$, often written as $-a$ and vice versa.

You may now be having nightmares from your early school days. I laugh at your distress, and shall now give the axioms of multiplication.

5. For all $a,b$, $a \times b = c$ for some $c$

6. multiplication is both commutative and associative.

7. $a \times (b + c) = (a\times b) + (a\times c)$. That is to say multiplication is distributive over addition

8. There exists a number called one, represented by $1$, such that $a \times 1 = a$
and for any $a \neq 0$ there exists a $b$ such that $a \times b = 1$. $b$ is the multiplicative inverse of $a$, and vice versa.

Notice that the additive inverse of $a$ is equivalent to $-1 \times a = -a$
As a notational shorthand $a - b$ is taken to mean $a + (-b)$ and is sometimes called the difference of $a$ and $b$, or the subtraction of $b$ from $a$

These 8 axioms establish the real numbers as a(n algebriac) field. Do not confuse these as the places you'd usually go to when you want to pillage, for that would be stupid. When you have gained more muscle mass, the discussion of fields in mathematics may be tackled in fuller detail.

Notice now that we know that some numbers are larger than others, or at least that was what your least favourite teacher told you. We shall now define what we mean by order on $\mathbb{R}$

We define a subset, $P \subset \mathbb{R}$, called the positive set.
$P$ has the the following properties:

$a,b \in P \implies a+b \in P$
$a,b \in P \implies a \times b \in P$
Exactly one of the following is true:
$a \in P$ $a = 0$ $-a \in P$

We define an order, $<$.
$a < b$ if $b - a \in P$ Also $a \leq b$ if $a = b$ or $a < b$
Now we can define what we mean by an upperbound of some set $A \subset \mathbb{R}$: $b$ is an upperbound of $A$ if $x \leq b$ for all $x \in A$

One final axiom is now required:

If $A \subset \mathbb{R}$ and $A$ has an upperbound in $\mathbb{R}$, it also has a least upperbound in $\mathbb{R}$ In order words: if for all $x \in A$ there exists some $a$ such that $x \leq a$ then there exists a number b such that $ x \leq b \leq a$ for any upperbound $a$ of $P$

This axiom is known as the Completeness Axiom, and is required when more technical arguments are met.

Remember to practice, or you will forever remain weak and unloved!