Did you think Olaf would forget that you remain a disgrace to the human species?!
Olaf is here to make your existence marginally less useless!
We have already seen velocity defined as $\frac{x_1 - x_2}{t_1 - t_2}$ as $t_1 - t_2$ becomes smaller and smaller.
We shall now present the necessary mathematics in order to make such a wishy washy wooly and wimpy statement rigorous enough to withstand a barbarian's bachelor's party!
We have already discussed the real numbers $\mathbb{R}$. Today I will introduce the subset $\mathbb{N} \subset \mathbb{R}$ defined by:
$\mathbb{N} = \left\{1,2,3,4,..\right\}$
We shall refer to $\mathbb{N}$ as the set of natural numbers.
Notice that $n \in \mathbb{N} \implies n + 1 \in \mathbb{N}$. This leads to a rich application in mathematics, computer science and mating, as it is the underlying fact behind the techniques of mathematical induction and recursion. However, these will not detain us at this stage.
Consider now an infinite sequence of real numbers, labbeled: $a_1, a_2, a_3,...,a_n,...$. For convenience, the entire sequence may be referred to as $(a_n)_{n \in \mathbb{N}}$
consider the sequence defined by $a_n = \frac{1}{n}, n\in \mathbb{N}$
As $n$ is chosen to be larger and larger, $\frac{1}{n}$ will become smaller.
Indeed, for any real number $\epsilon > 0$, I can find an $N \in \mathbb{N}$ such that $\frac{1}{N} \leq \epsilon$ In this sense, we say that the sequence $(\frac{1}{n})_{n\in \mathbb{N}}$ appraches 0 as $n$ approaches infinity.
We shall generelise this notion, giving the definition of a limit of a sequence:
A sequence $(a_n)_{n\in \mathbb{N}}$ is said to converge to $a$ if for every $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $|a_n - a| \leq \epsilon$ for all $n \geq N$. $a$ is also known as the limit of the sequence.
We can write this as $\underset{n \rightarrow \infty}{\lim} a_n = a$ or $a_n \underset{n \rightarrow \infty}{\longrightarrow} a$.
if $(b_n)_{n \in \mathbb{N}}$ is some other sequence which converges to $b$, then then we have the following result, whose proof would be up to you to produce! FOR ALL BARBARIANS PRODUCE THEIR OWN PROOFS!
$\underset{n \rightarrow \infty} {\lim} (a_n + b_n) = \underset{n \rightarrow \infty} {\lim} a_n + \underset{n \rightarrow \infty} {\lim} b_n = a+b$
$\underset{n \rightarrow \infty} {\lim} (a_n \times b_n) = \underset{n \rightarrow \infty} {\lim} a_n \times \underset{n \rightarrow \infty} {\lim} b_n = ab$
$\underset{n \rightarrow \infty} {\lim} (\frac{a_n}{b_n}) = \frac{\underset{n \rightarrow \infty} {\lim} a_n} {\underset{n \rightarrow \infty} {\lim} b_n} = \frac{a}{b}$ if $b \neq 0$
We are now equipped with the proper weapons to attack calculus!
But you now require rest. GIVE ME 1000 PUSHUPS NOW, WEAKLINGS
Thursday, February 3, 2011
Tuesday, February 1, 2011
Momentum, or, Why I will always beat you
At the risk of causing injury to you, I will now introduce some physics. If you find yourself unable to lift the required weight; resign yourself to a life of tedium and serfdom, for the free and great life of a barbarian is far beyond your abilities.
Every barbarian must have an acute understanding of physics, be it while fighting in glorious hand to hand combat, sailing your longship to ripe pillaging fields or calculating the trajectory a system undergoes through state space during a scattering process whilst remaining Lorentz -covariant.
Without being marred with philosophy, I shall make a few statements which are self evident (AGAIN, LOGIC!) and then construct the foundations of what may be called "Newtonian Mechanics", after one of the great barbarians: Newton the Madaxe (although most sources erroneously call him Isaac Newton)
Space can be considered as an ordered collection of numbers $(x,y,z)$ such that $x,y,z \in \mathbb{R}$. Upon fixing the 3 numbers representing some specific location in space, such as yourself, all other points may be unambiguously defined.
A simple way to do this systematically is to decide on a left and right, forward and back, up and down while the numbers $x,y$ and $z$ would correspond to a distance from the initial fixed point along those directions.
We can now know where everything is! However, bitter experience has shown us that things do not necessarily remain where they are right now. So therefore, we require a measure of time, $t \in \mathbb{R}$.
We also require another quantity mass, $m \in \mathbb{R}$, which encompasses "how much" of something there is.
From these we can extract the necessary physics.
An object of mass $m$ can therefore be labelled by a succession of different positions $(x,y,z)$ for every time $t$, with time constantly increasing.
Define the velocity $\bold{v} = (\dot{x}, \dot{y}, \dot{z})$ of an object by the change of positions divided by the change in times, as the change in time goes smaller and smaller.
In other words:
$\dot{x} = \frac{x_1 - x_2}{t_1 - t_2}$ where $x_1$ is the value of $x$ at time $t_1$, and so on, while $t_1$ is 'almost' equal to $t_2$
$\dot{y}, \dot{z}$ are defined in a similar way.
A quantity of particular importance in physics is momentum. It is defined as
$m\bold{v} = (m \times x, m \times y, m\times z)$. From now on, for convenience, the symbol $\times$ will be omitted. You should know enough by now when we mean multiplication because of the context! If not, don't worry. Not everyone can be a barbarian, someone has to be pillaged!
The great discovery of early barbarians was that if you have another body, and it hits another body, if you add the two momentums before and after the collision, it does not change.
Already, you can see the applications such a discovery has on axe fighting!
Keep at it, wimps!
Every barbarian must have an acute understanding of physics, be it while fighting in glorious hand to hand combat, sailing your longship to ripe pillaging fields or calculating the trajectory a system undergoes through state space during a scattering process whilst remaining Lorentz -covariant.
Without being marred with philosophy, I shall make a few statements which are self evident (AGAIN, LOGIC!) and then construct the foundations of what may be called "Newtonian Mechanics", after one of the great barbarians: Newton the Madaxe (although most sources erroneously call him Isaac Newton)
Space can be considered as an ordered collection of numbers $(x,y,z)$ such that $x,y,z \in \mathbb{R}$. Upon fixing the 3 numbers representing some specific location in space, such as yourself, all other points may be unambiguously defined.
A simple way to do this systematically is to decide on a left and right, forward and back, up and down while the numbers $x,y$ and $z$ would correspond to a distance from the initial fixed point along those directions.
We can now know where everything is! However, bitter experience has shown us that things do not necessarily remain where they are right now. So therefore, we require a measure of time, $t \in \mathbb{R}$.
We also require another quantity mass, $m \in \mathbb{R}$, which encompasses "how much" of something there is.
From these we can extract the necessary physics.
An object of mass $m$ can therefore be labelled by a succession of different positions $(x,y,z)$ for every time $t$, with time constantly increasing.
Define the velocity $\bold{v} = (\dot{x}, \dot{y}, \dot{z})$ of an object by the change of positions divided by the change in times, as the change in time goes smaller and smaller.
In other words:
$\dot{x} = \frac{x_1 - x_2}{t_1 - t_2}$ where $x_1$ is the value of $x$ at time $t_1$, and so on, while $t_1$ is 'almost' equal to $t_2$
$\dot{y}, \dot{z}$ are defined in a similar way.
A quantity of particular importance in physics is momentum. It is defined as
$m\bold{v} = (m \times x, m \times y, m\times z)$. From now on, for convenience, the symbol $\times$ will be omitted. You should know enough by now when we mean multiplication because of the context! If not, don't worry. Not everyone can be a barbarian, someone has to be pillaged!
The great discovery of early barbarians was that if you have another body, and it hits another body, if you add the two momentums before and after the collision, it does not change.
Already, you can see the applications such a discovery has on axe fighting!
Keep at it, wimps!
Monday, January 31, 2011
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!
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!
Sunday, January 30, 2011
Logic Primer
As I was saying, logic is skill, and if you have skill you can certainly kill. I shall demonstrate to you the first simple techniques that will put you on track to being as great a barbarian as me! Unlikely as it may be.
What is logic?
Philosophers can spend their entire careers debating this question while I steal their lands, their women, burn their books and defecate over their reputation. A true barbarian would simply take logic to be a system of rules by which one can conclude that a statement is true given than some other statements are also true.
Now this is all a little high brow and pompous. Indeed I shall need to armwrestle Surtr after this blog in order to regain some of my barbarity, but without it you would be little more than an unarmed peasant facing a horde of angry Vikings!
Ahhh, good times. Hint: I am not a peasant.
Let us begin. I make my first statement, assumed to be true:
The blood of my enemies is red. (S1)
What statements can I conclude from it? Can I conclude that the people I pillage are weak?
NO!
Even though experience verifies it to be true, its conclusion is not possible given my initial statement (S1)
One statement which can be concluded from (S1) is:
The blood of my enemies is not blue (S2)
Why? Because if something is red, IT CANNOT BE BLUE! Notice, that from (S2) I cannot then conclude (S1), for if something is not blue, then it may be green, yellow, purple, cyan, magenta, grapefruit and so on, none of which are red.
Notice, that given any statement P implying any other statement Q, one may not necessarily then conclude that Q then implies P.
This is a fundamental error that many make, and it makes Olaf ANGRY! Many a barbarian have found themselves slain after showing this lack of basic technique.
I need to go breathe in some mead now.
Until next time, wimps!
What is logic?
Philosophers can spend their entire careers debating this question while I steal their lands, their women, burn their books and defecate over their reputation. A true barbarian would simply take logic to be a system of rules by which one can conclude that a statement is true given than some other statements are also true.
Now this is all a little high brow and pompous. Indeed I shall need to armwrestle Surtr after this blog in order to regain some of my barbarity, but without it you would be little more than an unarmed peasant facing a horde of angry Vikings!
Ahhh, good times. Hint: I am not a peasant.
Let us begin. I make my first statement, assumed to be true:
The blood of my enemies is red. (S1)
What statements can I conclude from it? Can I conclude that the people I pillage are weak?
NO!
Even though experience verifies it to be true, its conclusion is not possible given my initial statement (S1)
One statement which can be concluded from (S1) is:
The blood of my enemies is not blue (S2)
Why? Because if something is red, IT CANNOT BE BLUE! Notice, that from (S2) I cannot then conclude (S1), for if something is not blue, then it may be green, yellow, purple, cyan, magenta, grapefruit and so on, none of which are red.
Notice, that given any statement P implying any other statement Q, one may not necessarily then conclude that Q then implies P.
This is a fundamental error that many make, and it makes Olaf ANGRY! Many a barbarian have found themselves slain after showing this lack of basic technique.
I need to go breathe in some mead now.
Until next time, wimps!
Sound the Gjallarhorn
Behold and quake in fear, for Olaf is among you! Olaf is very disappointed in you all, Olaf will teach you about the world and how to be a great Barbarian!
Physics and Mathematics, like the axe and spear, are things no true Viking blooded barbarian would be caught without! Far better to be naked in the frozen north, hunting polar bears, than to be without them. Logic is the skill with which you wield your axe in a crazed bloodbath! Before you whelps can be trusted with weapons, I'd best make sure you don't accidentally decapitate yourself while puncturing your lungs with them even though it would be very amusing to see.
WAAAAAAGH
Physics and Mathematics, like the axe and spear, are things no true Viking blooded barbarian would be caught without! Far better to be naked in the frozen north, hunting polar bears, than to be without them. Logic is the skill with which you wield your axe in a crazed bloodbath! Before you whelps can be trusted with weapons, I'd best make sure you don't accidentally decapitate yourself while puncturing your lungs with them even though it would be very amusing to see.
WAAAAAAGH
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