1. Assume that the statement to be proved is false.
2. Show that this assumption leads to a logical contradiction.
3. Conclude that the original statement is true.

Show that the product of any nonzero rational number and any irrational number is irrational.

1. Suppose that the product of the two numbers is not irrational i.e. rational.
2. Show that there exists a real number s and a rational number r such that rs is rational.

    rs = p/q p,q ∈ ℤ and q ≠ 0
     r = a/b a,b ∈ ℤ and b ≠ 0

    rs = p/q = a/b x s

Therefore s = pb/aq so s is rational. But s is irrational. Contradiction.

3. Therefore the original statement is true.

These three terms are related to set theory and describe properties of certain sets, more precisely a list of ordered pairs. They are often described in very difficult to understand terms but don't need to be.

Consider the following binary relation R on a set A.

R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(4,4)}

Reflexive means that you have to find the form (x,x) in the elements in R. In this case R is NOT Reflexive because there is no (3,3) in the list. Visually it is equivalent to having a loop at every vertex.

Symmetric means that if you have a pair (x,y) in a list, there also has to be (y,x) in that list. The above R is symmetric. Visually this means that two vertices are connected with two one way edges, one from x to y and the other from y to x.

Transitive means that if you have a pair (x,y) and a pair (y,z), you MUST have a pair (x,z) in your list. The above R is NOT Transitive as we have (3,4) and (4,3) but no (3,3). Visually if there is an edge connecting two vertices, and another going from the second to a third vertex, then there has to be one from the first to the third.

I'll put pictures up here one day to illustrate the point.

I came across this one when building a computer for a friend of mine. It kept us up most of the night figuring out as it is a pretty random problem and it makes a "bsod".

The error message is something like:

   A problem has been detected and Windows has been shutdown to prevent damage to your computer.

   STOP: 0X0000007E (0XC0000005, 0XF748E0BF, 0XF78DA208, 0XF78D9F08)

   pci.sys - Address F748E0BF base at F7487000, Datestamp 3b7d855c

This error occurs because early versions of the OS don't recognize pci-express cards properly. You need XP: Service Pack 2 or greater on your installation in order to get this to work. If you don't have an original disk with sp2 on it then you will have to make one with your old XP disk. This is called slipstreaming. Download the latest Service Pack from wherever and follow the instructions here to create a brand new installation for free!

While you are at it, check out nLite. It is a nifty program that lets you chop and choose the components of your XP install disk. It also creates the .iso for you. No more Windows Media Player if you don't want it.

This function seems pretty straightforward but I had some nasty little bugs with it in my code. Everyone I talked to said, "Yeah strtok is a weird one..." but never gave me an answer as to why.

Declaration:

    char *strtok(char *str1, const char *str2);

At a basic non technical level, str2 is a list of characters you don't want. Strtok checks each character in str1 until it finds one it doesn't want. It cuts str1 there and returns the word or sentence up until that cut.

The coding weirdness lies in the fact that it internally saves the pointer iterating over str1. This means that in subsequent calls, say in a loop for example, it starts from the last known position, NOT the beginning again. This can really throw out any outer loops; the ones used in fgets for example.

See strtok on the C Reference Page for a more detailed description of exactly how it works.

Find, if they exist, integers x and y which satisfy 91x + 221y = 676.

First apply the Euclidean Algorithm to 221 and 91 to find gcd(221, 91).

221	= 91 · 2 + 39	(equation 1)
91	= 39 · 2 + 13	(equation 2)
39	= 13 · 3 + 0	(equation 3)

Thus, gcd(221, 91) = 13. Since 13 | 676, we know that a solution exists to the Linear Diophantine equation 91x + 221y = 676.

Now work backwards through the equations of the Euclidean Algorithm to find a solution.

13	= 91 - 39 · 2	(from equation 2)
	= 91 - [221 - 91 · 2] · 2	(from equation 1)
	= 91 · 5 - 221 · 2

Therefore, 91(5) + 221(-2) = 13.

Multiplying both sides of the equation by 52 we get: 91(260) + 221(-104) = 676.

Hence x = 260 and y = -104 is one solution to the linear Diophantine equation 91x + 221y = 676.

Show that whenever you add two rational numbers you always get a rational answer.

Formally: ∀ r,s ∈ ℚ, (r + s) ∈ ℚ

Let r = a/b and s = c/d     a,b,c,d ∈ ℤ and b,d ≠ 0

Then r + s	= a/b + c/d
	= (ad + bc) / bd

(ad + bc) ∈ ℤ by closure of multiplication and addition, also bd ≠ 0

∴ ∀ r,s ∈ ℚ, (r + s) ∈ ℚ

In english: For all rational numbers r and s, r + s is also a rational.

Tip: Even an integer is a rational e.g. 5 = 5/1.

Right so because I hardly ever use this and it seems to be a pretty good place for it, I have decided to start posting my adventures around the computer / networking / math world. Despite the fact that this degree forces me to work within a ridiculously limiting framework (which is unnatural for me) it is starting to open up. Maths is not cryptic anymore, I know the difference between throughput, bandwidth and bus speed.

Moreover, I can't count the number of times that a forum or journal post has saved my butt. You know when there is this really annoying glitch in your system? Be it either an inconsistency that sets off your aspergers or something that causes your system not to work entirely. You sit up all night in a state of mild anxiety not knowing what could be wrong. Then you start hoping (rather selfishly) that someone else has not only had this problem before, but has written about it. I have had that too many times, and the only thing more selfish than hoping for someone else's misery is fixing your own and not letting anyone else know HOW YOU DID IT!

This then will be my attempt at absolution. I shall try to update as much as possible but my patience for typing really low. Stick it out mate, stick it out...

What's the word for it when you just want to not see something you love again for a while because you have been looking at it too much? I have been studying for the last week and I am really sick of it. There is only so many theorems, bases, transformations, series and vectors that one can realistically learn in 6 months. This is compounded by the fact that my brain really doesn't properly wrap around something until I understand it properly. Each thing in maths is built on a theorem before it so when you have been out of practice for a while learning one thing is actually learning five.

That said I think I have a handle on it now, which is good because the exam is in two days. I did just this morning have a calculus "moment of clarity" as I woke up. I suddenly understood perfectly the chain rule and many of it's implications. This led on to the fundamental theorem and the reason why Newton chose to represent it the way he did.

Oh no! This was supposed to be a break. Well I guess I better get back to it. Not long now.

W3rd! It is going to be great. Mason and I have been planning some ill sh*t for that weekend. It's pretty good because the feeling of being cooped up and snowed under is starting to become a real problem. Whoever decides to come down be prepared! Are we all staying basically in the city? Alan and I have found this hostel very close so we may have some backpackers join us on the escapades.

Furthermore, does anyone know where I can get my hands on original copies of Gridlock cd's in Australia? I have been looking around and they come from Amazon etc. which I am not too excited about...

Somebody save me!

I am here til 6:00pm and the time is dragging like a gypsy mule.  To make matters even more ass, my eyes are bleary from staring at a huge spreadsheet all afternoon.  If this is the dream, then I want reality; some soup, my girlfriend back and a large screen tv with movies all day.

I probably should take this time to mention how awesome she is.  Just throwing it out there.  Although it is likely that anyone who knows her already knows that.

...

Awesome

Yeah so because Carmen has gone overseas three days ago I find myself with all this wierd and confusing free time. Hence the new post.

As an update, I have been working at CITEC now for the last half year with Mason and Alan. This has been an excellent place and I have now managed to convince them to let me go to uni as well!

I was doing uni last semester but only a token 1 subject because I couldn't go to the lectures at all. This made it pretty full on (I have never done a subject without going to lectures) but I managed to pull through with a 7 haha!

This semester is MATH1051 and INFS1200 which are both fairly cool but I think the math subject will be really hard so I am a little worried. It makes me pine for the good old days of Alexander the Great and the First Crusade, things I am very comfortable with. Ah well, I guess if you never do anything that challenges you then you never get better.

Speaking of which, I have gone back to Gymnastics and because I now have an actual teacher I think I am getting better than I ever was. I am working on my twisting somersaults on the tramp so eventually I hope to do a roundoff, backflip, full twisting back somersault on the floor.

So all in all it has been a really great last 6 months and apart from missing Carmen like crazy (whilst simultaneously being very proud of her) I am really happy!

Here it is, the end of 2006. So many things have happened. Quit my job, went overseas, came back to job, went to uni, quit my job, got a new one...

So here is the alternative to myspace for all those people who I give this address to... Hello!

Well I'm back in sunny London. I have spent the last three days visiting my friends up in Leeds.

Leeds is a lot different to London. As my friend Michael was saying it is large enough to be a city but it still has a sense of community. London is so big that there is very little of that. Leeds also has a more distinctive style than the mish mash of the capital. They use sa particular style of sandstone that seems to go black over time causing this really cool melting/burning effect. I think that is what inspires the multitude of goths that live here. I'm serious there are literally throngs of them. We went to the Corn Exchange which is this huge domed building in the centre of Leeds and just outside was a large gathering of gothlings. Michael was saying that this is where the Sisters of Mercy used to hang out and it is probablyone of the seminal historical places for the beginning of the movement.

They also have the Royal Armouries in Leeds which are brilliant! They were moved from London out there a while ago and everything is free to view probably because it is still a public space. They gave live demonstrations and historical re-enactments every couple of hours and the armour thay have there (even the modern replicas) are second to none. I saw two performers in full plate re-enact the Talhoffer manual fighting 2 metres from me. There were also talks about the one day truce in WW1 where everyone played soccer and sang "Stille Nacht Heiliger Nacht".

The next day we went out to the Yorkshire dales which, despite being a very rainy day, were pretty spectacular. During the drive we also stopped off at the Bronte Parsonnage Museum where Emily, Charlotte anf Anne wrote their novels. I can see why their subject matter was so bleak given their situation. One of their windows faces a graveyard so go figure. We also stopped to go caving for a bit, utterly unexpected but fantastic.

The music here is awesome by the way, it has the best bar in England.

Did I mention how brilliant this place is?

Every now and then when I am looking on the internet for my next place to stay I see some airline deal which is a similar price so I go for it. I got the Icelandair holiday deal for not that much more than it would have cost me to stay in a hostel and they have put me up in a 4 star hotel! I have my own gym, sauna, spa, free huge breakfast, city views and free shuttle buses everywhere I want to go.

I went for a walk in downtown Reykjavik yesterday as I have become accostomed to doing. I think now thats it is actually the best way to see a city because you don´t just zoom past everything and you get to know your way around really quickly. Normally I look for clues as to why each city has been set up in a particular place but for Iceland everywhere is a great place. Besides which apparently Ingolfur Arnarson, the founder of the city in 874, used the age old viking technique of throwing the backs of your throne out of the ship and settling where they wash up. It took them 3 years to find it apparently.

I am going to a lake today that is geothermally heated. An entire lake!

Well I´m finally here!

This place is exactly how I imagined it and more. Not only does it rival Geneva for beauty but it has a really sweet culture. This truly is the land of fire and ice. I took a bus trip around and we went through volcanic lave fields into glacial runoffs and then to geothermally heated lakes.

It´s wierd to think that a people known historically for their violence have spawned such a beautiful and peaceful culture. Everyone here is so relaxed and serene that it is infectious. This peace is not the kind we are used to. I often associate peace with a sombre, often morose outlook (I think this is the English culture coming through) so when I see it here I am heartened. The whole grin and bear it attitude is not here in the slightest, and thats coming from a land with much harsher weather. Theres a saying here "If the weather is bad in Iceland wait 10 minutes" and that is no exaggeration. Yesterday started with a blizzard and ended with sunshine.

Another thing here is that after coming from the places I have been I don´t feel uncomfortable in the slightest. France was demanding, Italy fast paced, Greece chaotic, England had too many rules, but in Iceland there is none of that. It is just a really sweet place. One thing though, who would´ve thought the arctic circle would be so cold at the end of winter? Damn. I´m used to it now though and I like it heaps. The air is really crisp. It is apparently not as cold as other places on this longitude because of the Gulf stream so thats really something.

Oh yeah, I always though that it was a bit snooty to not introduce words into your language because they are a foreign source like France does, but being here I realise my error. In Iceland everyone speaks english so there is no need to import the word because everyone understands it anyways. English is good enough by itself at appropriating foreign words.

Yeah and its a perfect place to store 50 Glitter Boys so shut up!

To be brutally honest I really wasn´t expecting too much of this good old town. I had few reasons to go there and even less funding. Having said that I am glad because it has totally taken me by surprise. London (apart from New York though I haven´t been there) is the quintessential city of cities. Everything about it says a big "We are the right royal centre of the world and we know it".

Quite obviously built to be imposing, this city has the real empire feel, complete with "The Bill" style cops and fines (I saw 3 fines being metered out in the first day!). There is definitely something about this place. I am staying at a place called the Astor Museum Hostel which is right across the road from the British Museum. That is really handy for me as I have been spending some time there, but you quickly realise that there are literally too many things to do at any given moment. Compare and starkly contrast with Brisbane, where if there is anything to do you go out of your way to make it a big occasion. I have met a few old friends here and the night life is smashing.

Another thing, the people here are SO amazingly dressed. This is definitely the Goth capital of the world. I went out to the Camden markets and I couldn´t move between the people wanting to buy purple nail polish and hair extensions. There is a specific store that just blew me away. I know I have probably been told about it 100 times but you really have to see it. Everyone reading this probably knows Cyberdog, with its flourescent lights and kick ass clubbing gear.

Another cool thing is that it makes a really good base for Australians wanting to travel to the continent. If you can make it here, you can get a visa to work and it is just an easyJet or Ryanair flight to most places of interest. Really worth noting.

In case you haven´t guessed I am quite taken with London.

I know I people told me not to, but I just looked at the the pictures of Innisfail.

I am sitting here writing this after two of the most important days of my life. I went into the Mediterranean for the first time. It has been a focal point of my life. There are no words.

I arrived in Athens on Friday 17th. Since then I have visited the Acropolis, Thebes, Delphi, Salamis and the most important two, the Mediterranean and Thermopylae. It was not the intended plan to make it to many of the ancient sites on the trip but since I split it in two it had become an epic journey. Today I did both Thermopylae and Delphi. If you know how far that is from Athens you'll appreciate how difficult that was. For some crazy reason I decided to rent a car for the trip. This was good because it got me there and back in one day but driving in Greece is the most stressful thing I have done in years.

Breaking it down:

1. NOBODY drives in the lines and they even drive on the opposite side of the road in the city.
2. People pass on the highway over double lines as if they weren't there.
3. They consistently go over 160 and tailgate you at that speed. (What is that? No room for error)
4. I got a huge ass ticket for crossing the highway. There were no signs (I checked) and it was like the least illegal thing I saw all day and I get the ticket for it!?! I think the cop wanted me to bribe him but I didn't know how to approach that so he got pissed off.
5. Cars sometimes don't actually use the road. They drive on the very far edge of the road so if you want to overtake them illegally (see note 2) you can and it is only really dangerous (as opposed to bullshit dangerous).
6. The cars are backward and everyone drives on the right side of the road.

But here are the benefits.

I went to Thermopylae first. If anyone knows me they know that this battle, and the campaigns of Alexander (which, coincidentally follow a similar route as the first crusade. More on that later.) Nowdays it is just a hill but Leonidas' monument is still there. This battle, even though it ended in defeat, is probably the single greatest example of courage and determination in history. The 300 Spartans were well aware of their last stand, indeed when Leonidas' wife asked him what to do when he left, he said "Marry a good man, have good children". I am not going to go into the details now, you can find them out later, but suffice it to say that this was a VERY important place for both myself and for anyone who has ever heard of them. It was emotional.



Ω ξείν', αγγέλλειν Λακεδαιμονίοις ότι τήδε
κείμεθα τοις κείνων ρήμασι πειθόμενοι.

Oh foreigner, give a message to the Lacedaemonians,
that here lie we, their words obeying.



Then I went south to Delphi, the home of Apollo. Once you get there it is wasy to see why they chose such a place. It is specatcular! It is a small village tucked away in the central mountains of greece but it was a focal point for all of the influential ancient city states. Each of them had a treasury here to donate to the gods and anyone who was anyond consulted the oracle before making important decisions. The temple of Apollo overlooks clifranges as far as you can see and if you make one wrong step youll fall hundreds of metres down. This is definitely the place for a god. I think, now having seen it, that the site is extremely ancient and that probably a version of Apollo did originate from here. If you see it, you'll understand. It is impossible to put into words.

I see now what travel does to the mind, especially seeing the places you have loved from afar for so long. It is good I guess that it took me so long. Now I have images, smells, feelings and sound to associate with the romance of history.

I am weary.

Today has been a very special day. I remember my first ancient history lessons at high school and my teacher giving me a latin copy of Julius Caesar's "De Bello Gallico" as a present for doing well. I still have it. It wasn't that hard because I really liked Caesar.

Anyways today is the Ides of March. Julius Caesar was stabbed to death on this calendar day in 44BC, 2051 years ago. I went to his grave and spent a bit of time with the man. Nowdays his grave is just a mound of rocks but it was important for me to know that his body was buried right in front of me. I spent a lot of time there. It is good to just sit and relax rather than furiously wandering around trying to find things (I did that on the first day). I didn't really know what to say to the man, I mean, how could I, as a 21st century person, do anything to allieviate grief or effect him whatsoever. I came to this. It seems that as a mark of respect for the man I have promised to read his book, cover to cover, in his own language. Now he writes in very good high latin so I understand some, but I want to get it right. I know its not much but it is difficult and I am pretty sure he would have appreciated it. I also thought that I was the only one getting emotional about it but it as I was walking past his statue onthe Via Dei Forum Imperii I saw lots of flowers at his feet. It seems others are thinking of him too. Even after all these years he still inspires people.

Oh I also saw the Pope talk and the grave of the Apostle St. Peter in his basilica.

So much has happenned.

I hardly ever get time to get on these internets but ok here goes... I caught the train down to Roma to find the hostel I booked was significantly better than the previous ones I have been staying at. Compared to Paris it is like staying in a hotel. It doesn't stop there. I am at least able to understand people here because of Innisfail and I have become addicted to cappucinos in the morning (There you go Kai, but I can't help it, they are really good here). Trastavere also has some of the best Pizza in the world. I can officially vouch for that. It's not smothered in cheese like the american style we get at home. It is actually really healthy too.

I have been to most of the Important sites of Rome. Been into the Collosseum, around the Forum (where i spend most of my time now), Palatine hill, Circus Maximus, Spanish Steps, Trevi Fountain, Pantheon, Titus and Constantines arches, etc etc etc. I have only just realised that the ancient and medieval sites are all within walking distance because the towns back then were significantly smaller, duh.

I have had many specific moments here. Tomorrow I am going to Julius Caesar's grave for the Ides of March (totally unplanned, just really lucky). Most importantly I have settled into this travel thing much better so I am much more relaxed now. It really looks now like, both financially and practically speaking, that I will have to split this journey up into two chunks. There is just too much to see in Europe and not enough time / cash to do it all in one go. I can also work out those tricky middle eastern visas more before I go. It is awesome though because it means I shall be able to explore around here more thoroughly. Best of all however, is that it means I can go to Iceland, a place I have always want to go to but didn't know if I ever would. I am REALLY excited about that. Something tells me it will be an unforgettable highlight.

This trip has taught me so much already, mostly that travelling like this is not the demon I always worried it was. I should have known already, having travelled all around Australia, but now I know I can do it I am certain I shall factor many more trips into the plan. It is most definitely a matter of planning for me. I am not the spontaneous, polar person I always thought I was. Plans. I think my brain is waking back up.

I'm also going to the Vatican tomorrow to get my journey blessed by the Pope, just like they did. Him and I are like this; X.

Ciao

Oh yeah, they have SPQR written on the drainpipes here, thats teamwork.