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Offline jim360

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three60's Short introduction to... corner
« on: February 13, 2009, 11:37:15 AM »
I'm just starting this thread off with the sample I posted in General board. The next one should be arriving sometime next week, after it receives some peer-reviewing. Hope you enjoy it!!!

Suppose you're wanting to organise something secret. Then you'll want your friends to know, but not your enemies. Therefore, you'll have to find some way of achieving this. This is the aim of a code or cypher. There's a few types of these, each having their strengths and weaknesses. A code is usually simpler than a cypher. For example, one code might be: "Read each word directly after a punctuation mark."

"Hi there! My day has been quite a special one today! Friends from all over the country turned up, would you believe - Like the last time we spoke, to discuss plans for our holiday next Summer. Rob was a real pal as usual, the old rascal! Store up any prized possessions and he'll be there, tomorrow given the right form of transport!" 

Codes are usually hard to break because you
could read that message and not even notice anything unusual (obviously the example given is rather contrived, but with more time and thought, and a simpler message, the message can be more cunningly disguised), and they can be quite varied too - first letter of every second word, second of every third and so on...

The way to break them is normally to recognise that there IS a code to be broken and hope that you have some idea what it might be about, but these sorts of codes have been used for centuries quite successfully. The example I used was put to good use in World War Two, say. The weakness of codes is that initially you'd have to meet up to discuss how the code would work or produce a code book and if the enemies get a hold of this then you're in trouble.

Cyphers are, in a way, easier to break since they always follow a mathematically-expressible rule, but they can be made ridiculously complicated. The main problem with a cypher is that it's usually pretty obvious that the message being sent has been encyphered, since no sane person will send out a normal message "QTY RFH BMNCDOSVC".

The simplest form of cypher is a one-letter substitution, a nice example being "Caesar's Wheel" which is a rule that sends one letter to another in a standard order. e.g.

a b c  d e f g h i  j  k  l  m n o p q r  s  t  u v w x y  z
D E F G H I J K L M N O P Q R S T U V W X Y Z  A B C

so that "Iamhappy" becomes "LDPKDSSB" (it's usual to get rid of spaces since this makes the message just a little harder to decrypt; also conventionally the text to be coded is in small letters and the coded result is in capitals).

These cyphers are easy to break, just by trying all possible shifts until you get a coherent sentence.

Of course, another idea is to mix up the letters on my "bottom" line, so instead of  ABC...Z, I used a random sequence QDGEJFVOR...P or something. Somehow, though, you'd have to get this sequence across to your friends, so it's a better idea to try a simple codeword, e.g. DOGSABCEFHIJKLMNPQRTUVWXYZ" or the like, whihc is easy to remember.

The encrypted messages formed in this way are a right pain to break, but only if the messages are short. Long ones become almost a doddle, because there is a particular order of how many times each letter in the alphabet is likely to occur in a longish message. E is of course the most common, then T, A, O, I and so on right down to Q and Z. so you'd just count the most common letter in the coded message, replace it with E and so on, and it generally won't take long to crack the message, especially with a computer to try all the possibilities for you.

At this point, though, things get even more fun!! Instead of just one "sustitution cypher" you could have two of them running at once, so that "hello" could become "IDMKP" and notice that the double letter has disappeared. Or you could introduce special symbols for "the", "at", "to", "LL" and so on. At this point, breaking the code requires a lot of hard work and you'd need either to know the rule or have a lot of text encoded in the same way.

Codes and cyphers can be used for fun with friends, of course. Making your own secret code used, at least, to be a fun part of socialising for some people.

Well, that was just a short introduction to codes and cyphers, hope you enjoyed it!!


Next Time: The origins of mathematics.
« Last Edit: March 11, 2009, 07:40:06 AM by bfm_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: jim360's Short introduction to... corner
« Reply #1 on: February 20, 2009, 06:16:39 AM »
History of Mathematics.

You might wonder why I chose this subject to be first; after all, what could possibly be interesting about the stories behind how people came to work out that the cross-product of the tangent and normal is the binormal when taking the derviatives of a curve in R3 or some such incomprehensible mumbo-jumbo??

Good point. So I won't be talking about that, but rather how it all got started in the first place, which reveals something about human nature (ancient Greek human nature, anyway) and most importantly why mathematicians across history are the people who have most shaped the world.

For a full history, covering the beginning of maths until the present day, there are several books available, but by about the year 1700 the concepts discussed are well into university-level maths, so my *short* history ends in around 800AD, with many big gaps before then.

Mathematics begins, it seems, with problem solving. It's all very well saying, "I'd like a big city built here in my name, guys, get right on it!", but how will you know how many slaves/ materials/ days to finish/ food suppliers etc., will be needed to finish the job?

So early maths solves problems like:

- We're sending a 5,000-man army to conquer that city across the desert. How much water and food will we need, if it takes 10 days to get there and the siege will last at least 20 days?
- When will the Nile next flood?
- How should we divide up the late rich farmers' fields among his 12 sons, given that their shares should be not equal but in relation to their position in the family?

(yes, these examples all relate to Ancient Egypt, an important early centre of these sorts of things.)

And so on. The methods for solving these become remarkably advanced, but - and here's a thing that's rather interesting - we don't know how they worked out how to sovle these problems. This means that much of early maths is a set of instructions for how to solve a problem given these numbers, and very often the same basic problem can be restated ten or more times with different starting values (e.g 4,000 men and 200 camels instead of 5,000 men and just a single, solitary camel).

Dry stuff, so far, eh? Well done getting this far, but there's not that much interesting at the moment about maths of 4,000 years ago - except for a couple of things that suggest maybe a couple of people were doing Maths for fun.

Maths, you see, is about going beyond the real and entering the most abstract concepts in the world. So these problem-solvers, while very good, weren't really doing anything other than what they had to do. Given any
problem, they would look up the method and find you the answer, then sit back and count their wages.

Maths as we know it today starts, then, with the Greeks. For some reason, it seems that these guys just loved to argue with each other, for no reason at all other than they just enjoyed it [as explained by my History of Maths Lecturer). This eventually led to schools of argument being set up, e.g. the Sophists (hence "sophisticated"). One of
these blokes, a guy named Parmenides, steps up in around 600-530 BC and declares, "You can't have sure and certain knowledge of the world, because I can create uncertainty in it - see this stick? Well, *breaks stick" it's in two bits now - bet you didn't see that coming!!!" Or words to that effect.

Ouch.

The big deal of this is that you'd quite like to be able to know some things are absolutely true, otherwise it's not very easy to argue - you may be triumphantly hammering home your point when someone points out, "But you can't know all this so everything that comes after it is just a theory!!"

This is where mathematics steps in, in the form of Geometry. Here, the Greeks spend a long time essentially working on establishing facts in Geometry that you can't argue with no matter how hard you try - in other words, they try to prove things beyond all possible doubt. Out of this time you get Archimedes, Euclid, Plato and Aristotle (philosophers by trade, but Plato at least made some big steps in Maths on-the-side), and a couple of others no-one seems to have heard of - Eudoxus and Hippocrates of Chios. Look them up to see what they did, if you like. Over this time, the idea of something having been "proved" develops, from the simple, "Well man, it's obvious innit?" to the more formal language of today.

So mathematicians now started to churn out results that are actually rather incredible - Ptolemy much later managed to construct a model solar system that fitted the known facts and, though it's wrong, it's still an incredible achievement since, if my History of Maths professor is right, one of his predictions is that the Earth (which he places at the centre of the solar system) isn't quite at the centre of the Solar system. In other words, his model essentially predicts ellipses long before Kepler came along.

Essentially, the History of Maths can be explained like this: "Yes we CAN know certainly things about the world, see??", and they look for more and more things to prove even today.

The final part of this little tale is about why we have to thank the Middle East for our modern way of life:

 - All the stuff that came before them would be lost were it not for an incredible amount of time they spent collecting, preserving and combining it all into their libraries.
 - The Arabs invented modern algebra, without which we could go nowhere in maths.

Just to show how important algebra is:

Suppose you have a garden 20 metres long and 50 metres wide. Then you'd like a path around the edges of it to get to the shed at the far side. You'd like, of course, some garden left and the house owner wants about 800 square metres left after the path is put in. How much concrete will we need to make the path, given that it takes two bags per square metre and that the path will have the same width throughout?

Oo-erg??? After frantic measuring and alot of adding together rectangles you might find the answer, but:

width of paths is, area wanted to be left is ab - (a-2x)(b-2x) [ab=1000, ab
- (a-2x)(b-2x) = 800, a=50, b=20]

=> x2 - 35x + 50 = 0 => x = 1.5m

area of path = 2x((a)+(b-x)) [a=20, b=50] =>. area is 204m2

area*n [n=2] = 408 bags.


Four lines of working solving any problem of this nature. Brilliant. You may not be able to follow it, but you gotta admire how quick that was.

Hope you enjoyed that!! It's a bit of a crash course, and there's a lot more out there about this period in Maths and what follows.
« Last Edit: March 11, 2009, 07:43:14 AM by bfm_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline fLipSIDe

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Re: jim360's Short introduction to... corner (UPDATED)
« Reply #2 on: February 20, 2009, 08:27:08 AM »
All I can say is :liljawdrop:

 :hrmbig: :interesting: TGIF 

 :LOL:

Regards,

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Offline Bowser

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Re: jim360's Short introduction to... corner (UPDATED)
« Reply #3 on: February 20, 2009, 01:20:00 PM »
My head, oh my head!  :LOL: Very nicely done Jim! ;D

Offline BFM_Edison

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Re: jim360's Short introduction to... corner (UPDATED)
« Reply #4 on: February 20, 2009, 07:24:57 PM »
Topic suggestion: axioms. I honestly don't know too much as to how they first came about and how they were chosen and stuff like that.
52.87   60.07   46.40   72.73   68.23   55.10   98.27   84.73

Offline jim360

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Re: three360's Short introduction to... corner (UPDATED)
« Reply #5 on: February 20, 2009, 07:50:40 PM »
Interesting suggestion Edison, I'll get round to that right away - make it my next topic but one I think.

I hope you liked this one!!
« Last Edit: February 24, 2009, 07:07:13 AM by bfm_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline Vincitore

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Re: three60's Short introduction to... corner
« Reply #6 on: February 25, 2009, 03:26:51 AM »
Your sig is totally right, you made my head spin! :XD:

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #7 on: March 04, 2009, 06:40:54 AM »
Axioms and Proof

This topic, to a certain extent, logically follows from the last one, since once again the origins of axioms are Greek, and I’ll glance very briefly at proof since it too links in nicely with axioms.

So, without further ado...


How do you prove something to be true?? If you think about it, you can find a way to poke holes in many arguments – that’s what philosophers spend their time doing (and trying to create un-poke-hole-in-able arguments). This can get quite ridiculous. If I said to you that “2+2=4” you’d say, “Yeah, well duh…” but if I asked you to prove it, how would you go about doing so? Again, not an easy question. You might try:

-   “Here’s two pebbles, mate, and here’s another two, now count them all up and you get four – stop asking silly questions now!”
-   “What if those were dogs instead? Or apples? Or grains of dust in the Sahara?? What even is the number 2 and the number 4??”
-   “Well 2=1+1…”
-   “And what’s ‘1’ then??”
-   “Erm…”

A little silly, you might think, but the basic point is that somewhere along the line you’ve made some assumptions, such as 4= 1+1+1+1, and 2=1+1, where 1 is … what?

Now the problem basically comes here because, at some point along the line, you’ve got to decide that some things are just what they are defined to be, or else you could never get started. This is, basically, what an axiom is: it’s a fact that is accepted to be true without proof.

For example, one lot of axioms that sets up basic arithmetic:

i)   If x is a number then x = x
ii)   If x = y then y = x.
iii)   If x = y and y = z, then x = z
iv)   If a = b and a is a number, then so is b.
v)   x + 0 = x
vi)   a * 0 = 0

These apparently self-evident facts help save a lot of bother later, because they define what “=” means and what “0” is. You might think theses are self-evident, but the problem is that “=” is a special case of what’s called a relation written xRy. This applies to set Theory; don’t worry about it too much. Just appreciate that when you started learning to add, you accepted these facts as self-evident.

Equally, however, I could set up a new system as follows:

i)   x + 0 = 1/x
ii)   a * 0 = a-1

And work with those and, so long as I stuck rigidly to these definitions, I could prove things that are true in this system. What axioms do is give you things to work with – once you’ve decided what you’re going to work with what follows need only be true with what you started with, even if it’s “not” true in “the real world”.

That’s the background on what axioms are for people not called Edison, now for a little history.

Funnily enough, back when Greek maths got started, exactly the same problem presented itself: once Parmenides came along and said that nothing could be said to be certain, how would you go about proving him wrong? In other words, how could you make an argument completely, totally, and utterly immune to any disagreement whatsoever for all time??

In the usual manner, it started with setting up ways of showing things are “obviously” true – “Look man, it’s obvious innit?” – where adding “innit” to a proof is presumably equivalent to “Q.E.D.” – and then rapidly moves on to more formal ways of doing things: “I just drew a diagram using circles and lines, it works, it’s true, done! :P

But this geometry hit upon a few problems, three of them to be precise (all of which to be solved using pencil, straigt edge and compasses only):

i)   How do you find a cube twice the size of another cube?
ii)   How do you trisect an angle?
iii)   How do you construct a square the same area as a given circle?

The third problem in particular ultimately led on to the Integral calculus and is thus very important, while the first and the second linked to irrational numbers and frankly were pifflingly trivial by comparison. In each case it was found that the problems were impossible, but they couldn’t PROVE it to be so. In one Greek play of 4th Century AD or something like that, one character rudely insults another with the name: “Circle-Squarer!!” – this after about 1,000 years of trying to solve that problem. One obvious point to make – if you can’t find a square that is the same area as a given circle, does that circle even have an area? Note that we think of area, even today, in terms of squares, so this question isn’t as silly as it sounds.

Skip forward a bit, to Hippocrates of Chios (NOT the Medicine man, who was of Kos)
in around 440BC, and we find a man who decided that he’d give up trying to prove that it was possible to quadrate a circle, and instead assume that it was possible. He then managed to show that things didn’t go wrong and in fact you could find the areas of a lot of curvy things called “lunes”.


Then Eudoxus developed a technique now called “Reductio ad absurdum” or basically “assume the opposite and show that it’s false”. This allows a heck of a lot of things to be proved more easily, since very rapidly the opposites can fall down, whereas showing that something is true can be much harder.

The Greeks worked in Geometry, and so even arithmetic was related to geometric problems, sometimes “clumsily” when compared with modern techniques, but they got pretty far as stated before.

With the writing of Euclid’s axioms, High-School geometry stands on a firm footing. Out of the following five axioms everything you know and love can be shown (and Euclid did show in fact) to be true, taking these statements to be true:

1)   It is possible to draw a straight line between any two points.
2)   It is possible to extend any straight line segment indefinitely.
3)   Given any straight line segment a circle can be drawn with the line segment as the radius and one of the endpoints the centre.
4)   Right angles are equal to each other.
5)   Given a line and a point not on the line, only one line through that point is parallel to the first line. [equivalent statement]

From these rapidly follow everything in standard geometry. In fact, almost all of the results had been already proved, but Euclid linked them, collected them and ordered them, and showed how they followed from the simple assumptions.

Maths was at last on firm footing, capable of being unchallenged. You could debate the starting points, (and in the case of point 5 you really MUST), but you cannot disprove anything since these were assumed true anyway.

Axioms thus allow you to move on from the mundane and into the (relatively) exciting. No more must you show that 1+2=3, it is so because I define it to be!
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline xsix

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Re: three60's Short introduction to... corner
« Reply #8 on: March 05, 2009, 04:23:25 AM »
Wow! that was confusing >.< but made sense some how thanks Jim! :)

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #9 on: March 14, 2009, 04:46:38 AM »
To all those waiting for my next article (i.e. nobody probably) then it'll be out soon, I've just got a bit of work to do first.

The topic will be the Enigma Code and the significance of its being successfully broken in WWII

By the way, has anyone a suggestion for a non-maths related article they'd like to see??
« Last Edit: March 14, 2009, 06:58:33 AM by bfm_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline Fraggle

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Re: three60's Short introduction to... corner
« Reply #10 on: March 14, 2009, 10:49:45 AM »
The topic will be the Enigma Code and the significance of its being successfully broken in WWII

Well I wasn't waiting eagerly for the next article, but I certainly will now!!!

Maybe at the bottom of each article, let us know what the next one will be. that would be awesome and I know it will take some planning, but it will whet the appetite somewhat and keep us all interested. :P

Seriously Jim, I've been meaning to post on the articles. I was Hugely interested in the whole concept of the axioms and proofs.

Brilliant thread, Keep it up!

Fraggle
« Last Edit: March 26, 2009, 04:35:29 AM by bfm_Fraggle »
Many thanks to BFM_MiG for the awesometastic siggy!!
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Offline Marty

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Re: three60's Short introduction to... corner
« Reply #11 on: March 15, 2009, 09:01:03 AM »
Like the thread! IMO topics should be about general topics- Maths, History, Geography, Music, Science, English - each focusing on a more specific part of that topic.


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Offline BFM_Nemesis

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Re: three60's Short introduction to... corner
« Reply #12 on: May 11, 2009, 03:05:33 AM »
o.O that last 1 like... yeah.. lol


Offline jim360

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Re: three60's Short introduction to... corner
« Reply #13 on: July 08, 2009, 03:06:09 AM »
Oops! A combination of exams, illness and lethargy have led to this thread being somewhat neglected. The next article that my keen readers (both of them) have been waiting for should be out sometime by the end of the week.
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline Racr

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Re: three60's Short introduction to... corner
« Reply #14 on: July 08, 2009, 02:18:06 PM »
THREE60'S FRIENDS ARE GONNA ROB A STORE!!!!!!!!!!!!!!!!!  :evil: :evil: :evil: :siren: :siren: :cop: :cop: :cop: :toughguy: :toughguy: :toughguy: If you want me to remain silent about your friends' plans, I require half the loot. :P




Thanks to Manticore for that!



Thank you BFM_Arya for those!!



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