BFMracing
General Category => General Board => Cogitative Corner => Topic started by: jim360 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.
-
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.
-
All I can say is :liljawdrop:
:hrmbig: :interesting: TGIF
:LOL:
Regards,
~ FLipSiDe
-
My head, oh my head! :LOL: Very nicely done Jim! ;D
-
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.
-
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!!
-
Your sig is totally right, you made my head spin! :XD:
-
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!
-
Wow! that was confusing >.< but made sense some how thanks Jim! :)
-
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??
-
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
-
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.
-
o.O that last 1 like... yeah.. lol
-
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.
-
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
-
good read but plz make it more less technical plz in the next one for people who dont know the stuff good.
-
Epic bump...
I kind of never got around to the last article I was planning to write, since I felt it was going to be either a very boring read or a possibly inflammatory one.
Having said that, I may actually write another article soon! Is there any particular topic people would like me to write on, otherwise I'll pick my own.
-
Awesome, I remembered this topic from a long time ago and I love to see it come back. I like these kind of scientific facts ;D .
What about something with physics (maybe newton or sombody else), something about the universe would be cool too.
-
Any chance you could be more specific, xen0n?
Having said that, I have a great idea about one article that links with the last one I wrote.
-
maybe something about stars or black holes, I've always found astronomy very interresting.
-
Well here is my next article. It's a bit of a mixed bag, and many of you who read it may have trouble following since - make no mistake - this is hard stuff. I don't follow some of it myself. But hopefully there is some interesting stuff in here.
Space-time, relativity and all that
I expect you’ve all heard of Einstein’s theory of relativity. Some of you may even know that it comes in two parts: the “special” and the “general”. And you must also know E=mc2 too! But what does it all mean, exactly (well, roughly)? What is “space time”? And why do we need this?
In many ways, Einstein’s role in Relativity is a little overstated. It’s been said, at least of the “Special” theory of Relativity, that the whole thing was just waiting to happen and someone somewhere was going to come across it. This is because a lot of the ground work for Relativity Theory had been finished – done and dusted – some 20 years beforehand by mathematicians such as Riemann (who may well get a mention elsewhere) and Lorentz.
I’ll start with the last question I asked – why do we need relativity theory?
The simple fact of the matter is that Newton’s Theories of mechanics (how things move) and Gravity are just not good enough. To be fair, you can work out how the moon moves, how an arrow flies and get a good idea of where the Earth is going to be in a year down to the 3rd decimal place, but the motion of the planet Mercury you can’t get right in Newton’s world. Not only that, but another, at first glance entirely unrelated, field of science called electromagnetism, has some strange stuff too.
You see, electric forces are created when you have charges, and magnetism comes from current, which is just a charge that is moving. So shouldn’t these two essentially be the same? In the 19th century it was known that they were linked – through a set of equations called Maxwell’s equations – and that one of the linking constants was the speed of light, c. Perhaps the most bizarre part of the whole thing was that this speed of light was an absolute constant. That means that it doesn’t matter how you move, the speed light travels at is always the same. Everywhere. You could be moving alongside a beam of light at almost speed and it would still head away from you at the speed of light.
What?? Think about it for a second, that can’t make sense, can it? If you’re running and fire a gun, the bullet travels at the speed you run and then how fast it would travel if you weren’t running put together. But this doesn’t happen when you travel at fast speeds.
Clearly something else, other than Newton’s mechanics, is needed. Enter special relativity, which among other things, says that instead of the normal way of adding speeds (that is, “new speed”=”speed you’re moving at”+”speed the object was moving at before you started running”) you have to do things in a slightly more complicated way. Specifically, you use what’s called the “Lorentz factor” which looks like this:
γ=1/√(1-v2/c2 )
Now the “curly y” bit is just the symbol used for the Lorentz factor (it’s the Greek letter gamma), c is the speed of light, and v is the difference in speed between two different frames. A frame, I should add, is the universe as seen by any particular point in space. So, for example, you could talk about “the frame of the spaceship”, “Earth’s frame” and so on.
What this factor means is:
- 1. You cannot ever actually travel at the speed of light. If you did, this Lorentz factor becomes 1/(1-1)= 1/0 which works out to be infinity.
- 2. When v is small the factor works out to be about 1. This means that you can ignore it for small speeds. And in fact Newton’s mechanics only works properly for small speeds.
- 3.At high speeds the Lorentz factor becomes bigger and bigger. This has the effect, when you work through the maths of the thing, of slowing time down and squashing space. In fact if you were to travel at the speed of light the entire universe would be in the same place and everything would happen instantly. (This idea also pops up elsewhere, in Quantum Mechanics.)
Someone called Minkowski said that this means that space and time are linked. This leads to what’s called “space-time”. It also means that the universe is not three-dimensional after all, but in fact has (at least) 4 dimensions: up, forward, left and time.
Even all this exciting stuff still has problems – why does Mercury still not move in the way we want it to?
Let’s step back even further, all the way back to the ancient Greeks, in fact, and Euclid. I mentioned before that all the geometry you’ll have learned, or will learn, at school is Euclidean. That means that the following 5 things are assumed:
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]
These may seem obvious, but I am now going to construct a space where they fail:
To break number (2) is easy. Just draw a straight line on a piece of paper and you’ll find that you have to stop at some point.
To break number one is a little trickier, but here’s one way:
(http://i48.tinypic.com/33kf3nt.jpg)
The challenge here is to draw a straight line from point A to point B while keeping the line inside the yellow region. You can’t, it’s impossible. So in fact it’s not always possible to draw a straight line between any two points.
Now, in fairness, you aren’t going to walk along and suddenly find this whopping great hole that won’t let you across it because technically it’s not even there. But the point about this is that it’s possible to imagine and create places in which, after all, Euclid’s geometry fails.
The most interesting part of Euclid’s 5 axioms is the fifth one. This says that lines will meet at some point if they aren’t parallel to each other, and if they are parallel then the lines won’t meet, and if you have a line, and any other point in space, there is only one line through that point parallel to the first line:
(http://i45.tinypic.com/21j9m9s.jpg)
That diagram may help to explain things. As you can see, there’s only one line that is truly parallel.
But now, let’s recall an interesting riddle:
“A man got out of his tent, walked a mile south, tracked a bear eastward for another mile, finally shot it, then took the bear home by walking due north for a mile. What colour was the bear?”
The answer is white, because you were camped at the North Pole. But let’s think about this for a bit: You just walked along the sides of a triangle, taking right-angle turns all the time – that means that the angles in the triangle add up to 270 degrees and not 180.
This means that the geometry on the Earth is not Euclidean either. In particular, all lines joining the North Pole and the South Pole are parallel to each other yet they all meet. So that fifth Axiom of Euclid’s fails on the Earth.
It took about 2000 years for people to realise this. Up until Gauss and Riemann, all geometry was Euclidean. Then those two developed new geometry that wasn’t Euclidean – the two main ones being spherical (where parallel lines can meet) and something called “hyperbolic” (where through any point there are infinitely many parallel lines). These two geometries, and the whole idea of non-Euclidean geometry, allowed Einstein to describe Gravity.
We come back to Relativity now. You might even have heard of this bit: “every object bends space-time around it”. You can picture this two-dimensionally by taking a net, stretching it out tightly then putting something like an orange on. The net gives way under the weight of the orange. Now look down at the net and you might see something like this:
(http://i45.tinypic.com/2wdnl1v.jpg)
This helpfully shows that space-time (represented by the netting) has been bent and distorted by the mass of the orange (which is in the middle of the right-hand diagram).
With some horrendously difficult mathematics it’s possible to predict the shapes that this bending will take. And that’s exactly what Einstein did. What General Relativity says, in a nutshell, is that you can view gravity not as a force but as the bending of space around anything with mass. Remarkably, this is just what is needed so solve the problem of why Mercury moves the way it does.
Now xen0n asked about black holes. They fit into this too. The "black hole" is a solution to the equations of general relativity when, essentially, there is a point where you have to divide by zero. That means that instead of space-time (the netting) gently bending under the weight, you have such a heavy weight that the netting - space-time - collapses, breaks and you get a hole. It would look something like this (this image has been grabbed from a website, and also gives the equation for the event horizon of a black hole):
(http://vega.bac.pku.edu.cn/~wuxb/bh.gif)
As you can see, there is literally a hole in space-time at the point of the black hole. That's why anyone going too mear a black hole gets sucked in - they fall into the hole.
The great power of General Relativity is that it can help us understand what gravity does near the planet earth much better than simple Newtonian gravity can. That means that space flight, satellites and the Global Positioning system work.
Hope you enjoyed this! Once again, any more ideas, please let me know.
-
Very nice jim,I realy enjoyed this one ;D I always wanted some explanation on relativity, we're gonna see it soon in school I think (now I'm gonna have a head start :P ). Anyway, keep up the great work, maybe I'll give a new suggestion later ;D
PS: I think you missplaced a diagram :ninja:
-
Glad you liked it.
As for the diagram, good thing too, since I thought about it some more and decided I got it wrong. The corrected image has been inserted.
-
I've written an article on cryptic crossword clues, which will be put up here sometime tomorrow. Those of you who spend your time attempting to solve my BFM-theme clues (more of which to be also put up shortly!) may find this useful!
-
How to write a cryptic crossword clue (or at least how I try to)
Recently you may have noticed that in the “riddles game” I’ve been posting cryptic-style clues, taken either from the Times or that I wrote myself. I also found that at least one person was interested in writing them him- or herself. So, here is a rough guide about writing the clues. People who read this may also find useful tips to help them solve the clues I write.
Firstly, what is a cryptic clue? Well, let’s pick a simple example. Here, for example, is a clue taken from the Times March 3rd, 2010:
Hero has sport, turning out a marksman (12)
And here is the same clue as it might appear in a non-cryptic crossword:
Expert marksman (12)
Now, clearly the first thing you see is that both clues share the word “marksman” so this is the important bit. The word that answers both clues is “sharpshooter” which is another word for marksman.
Now, have a look at the letters in “sharpshooter” and rearrange them a bit, and you’ll find that you can make the words “hero has sport”. So what the cryptic clue is trying to say that, “if you were to rearrange the letters of ‘hero has sport’ you would find that you could make a word that means ‘marksman’.” It has done this by the phrase “turning out” that is telling you to rearrange “hero has sport”.
Not all cryptic clues are anagrams – despite the huge number of possible anagrams there are some words which refuse to be rearrange-able (new word ;D) into other words – and even if they did it would be boring solving anagrams all the time. So we might have something different:
One crows, seeing head of organisation in duty list (7)
If you thought of the word “roster” when you saw “duty list” then you might be able to guess at the answer “rooster”. You would be right. The clue is saying, “Put ‘o’ (the first letter of the word ‘organisation’ into a word meaning ‘duty list’ and you make a new word that means ‘something that crows’.”
Hopefully you can see that in general a cryptic clue is trying to describe what happens when you play around with words. At the same time, though, the clue is designed to mean something completely different at face value.
So how would we go about writing such clues? Clearly what we have to do is work backwards – take any word and find some clever way of forming that word, telling the solver what you have done and confusing anyone who doesn’t recognise what’s going on. How to do this is described below.
The first point is that any clue that is written should be solvable. It’s no use having a really clever clue that nobody can possibly solve unless they were sat next to you as you were writing it. So this means that you have to write the clue in such a way that it’s clear what you have done.
Here is an example of a bad clue:
E? (13)
In fact most cryptic crossword enthusiasts would get this instantly since it is a “well-known” example. The answer is “senselessness”. Why? Well, if you write “senselessness” as “’sense’ less ‘ness’”, then you do this “word sum”, you find that you are left with the letter “E”. Hence the clue. But the problem is that, frankly, there’s no way of being able to recognise that based on the clue itself. You would need a leap of faith and a brain wave to solve that clue – it is badly-written. (There are in fact ways to “rescue” this clue but sadly they aren’t really appropriate for these forums.)
So what is allowed? Here is a list of the most common ideas used in cryptic clues I’m used to:
1. Anagrams. As discussed, if you have a word, say “sharpshooter”, and you can rearrange the letters to something that looks fairly sensible, then an anagram-based clue could be worth a try.
2. “Buried answer”. What this means is that you have a word, say “happy”, and you put other letters around this word to make new words. In this example, you could add “mis-“ to the front and “-rotechnic” to the end and you get the phrase “mishap pyrotechnic”. Not too promising, but you now add more into the clue to make it “work”. In this example, you could add “despite” to the front: “Despite mishap, pyrotechnic display” – a phrase that makes sense. I’ve also added a comma to break up the answer a bit. Indeed, looking near any punctuation in a clue can be a good trick to discover the answer. We’ll complete this clue later.
3. Other “buried answers” can have the answer written backwards: as an example you could take the word “tea”, reverse it (“aet”) and write something like “praetor” (a Latin word) that has the letters in the right order.
4. Another good trick is “acrostics”. Take a word, say “pain”, and think of words that begin in “p”, “a”, “i” and “n” – say, “Paper and ink needed”. Again, this can work backwards.
5. Synonyms and homophony. These work in the same way – you either think of two words that can mean the same thing, such as “pricy” and “darling”, and find a word that means both of these – “dear”. Then the clue could be “pricy darling”. Homophony is when things sound the same – “dear” and “deer”. Then the clue could start “expensive venison” (which would read “dear deer”).
6. Word-building. This is the most complicated type of clue. What you do is you take a word and disassemble it into smaller words and/ or letters. You then find synonyms of these smaller words. This is very useful for compound words. One example might be “shoplifting”, that we can say is “S” plus “hoping” plus “lift”. Then a lift could be a boost, and “hoping” could be “expecting”. The problem now is that we have to be able to connect these words into a sensible phrase – see later.
7. “Pedantry”. Here you take a word or phrase that is well-known and look at what it might mean if you took it literally. For example: “A rolling stone gathers no moss” could mean something like “Mick Jagger won’t pick Kate [Moss, the model] up”. This clue needs a couple of extra touches but it’s almost complete as it stands.
8. Clues which work backwards. You don’t often see this but what it means is that you look at the answer in a way that could make that the cryptic clue. The clue you write down is then the “answer” to the clue suggested by the answer. This is giving me a headache so let’s find an example: “officer’s mess”. The word “mess” might mean “make a mess of” so we’ll “make a mess of” the word “officer”. We find that “rice off” works.
9. The final clue type is what is called “& lit.”. Most clues contain a part that is “wordplay” and the clue itself (anagram plus definition, for example). In some cases it is possible that the clue is contained inside the wordplay. Take “hairdresser”. By some amazing coincidence, “hairdresser” is an anagram of “shears drier”. We can then write “I manipulate shears and drier” as our clue. The clue is to be read both literally (the “& lit.”) and cryptically.
There aren’t many more techniques used than the ones listed above – and the key one is number 6, which you see all the time.
Now that we have some idea how clues work, we can set about trying to write some. Let’s start by completing the example clues I have given so far, starting with happy:
- When we moved on I had got as far as “Despite mishap, pyrotechnic display”. This is good, but to complete the clue we now need to put something in explaining to the solver what we have done. So we look for words that mean things like “contain”, “inside” or something similar. I think that “in part” might be a good fit here. That gives is “Despite mishap, pyrotechnic display ...in part...”. Still not done yet, we now add onto the end of this a word that means “happy” – one example might be “pleasant”. Even now we are not quite done because we want the clue to make sense. So we finish the clue as follows: “Despite mishap, pyrotechnic display partially pleasant”. That’s it!
- The next example was “tea” which I had put backwards in “praetor”. To finish this clue we need words that mean “hold backwards”, and a word that means “tea” – say, “drink”. One way of doing this is to write, “Praetor holds back drink”. That’s all you need to do to finish that one off!
- The next example was “pain” which we had started to clue by writing “Paper and ink needed”. Now we need a word that tells the solver to “look at the first letters of...”, say “at first”, and a word that means “pain”, perhaps “torture”. A clue might then be “Paper and ink needed at first to give torture”. Another finished clue!
- The synonym clues tend to be very short – for our first clue we can just say “Pricy Darling” and leave it at that. For the second one we need one more word to make it clear that a homophone is involved: “Venison is very expensive, reportedly”. The “reportedly” is the indicator that makes this clue read as “word that means ‘venison’ and sounds like a word meaning ‘expensive’.” The important thing is that if instead we write, “Venison, say, is very expensive” then the answer would be “dear” not “deer” (the clue now reads “a word, that sounds like a word meaning ‘venison’, meaning ‘expensive’”).
- Now we come to the clue for “shoplifting”. We disassembled this to “S”, “hoping”, “lift”, and said that “expecting” means “hoping” and “boost” could mean lift. The “hoplifting” part of the word has the word “lift” inside the word “hoping”, as you can see. So we put this down as “[hoping] receives [lift]”, or “wishing to receive boost”. Now what about the “S”? We can get round this problem by, say, choosing a word that begins with ”S” and saying “take the first letter of this word”. Now let’s also remember that “shoplifting” is a type of Crime, so our clue might look like “Crime S____ at first expecting to receive boost”. Why not pick “Squad”? This creates the final clue as: “Crime squad at first expecting to receive boost”. This is, in fact, the clue as it appeared in the Times a few months ago.
- To complete the clue for “A rolling stone gathers no moss” based around “Mick Jagger won’t pick Kate up”, we add a question mark at the end, and insert the word “perhaps”. The clue in its final version reads, “Mick Jagger, perhaps, won’t pick Kate up?” Why the “?”? This clarifies that the clue should not be read in a “normal” cryptic way – words like “up” and “pick” could be read cryptically as, say, “read from bottom to top” and “put one word inside another”. The “perhaps” says that we think of groups of which Mick Jagger is a member. This completes the clue.
- Our final example was “officer’s mess”. We saw that “rice off” was an anagram of “officer”, so let’s write the clue as “rice off, here?” The “?” again indicates that the clue works differently from normal cryptic clues, and the “here” is the “clue word”. I think anyone getting this clue would be a cryptic genius. I certainly wouldn’t.
Those are the basic forms of clues. Now let’s give a few more examples. Here I have picked 3 words that I don’t think I have seen clued before, and tried to write clues for them. The thought process is included throughout.
1. Thoroughfare. (This means “main road or highway”). “Thorough” sounds like “furrow, “fare” like “fair”; “trench” means “furrow” and “even” means “fair”, so we can say “Trench even, we hear, in highway”. “we hear” is the part that indicates the homophone.
2. Dragonrider. I have already used this clue in the Riddle thread. How I got to it was as follows. I noticed that “roaring red D” was an anagram of “dragonrider”. This didn’t seem too promising, but dragons can be red and do tend to roar, so I saw potential in this. The clue was completed because “D” is the first letter of “Dstroyr”. So I had “Dstroyr’s first ____ roaring red _______”. The words “to sit” tells you that “D” is the first letter of the answer. So I put that in. Then I used “strange” to indicate the anagram: “Dstroyr’s first to sit on strange roaring red thing”. To complete the clue to Times standards I would have to add one other word to make it clear that the answer is indeed a BFM admin. Hence the final clue should read something more like “BFM Dstroyr’s first to sit on strange roaring red thing”. The word “thing” is surplus and means that this clue, perhaps, is not quite perfect. But I enjoyed it and that’s what matters.
3. Gordon Banks (England Keeper, 2nd best in the 20th century according to one poll). Gordon Brown being the current UK Prime Minister, and “banks” being a verb, we might try something that means “Gordon banks”. Something simple that works could be “Brown bets on a great saver!” The exclamation point? – let’s face it, Banks deserves one. Also exclamation marks tend to indicate double meanings.
And that’s it! To write your own clue, find a word, look at it in another way, write a short sentence that contains all the information the solver will need to solve the clue, and try to make the sentence make sense in another way.
It may be hard, but it’s great fun!!
[readers may be interested to know that there were exactly 2337 words in this post until I added this comment.]
-
I hope people enjoy both of the last two articles. Has anyone any suggestion for the next one I could write?
-
In lieu of anyone having other ideas, over the next few weeks (probably around Easter time when I have nothing better to do) I'm hoping to write a series of articles leading to a very famous equation. Each one will focus on mathematical topics that were covered when I was at high-school, so in theory at least they should be accessible to a lot of people. The final article will link the first articles in a single equation.
Edison will probably be able to guess what equation that is. :P The rest of you who may not know of it yet...I hope you look forward to it!!
-
Update for those interested: the first of the three or four articles in this series, on trigonometry, is due out tomorrow or Sunday, with the next one to follow within the coming week.
-
This article is the first in a series of four articles with the hope that at the end I will be able to introduce to you the “most beautiful equation in mathematics”. How I will go about doing this is by taking the three subjects in mathematics that lead to this equation in turn, all of which seem to stand fairly separately, and finally tie them together. These topics are: trigonometry, calculus, and complex numbers. The equation at the end is known as “Euler’s Identity”.
Let’s start at the end. Euler’s Identity states that:
ei*(pi)+1=0
All very well, but... what is e? What is i? And what, for that matter, does having i and pi to the top right of the e mean? Why does this have anything to do with triangles?
All these questions and a few more I hope to be able to answer for you in the course of these four articles. So let’s begin!
Trigonometry
When you get around to starting your own civilisation, sooner or later you’ll find that you need to be able to measure something. Not only that, but a bit of experiment shows that in fact one of the “strongest” shapes there is is the triangle. So, the early civilisations concluded, why not see if we can figure out everything there is to know about triangles?
This is the reason why trigonometry is so important. Even for the layman measuring is useful, in DIY, etc.. In fact as we shall see later on trigonometry has far more uses than just measurement.
Triangles come in several varieties, though we’ll start with (and indeed stick to) right-angled triangles. It’s a good place to start because for every rectangle there is a right-angled triangle that goes with it, so we could also find out stuff about rectangles into the bargain.
First, a few definitions:
(http://i39.tinypic.com/2dj6i34.jpg)
The symbol next to the marked angle is the Greek letter theta, which is the usual symbol used to mean an angle.
In any triangle there are seven measurements – three sides, three angles and the area. Cool thing we shall find out is that so long as we know at least 3 of these measurements (of which at least one is a side) we can find out all the others. The answer to the question how is trigonometry.
The easiest way to start trigonometry off is by noticing that when you measure the sides of a triangle and find them to be, say, 3cm, 4cm and 5cm, then drew a bigger triangle and measured its sides to be 3m, 4m and 5m, then both triangles look exactly the same.
(http://i40.tinypic.com/t05dgh.jpg)
This suggests that what matters isn’t so much the length of any side, but its length compared with the other sides. In fact, the angles in both these triangles are the same and it’s not too hard to see that the sizes of the angles depend on the lengths of the sides.
Because of this what we do is define the following three ratios:
(http://i42.tinypic.com/qx8ba0.jpg)
In days of old, before calculators, there were also several other ratios used, with glorious names such as the arc cotangent and the half versed sine, that now aren’t so useful because we have calculators.
These ratios link the angles to the sides and we could now work out what they are by drawing triangles with different angles, measuring the sides and dividing these answers – and that’s how it was done, in fact, for thousands of years. It’s more sensible, though, to fish out a calculator. The ratios are usually abbreviated to sin, cos and tan.
What can we do with these? That’s what High School maths spends some weeks doing. The basic gist, though, is that now you can work out how high buildings are, how far away stars are, and so on. To work out the angles from a given ratio you use the "inverse ratio" which appears on calculators as "sin-1", for example.
Since the world isn’t made up of right-angled triangles you might wonder how we deal with other ones. Actually it’s not too hard. What you can do is split up the triangle into smaller, right-angled triangles and work from there. This simple trick means that any and every triangle can be “solved” in almost no time at all – so long as you know the length of a side, of course.
But what if you don’t? It’s not the end of the world, because what you can still do is find out the values of the ratios of the sides.
There are one or two other useful facts we can find out about triangles, in fact, by ignoring how long the sides actually are and choosing the length of the longest side to be equal to 1. Now if you remember that in a right-angled triangle with sides a, b and c (c the longest side), a squared plus b squared equals c squared, then we get the following:
(http://i39.tinypic.com/25s7gvd.jpg)
This neatly links all the ratios together. In High School again, you’ll see that these two facts allow you to prove all sorts of other interesting, if generally useless, relationships.
So far there’s not been anything that gripping, and I’m sorry about that because the details are a bit boring. However the idea of making the hypotenuse have length 1 is quite useful as I shall now show. If you increase one of the angles all the way from 0 degrees to 90 degrees, and look at the path the opposite corner follows, you get this shape:
(http://i44.tinypic.com/jfaurk.jpg)
Yes, it’s circular! This quirky-looking coincidence says something far deeper – that in fact triangles and circles are very deeply related. We’ll come back to this in a couple of articles’ time, but for now keep this fact in mind.
The most immediate use of this is that we can now extend our definitions of sine, cosine and tangent to angles of any size, though only if you allow the confusing idea of "negative lengths". What you do is keep draggin that far corner round and measure the lengths of the sides, while saying that all sides to the left of and below the central point are "negative" in length. This leads to the “sine curves” and “tan waves” that will be or have been introduced at some point in High School. There are several uses of this, the most appealing being the cool fact that if you turned a sine wave into a sound it would be – it is – the purest form of musical note you can have.
You’ll find that, far from being restricted to triangles and measurement, these ratios have a use in anything involving waves and motion! The shape of a wave on the sea is, roughly, a sine wave, while the motion of a pendulum in a clock is also described in terms of sine waves - in fact you need sine waves to tell the time!
There is one last thing I will introduce into this article and that is the idea of a slightly nicer way of measuring angles. Degrees, you see, are all very useful in their way but are very artificial.
Instead, a far more useful method of measuring angles is as follows:
(http://i43.tinypic.com/35izx2g.jpg)
A well-known fact about circles is that the circumference is twice the radius times pi, so there are 2*pi radians in 360 degrees.
That concludes the article on trigonometry. The main conclusions I would like you to draw from it are:
- That despite the apparent fact that trigonometry is useful but not interesting, it turns out that even this subject has opened up all sorts of new possibilities.
- That rectangles and circles are somehow linked through trigonometry.
The second fact is the most profound. Mathematics is a lot about linking areas that seem unconnected, and the more links the better!
That concludes our brief look at trigonometry. There is of course a lot more to this that you will explore or have explored at High School.
Next article we will be taking a brief look at another area of high-school mathematics, calculus.
-
Things have come up over the last few months that have kept me from posting here with the second, third and fourth articles in this sequence. Sorry about that, but can't be helped. Hopefully over the next week or two I'll dig up the files for the next post. Until then, stay tuned...
-
The next article in this series is on calculus. Since a good deal of this stuff is, shall we say, fairly horrific, I'll try to cut out most of the working and equations - but unfortunately some of that can't be avoided.
As is usual with almost every subject in mathematics, it all starts with the Ancient Greeks. This time it was one of those three problems I posted earlier:
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 is the only one that leads anywhere seriously interesting. Why? Because it's about finding areas contained by curved lines rather than straight lines.
The Greeks quickly found that all three of these problems were impossibly, at least when using their favourite tools of a straight edge, pencil and compasses. So impossible, in fact, that in the 3rd or 4th Century AD, in one Greek play one character insults another by calling him "circle-squarer!", or, in modern speak, "time waster".
Flash forward 2,000 years and we know, of course, that the area of a circle is (pi)*r2, or roughly 3.14 times the radius squared. But, as we also know, the circumference of a circle is given by 2*(pi)*r. These results were known to the ancient Greeks too. Two questions come from this:
- Why, to move from the formula for the area to the formula for the circumference, have we only needed to move the 2 to the left and down a bit?
- And, for that matter, why is pi in the area formula at all? After all, it comes specifically from the circumference - pi is how many diameters there are in the circumference - so you might think it a bit odd that it appears again.
These quirks don't end there. For a sphere, the surface area is 4*(pi)*r2, while the volume is (4/3)*(pi)*r3. These too are uncannily similar. It's as if you can get one directly from the other and back again - that, in a way, they are the same problem. Why should this be?
The clue lies in the last article I wrote. We saw there that a right-angled triangle with a fixed hypotenuse and a fixed point, if the other two lengths are allowed to change, traces out a circle:
(http://i44.tinypic.com/jfaurk.jpg)
Now, we remember Pythagoras' Theory, that "the square on the hypotenuse is equal to the sum of the squares on the other two sides in a right-angled triangle" and write this using algebra as:
x2 + y2 = r2 (1)
where r is our hypotenuse or radius, x is the base of the triangle and y is the height.
Why have I done this? The point is that Pythagoras' Theorem in fact describes a circle, and by writing the circle in this way we can deal with the problem a lot more easily, basically because now we can mess around with the formula labelled (1) and get:
y = sqrt(r2 - x2) (2)
Where "sqrt" is the square root. We'll come back to this a bit later.
The fact that the area of a circle is given by (pi)*r2 suggests that it might be handy to look at the graph of y=x2. This can be thought of as "if we choose a number x, then the number y is the square of x". So, for x=2, y=4, and for x=4, y=16, and so on. If we draw this on a graph we get:
(http://t3.gstatic.com/images?q=tbn:ANd9GcRzLtghq7k6hVHYL5P3lPLRcuf4nRJfm5gWQjvP1uARRHT8E5tt8g)
Now it would be nice to know a bit about this graph, and in particular we want to know two things: how steep it is at any given point (for curved lines this is obviously changing all the time), and what areas we get if we draw three straight lines like so:
(http://i55.tinypic.com/11k82ki.jpg)
The reason we might want to know these things is that, for example, when a car is travelling along a motorway, then you could track its journey in terms of how fast it's going at any given moment, and draw this on a graph. In this example, the steepness (or, from now on, gradient) of the graph tells you how much the car is accelerating, and the area under the graph tells you how much distance has been travelled. So these are useful problems to solve.
The area problem links to the area of a circle, and will be looked at later. The gradient problem is the one we'll solve now.
You may have looked at "speed-time" graphs in school, and these would probably have straight lines in them. The gradient of the line is found by working out how much the speed increased in, say, one minute. We can write this as:
(http://i55.tinypic.com/2wolgtg.jpg) (3)
For a curved graph, we can get a reasonable value for the gradient at any point by using straight lines like so:
(http://i52.tinypic.com/2ufq907.jpg)
To get a better value for the gradient, we just shrink the size of the line:
(http://i52.tinypic.com/2pspwyg.jpg)
until, eventually, our line is so small that it has no length. We can't work out how steep that is, obviously, but if you take several measurements of gradients and get results of, say, 2.005, 2.0033, 2.002, 2.000001, and so on, then it makes sense to say that the gradient at that point is almost certainly 2. How do we prove this result?
Think about what we were doing. We took two points on the graph and drew a line between them. If the two points lie on the graph, then we can work out the coordinates of those points. For the graph y=x2, we can say that the gradient of that line is:
(http://i51.tinypic.com/o6zjvt.jpg)
The numbers in subscript are just a way of saying, for example, “the second value of x”.
This isn't too hepful yet, but we can make things simpler by, instead of using x1 and x2, we could just say that x2 is "x1 plus a little bit more". Call that little bit h, say, and now (the clever bit) think of y2 as (x+h)2. Now our gradient formula becomes:
[(x+h)2 – x2 ]/ h (4)
Expand the brackets, cancel a few things out, and you get:
Gradient (of the line) = 2x + h
And now all we do is let h get smaller and smaller, and so the line we’re using gets smaller and smaller too – and when h is 0, we are left with:
Gradient (of the curve y=x2) = 2x
Success! not only have we found the gradient of a curved graph, but we've also answered the problem about why the area of a circle and circumference are so similar. It comes from this result.
We can use this method to find the gradient of all sorts of curved lines, from the simple to the ridiculously complicated, so long as we know the equation that matches the line.
In the first part we left off with the formula y = sqrt(r2 - x2). This traces out a circle:
(http://i55.tinypic.com/xarm6q.jpg)
What is the area of this circle? we'll answer that by taking the top semicircle, finding the area of that, then doubling it.
Again, from high school, you might be given a curved shape and told to find the area under it. You can't do this exactly but you can get a good estimate by drawing the shape on squared paper and counting up the squares inside the shape. We'll do a similar sort of thing, only using rectangles, and we'll make sure these rectangles have the same base length:
(http://i56.tinypic.com/23vwz2h.jpg)
Obviously the smaller those rectangles are the better the estimate we'll get:
(http://i55.tinypic.com/99ihye.jpg)
Until, as with the gradient problem, we get rectangles that are so small they have virtually no area.
How to represent this in algebra? Since the area of a rectangle is the base times the height, call the base h, and the height can be found by the value of y at the top-left corner of the rectangle. We then sum up the areas of these rectangles. The next steps are fairly ugly (or at any rate messy) so I will skip them, but the end result is that the area depends, a little surprisingly, only on the shape the curve takes and the two points at either end:
(http://i51.tinypic.com/2vwg1na.jpg)
Here the "dx" is the length of the base of the rectangles, the curly line stands for "integral", which just means a sum, and the values a and b are the values of x at either end of the "curve segment" (part of the curve) we're interested in. During the working we would find that this problem is effectively the same as the gradient problem, only backwards! As a result we can start from the other end, which is usually much easier. That's what the third part of the expression means, where a capital Y is a sort of "anti-gradient" - usually we can find this by looking it up in a handy table somewhere. If you can find that then you can find the area.
For our circle, we replace a and b by –r and r (because we want the whole semicircle and that’s the value of x where the circle meets the x-axis) the result is rather strange-looking (you can work this out but I'd rather not bother):
(http://i52.tinypic.com/2ir9em0.jpg)
You may not be able to make much of this, but you should notice that there's a "sin" appeared. This is the same "sin" as in the last article, only again backwards. "sin-1" means that we know the value of the sine ratio and want to find the angle to match this.
For our problem, we're somewhat helped by the far that we've used x=r and x=-r. Squaring these gives r2 both times, so that horrible-looking first bit disappears, thanks goodness. The second bit just becomes:
(http://i56.tinypic.com/iz87xt.jpg)
so now we're left wondering what sin-11 might be. Look again at this diagram:
(http://i42.tinypic.com/qx8ba0.jpg)
So for the sine of some angle to be one, we need the opposite side to be equal to the hypotenuse. The only way we can get this is for the right angle itself! Now, we write this angle in radians, and if there are pi radians in 180 degrees (see last article), then 90 degrees is (1/2)*pi radians. So we get sin-11 = (1/2)*pi. Meanwhile, sin-1-1 is just -pi, and so the area of the semicircle turns out to be (1/2)*pi*r2. Double this to get the area of the circle...
So there you have it. The area of a circle looks similar to the circumference because one gives you the other - they are effectively the same problem.
What does this have to do with ei*(pi)+1=0 ?
Suppose, instead of squaring x, x2, you swapped the 2 and the x round, 2x? What would that look like, and what is its gradient?
Here's what it looks like:
(http://i53.tinypic.com/dnkl0h.jpg)
And using the gradient formula (4) above, with all letters meaning the same things as earlier but swapping a few things around, we find that:
Gradient = [(2h – 1)/h] * 2x
Using a basic power law. If we now let h get smaller and smaller, this eventually becomes:
Gradient = 0.693… * 2x
So the gradient of 2x is almost the same as what we started with. What’s more, if you do the same thing with 3x then you get:
Gradient = 1.098…*3x
From this it makes sense to think that there’s a number between 2 and 3 where we’d just have got the result that the gradient is the same as what we started with. And there is, this number is called e.
What’s e for? Well, without going into details, e appears in finance, probability, calculus, radioactivity, and indeed anything involving exponential changes. So it turns out to be a very useful number indeed.
We’ll see in the final article how e even turns up in trigonometry, despite its only popping up as a sort of fluke.
-
:Way Cool:
-
*applause*
Well done three60!
-
Forgot to mensh - there are two more articles on their way in this "ei*pi+1 = 0" series, the first of these about "i", and the second about the whole equation.
-
*applause dies out abruptly*
-
Hmm... I think the next article will be more likely easier to follow though... fewer equations for a start.
-
Methinks the forum should support LaTeX.
-
Why going to school when you have three60 here? ;D
-
I'm currently busy grinding through the diagrams for the next article - on the origins of i and the mysterious nature of numbers themselves...
I think I may have it ready this week. Eyes peeled, please!
-
Last time I explored the origins of calculus, and also cleared up the meaning of “e” in the equation ei*pi+1 = 0. Now it’s the turn of “i” to be explained.
The number i is usually introduced in terms of quadratic equations, though I’m going to be more historical about it. Having said that, let’s just explain what a quadratic equation is, what it is used for, and how it is solved. I’ll also look at something rather curious that you might not have realised was even worth noticing.
“Sqrt” means “square root”, throughout.
Let’s say that you have a rectangle-shaped field with an area of 600 square metres, and you are told that one side is 10 metres longer than the other. How long is the shorter side?
The usual first step is to draw a diagram:
(http://i52.tinypic.com/28sa7tj.jpg)
When you look at the problem like this, hopefully it becomes a lot clearer. We’ll call the shorter side “x” and the longer side “x+10”, and because the area is found be multiplying the lengths of the two sides together:
x(x+10) = 600.
This is a quadratic equation, which just means that there’s an x2 term in the equation, but no higher power of x than that (such as x3). At this point you have to expand the brackets (parentheses for my US readers), take everything to one side, and then factorise the equation, but I’ll skip straight to the answer since you can learn all about the rest of that in school. We find that x is 20 metres, and if you like you can try this.
For reasons that will be clear shortly, I’ll also introduce the so-called “quadratic formula”. Earlier I went from a real problem straight to how it would be written in algebra. So let’s take the most general quadratic equation possible:
ax2 + bx + c = 0
This means that we are trying to find x, while the new letters a, b and c are given to us in the question. So you might see something like “Solve for x if 3x2 + 10x + 3 = 0”. a has been replaced by 3, b by 10 and c by 3.
The quadratic formula says that if you see this sort of equation, you can find x by putting in the values of a, b and c into the following equation:
(http://i54.tinypic.com/2rdg0lg.jpg)
Don’t worry about the thing for now, though it just means “plus or minus”. Also the thick line is a division, and the squiggly line is a square root, just to clarify. The main thing to notice is that, with a little bit of work (and a calculator), you can solve any quadratic equation you can think of.
Except for these ones:
(http://i52.tinypic.com/33f3gxv.jpg)
For all of those equations, the quadratic formula goes wrong. Let’s take the simplest of those, x2+1=0, and we’ll bang this into the quadratic equation:
x = (1/2)*sqrt(-4) = sqrt(-1)
Which doesn’t exist. Alternatively, if we look at the graph of y=x[sup2][/sup] from last time:
(http://i51.tinypic.com/29dhkwi.jpg)
The curve doesn’t go below the x-axis, which means that there’s no value of x that, when squared, will give a negative number.*
So that’s quadratic equations for you. They solve problems mainly to do with areas, but it turns out that they crop up in other places.
If we return to the problem we started with, and go through the steps, the last step before the solution turns out to be:
(x+30)(x-20)=0
Now this is true when at least one of the brackets works out to be 0, and the “x=20” solution comes from the second of these. But if you look at the first bracket, you’ll see that there is another possible value of x, and that is x= -30. No trouble there then… except that the problem we were thinking about was that of a field, and there’s no way you can have a field with a side that is –30 metres long. So… what is that solution all about then? What does a negative number even mean?
Think about that, dear reader, while I move quickly into the world of cubic equations…
*For those of you who already know what i is, and are wondering how I can possibly get there when I just said that it doesn’t exist, read on.
So much for problems with areas, but what about volumes?
Let’s take a box that has a volume of 168 cubic metres. The longest edge is 10 metres longer than the shortest edge, and 5 metres longer than the third edge. What is the length of the shortest edge?
Again, this becomes clearer with a diagram:
(http://i52.tinypic.com/2ywibko.jpg)
And, since the volume is given by multiplying the lengths of the three edges together, we find that:
x(x+5)(x+10) = 168
With a bit of guesswork you’ll find that x is 2 metres. If we expand the brackets (multiply everything out) we find that:
x3+15x2+50x = 168
This is a cubic equation because now the highest power of x is the x3 term. Of course, to solve problems involving volumes we might want to study these equations too, but they turn out to be more complicated than the quadratic equations were.
Why else might we need cubic equations? Let’s return once again to those three “unsolvable” problems that the Greeks were worried about:
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?
To be clear about this, these problems can only be attempted when using a pencil, a straight edge and compasses. Last time we looked at the problem of the circle, but what about the other two?
The problem of the cube is obviously about volumes, but doesn’t go very far: say that we had a cube with all sides a metre long. Then to double the volume of the cube we’re looking for a cube with sides of some length x where:
x3 = 2
Then we just take the cube root of both sides and find that x is the cube root of 2. The only reason the Greeks couldn’t solve this problem is that there’s no way, using just a ruler and compasses, that you can draw this length. So let’s not worry about that one.
Instead, let’s look at the problem of trisecting an angle. This problem is about dividing any angle into three equal parts:
(http://i53.tinypic.com/2meyfcp.jpg)
Let’s call the angle we’re given 3x, and the angle we’re looking to construct (draw using compasses, ruler etc., but NOT a protractor) is x. Note that we don’t even know what size the angle 3x is.
How is this helpful? Let’s draw a few more lines on the diagram…
(http://i53.tinypic.com/30shev8.jpg)
Within this mess you should see a few things, most importantly the fact that triangles have appeared. This means that the problem of trisecting an angle is also a problem involving triangles, and those things we know about. Because of this link we can re-write the problem using trigonometry, and specifically by talking about the cosine (cos ratio from my first article) of the angle. Why? Well, with a few rules, such as “the addition formula for the cosine ratio" (whatever that is) we’ll eventually find that:
cos(3x) = 4*(cos(x))3 – 3*cos(x)
This ugly-looking thing can be made a little simpler if, we write, say, q for cos(3x) and t for cos(x):
4t3 –3t = q
By this point I should say that we’ve left behind the world of compasses – which is why the Greeks couldn’t solve this problem either. In the meantime, if we even wanted to solve this equation then q would have been given to us, so we might see “find t if 4t3 –3t – 1 = 0”. We aren’t going to solve this, but you can see that understanding cubic equations might help us understand this problem too.
In the same way that the quadratic equation has a “quadratic formula”, the type of cubic equation that looks like:
x3 + px + q = 0
(in other words the sort of equation that we just saw pop out of trisecting an angle) can be solved with its own formula. Again, p and q are given and you’re trying to find x. This formula, though, is a little more complicated…
(http://i51.tinypic.com/2ljnjg4.jpg)
Ouch. However, again, the main thing to see is that you can solve this if you wanted to.
Let’s think about a simple question: “Are there any numbers who are their own cubes?” This question can also be written:
Find x if x3 = x
We can solve this first by just looking at it, and you should see that x=1 works, and so do x=0 and x=-1. But we could also use the formula for solving cubic equations above…
(http://i53.tinypic.com/rh0nsm.jpg)
But here’s a problem, what’s that square root of –1 doing there? I just said that there’s no such thing, after all.
To make sure that we haven’t messed up it might be worth while checking that this hideous thing fits in with the equation we started with. We were looking at x3 = x, so if we cube this we should get what we started with (you’ll have to trust my working here, I’m afraid…):
(http://i52.tinypic.com/2wok2kw.jpg)
That looks fairly promising since the same ugly expression that we started with is back on the right. There’s just the small matter of that sqrt(-1)+1/sqrt(-1) bit. But wait! –
(http://i56.tinypic.com/1ghl3b.jpg)
So that first bit just disappears and (when you cancel out the 3's) we’re left with what we started with! This means that this non-existent number is actually a solution to x3 = x, and what is more it must be one of 1, 0 or –1, because there are never any more than 3 solutions to a cubic equation. So how can we have got this result, a number that doesn’t exist working out to be 1 or 0 or –1, numbers that we’re happy to work with?
Let’s backtrack for a minute, though, and think about the poser earlier: what does a negative number mean? After all, it’s no use when talking about lengths, or areas, or the like. For that matter, even when they do appear usefully such as when dealing with money, what does -$5 look like?
For an even more confusing thought, think about the following joke:
“Three people, a biologist, a physicist and a mathematician, are made to watch a room with only one entrance. They are told that the room is empty. Two people walk in, and three people walk out. How did they explain this?
The biologist says, “Maybe they reproduced?”
The physicist says, “Perhaps I counted wrongly.”
The mathematician says, “Send one more person into the room and it will be empty again.”
:LOL: :LOL: :LOL: :LOL: :siderofl: :siderofl: :siderofl: :siderofl:
When you have finished laughing your head off, think about the mathematician’s answer. If you write what happened as a sum:
0 (the empty room) + 2 people – 3 people = -1 person.
So his answer is to add one more person to make the room empty again, because at the moment there are –1 people in the room. Which is a silly thing to say, really.
The ultimate point of this is to make you realise something: Negative numbers, that many people are happy to work with daily without a second thought, don’t really exist. They’re just useful to help us explain what happens when shops (or banks) make a loss, or how you slow down, or any time you’re reducing something. They’re just a tool, in other words, to help us make sense of the world.
It’s not even that silly to doubt the existence or use of negative numbers. In fact, for many years mathematicians had arguments about whether they were useful or not, whether they existed or not. You might see people talking about (to go back that that field problem earlier) the “real solution” (20 metres) and the “imaginary solution” (-30 metres). Then, later, when that cubic formula turned up, the person who first wrote it up still didn’t even accept negative numbers as being worth thinking about. You can imagine how hard it was to contemplate square roots of them!
Eventually, out of sheer curiosity, mathematicians started looking into this quirk that was the square root of –1. Where this took them we’ll see in the next article.
Does the square root of –1 exist? Yes, it exists in the same way that negative numbers do. That means that it exists only if you want it to and if it’s useful. Which in turn means that any number exists if you can find a use for it.
This all goes back to another earlier topic: axioms. An axiom is something that is assumed to be true without bothering to prove it. In High School it’s assumed, for example, that there’s no such thing as the square root of –1. This is true. In more advanced maths, it’s assumed that there is such a number. This allows you to do a lot more, as we shall see.
So the number i is just the letter we use to represent the square root of –1. I stands for “imaginary”, by the way.
We’ll be thinking about what we might be able to do with this new “number” in the next article.
-
The next article is still not ready yet, I expect it to be finished within a fortnight though.
Any thoughts on the last one?
-
Something sciency would. Be interesting, maybe how electrons shufflefrom shell to shell during fireworks (or something like that...)
-
They do?
-
Ah huh, i dont really understand.
But i get this bit:
"Does the square root of –1 exist? Yes, it exists in the same way that negative numbers do."
:D
-
Maybe the next could be something about how an audio signal travels from the input ex. the computer to the output device ex. the speaker?
-
Certainly when I have time to fine-tune the article on complex numbers and write that one I might look at that idea.
-
Write something interesting, that I can actually understand. :)
-
Well thanks for the constructive criticism there.
-
Well, what with Exams round the corner and the fact that it'll take a while to put the article I'm planning to write together, it's looking likely that my next "proper" post here will be in a couple of months. I'm sure you're all devastated.
-
:neckbeard: :neckbeard: :neckbeard:
This is the first time I check this thread and I love it! Although I still disapprove the usage of paint-like diagrams :P I agree with Edison about LaTex...
“Three people, a biologist, a physicist and a mathematician, are made to watch a room with only one entrance. They are told that the room is empty. Two people walk in, and three people walk out. How did they explain this?
The biologist says, “Maybe they reproduced?”
The physicist says, “Perhaps I counted wrongly.”
The mathematician says, “Send one more person into the room and it will be empty again.”
You nub...
I actually laughed at that :D
Although the explanation was uncalled for...
Maybe the next could be something about how an audio signal travels from the input ex. the computer to the output device ex. the speaker?
Could be interesting. Dealing with the transformation of a continuous-time sound signal to the electrical discrete-time sound signal representation, Nyquist frequency, Fourier Transform (FFT actually), use of common filters as Butterworth or Chevyshev (depending on the frequency you want to transmit) and the basics of informatics is always interesting.
-
I think the next one should be on QCD.
-
I think the next one should be on QCD.
Maybe in about 5 years.
-
Maybe the next could be something about how an audio signal travels from the input ex. the computer to the output device ex. the speaker?
Could be interesting. Dealing with the transformation of a continuous-time sound signal to the electrical discrete-time sound signal representation, Nyquist frequency, Fourier Transform (FFT actually), use of common filters as Butterworth or Chevyshev (depending on the frequency you want to transmit) and the basics of informatics is always interesting.
Hmm. I might look into that but would find it hard to go into it too much. The problem is that Nyquist sampling depends on understanding the Fourier Transform, which itself depends on integral calculus and complex numbers, and all sorts besides. There's a lot of ground work behind it and I'm not sure that without that it would make sense.
-
Those of you who have died of boredom waiting for the next post, i.e. nobody, will have to wait some time more I think. Still, readers may be interested to know that the General Relativity post (http://www.bfmracing.net/forums/index.php?topic=27579.msg316279#msg316279), with a few typos and small additions, recently won third prize in a lucid Science Writing competition. FYI.
-
-still waiting-
-
- and waiting -
(http://news.bbcimg.co.uk/media/images/59883000/jpg/_59883143_cobwebs_thinkstock.jpg)
-
(http://i.imgur.com/IMGijDt.jpg)
-
Nubs.