Mathematics

To be honest, mathematics makes my head spin faster than an electric fan! Maybe because I’m not good enough in it. But I still appreciate it.

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YOU RIGHT DUDE !!

math is easy its everything else thats hard

the good thing about maths is that there is a right and wrong answer – not like most subjects which are about opinion

That's right 2+2 will always =4. No arguments. I did like it math class when I got the wrong answer, BUT, the taecher saw that I knew what I was doing, I just made a mistake in one of the steps. So I got 1/2 credit, Yea!!!

But there are examples in maths where (i times j) is not the same as (j times i), so it's not quite as clear-cut as you might think…

true but that kind of math doesn't count haha   in most cases of math you can always find the answer n its either right or wrong  like A + B (always) = C 

Let's assume, for arguments' sake that i=4 and j=3, now substituting these new values into your original equations, we now have: (4 times 3) and (3 times 4), both of which equal 12. Maybe I've over simplified, I don't know, but it seems pretty clear-cut to me.

exactly!  im all for the simplicity that gbman has shown, n thats y math is easier then other subjects

No, no dumbbrunette, you gave me the idea, it's all you.

How kind, what a gentlman, but how about we share the idea and call it even

Okay, we'll share it, we may need to stick together on this one. Finrod is not to be taken lightly, I know he's got one or two more tricls up his sleeve. 

Very true

minus's  and fractions mess it all up = try the sums including these – Finrod teaches this stuff so i aint arguing with him on this 

I know he's going to tell us anyway, so why not have a little fun?

fractions dont matter say i= 1/5and j= 2/3then

1/5 x 2/3 =2/15

2/3 x 1/5 =2/15

see so i say we challenge finrod

maths was easy and fun, until i was introduced to multivariable calculus, linear algebra, vector calculus, numerical analysis and discrete maths.

ok well that doesn't sound fun at all

Oh me oh my! I've a lot to read, here...

Right... Here we go.

gbman, if you pick quantities from the set of real numbers then what you say is true. But what is true is not always obvious, as mathematicians have found - often to their cost - over the centuries.

Gauss (I can't find a useful link for him! I'll come back to Carl Friedrich G. later) discovered a wholly different type of number, one which is called 'imaginary' but which is just as real - or illusory - as any number you know such as one, thirty-seven, minus ten, pi, three point three (recurring).

Essentially what he did was discover that with one addition he could step off the 'line' of numbers: think of a ruler and imagine the zero point. Now extend the ruler infinitely in both directions. There, you've caught all the numbers, haven't you? They must be all there somewhere on the line. Mustn't they?

No.

Most definitely not.

Look at this problem: "x squared = 4. What are all the possible answers to this?"

Answer: Plus two and Minus two

There are no others.

Now, how about this problem: "x squared = Minus 4"

That really stuffs it - cos we've established that +2 and -2 are the roots (sorry, square roots) of the first problem.

OK, said Gauss... let's discover a 'new' number that's the square root of Minus 1. Let's call it "i".

And with that Gauss leapt off the 'number line' and began to fly...

More later if interest is shown!

P.S. If you think i was being arch when I described numbers as 'illusory' above, then answer me this: what is 'seven'?

ok i've heard of some of the stuff you talked about and some of the stuff I didn't get, but sill the answer to any proble is there tofind whether its imaginary ornot thereisa rightanswer to find and a wrong one (actually several wrong ones)

ps. 7 is one more the 6

Yes... but what's six?

This is at the very heart of mathematics... you have to define exactly what you mean. It has been described as the language of science; but you have to use terms - like lawyers do, only more demanding - which say exactly what you mean.

Yes, Finrod I knew you were being arch when you typed your initial comment. That is why I pursued the course that I did. I have a problem (no pun intended) with "imaginary" number i (I am assuming that is what the i stands for. When I pose the equation of the square root of -1 to my trusty adding machine, I do come up with the value of i, but I do not see how you can use it in a mathmatical equation.

I came up with a similar solution in my accounting class in college, I developed a category called "net over/short" and I applied this to the appropriate line whenever I had $ left over. It was a simple yet ingenious solution, that ultimately resulted in my repeating accounting, but that's neither here nor there.

You can take this as interest Finrod. 7 is also 4+3, or the square root of 49, there are an infinite number is answers when, of course, the question is viewed mathematically...

Don't worry dumbbrunette, I got your back. I just hope someone has mine!! Yikes!

The quick answer, gb, is that the damn things are required in solutions to equations that arise quite naturally in, say, physics problems. I'll have to do some digging to dredge up suitable examples... the monkey and the negative coconuts springs to mind; but I'll have to abide in my ivory tower a while and reflect on my inmost thought.

God, that last bit sounded pretentious! Gotta go now, but I'll post more later today.

gb i recon you scared him off

No he didn't. I was busy doing this and forgot all about this thread.

I'll be back! 

If you have a strong constitution, the best Web version I can find of the Monkey and the Coconuts problem I mentioned is here.

The point I'm trying to make (it's on p.3 of the link) is that if you allow negative coconuts into your world a horrendous problem becomes ridiculously easy. Well, to a mathematician, anyway.

And has anyone here ever been overdrawn at the bank (same idea)?

So if it makes the problem easier, mathematicians will use it. After all, who would cut off their nose to spite their face?

Now, gbman, there exist equations such as (I wish I could use proper notation!) :

"1 + (x squared) = 0"

Are you going to argue that we should leave them unsolved, when Gauss has shown us what technique is needed to solve them?

I think the essence of what I'm trying to say is that "i" is a creation of the mind just as much as "7" or "6" are (which is why I posed the question "What is 7?" earlier).

Just because (3 times 7) is equal to (7 times 3) does not imply that this is true for all numbers - you can call it a conjecture, because of all the evidence you can gather together to back it up - but it does not constitute proof.

Incidentally, it is true for the 'imaginary' and 'complex' numbers - I'm only using these as stepping stones off the 'real' number line that I mentioned earlier. More later if your interest is piqued!

ok gbman i got your back but im not sure what good ill be b/c my mind is boggled by this negative coconuts thing!

1st of all Finrod ill have to take your word that negative coconuts solves the equation b/c your link doesn't show page three. So maybe what I'm about to ask was explained there, but i dont know. So my question is where did these mathmaticians come up with "i" or "e" for that matter? And I understand what you said about "i" not being on a numberline (ruler) but what about on a number plane?

Then your arguement about " "i" is a creation of the mind just as much as "7" or "6" " i can understand that, but then you contradict your self when you say that they are the same but different "Just because (3 times 7) is equal to (7 times 3) does not imply that this is true for all numbers ". So if "i" and "7" are the same just a creation of the mind to represent something then why wouldn't they work the same and have the same rules to follow?

Sorry if the link doesn't work - it works OK when I try it. If it doesn't work for you, the horrific equation that the very innocent looking puzzle generates (in one of its versions) is:

1024n = 15625f + 11529

Now, before anyone from the back shouts out "Two unknowns, one equation?" I'll slap them down with the condition that the (as yet) unknown numbers n and f have to be integers - whole numbers.

I'll tell you straight, I couldn't do that by trial and error. Apparently there are techniques (probably as obsolete as the slide rule skills I have), but I don't know them.

Anyway, in the problem, the sailors divide up the coconuts into five piles six times: this means the answers to the problem - there is an infinite series of them - relate to the number 5 to the power 6:

5*5*5*5*5*5 = 15625

Now consider starting with a pile of precisely minus 4 coconuts.

Minus 4 = minus 5 + 1

(I hope you agree)

- so the sailor can divide the ghostly pile into five shares of minus 1 coconut for each of them and have one (positive) coconut left over for the monkey.

So after he hides his share, he puts the other minus 4 coconuts back.

Hence the next sailor goes through exactly the same process, and so does the next, and so on...

Hence (you know, that's a very satisfying word when you've just worked out yourself that it's true!) the smallest positive solution to this monstrous yet innocent-looking problem is:

15625 minus 4 = 15621

If at this point your mind is saying "Stop reading this and go and watch the television" then I will thank you for having the patience to read this far. Me... I'm going to go lie down on a darkened railtrack for while.

I'll come back and address more points later.

My point about the 'number line' was that most people think that if you start with zero and keep counting, then you'll cover all the (positive) numbers that exist e.g.:

1, 2, 2.35, pi, 7, 36.9, 100, 10000000, googol, googolplex... (the last two are very large indeed)

-so then if you include all the negative numbers as well, you have an infinite line of numbers, and any number you care to dream up is somewhere on that line.

Not so.

What you have there is called the set of real numbers. The rules of arithmetic that we all use apply (of course!) to any numbers on that line.

The square root of minus one (labelled "i" by Gauss) is not on it. You can picture it as being on another line, at right angles to the first one. This crosses the 'real' line at zero and can be numbered:

zero, i, 2i, 3i, 4i 5i... and so on.

Add their negative equivalents and you've covered all the possibilities of that line.

Do the rules of arithmetic (I won't define them all here) apply on this line? Yes they do.

Can you have a mixture of 'real' and 'imaginary' numbers, e.g. (3 + 4i)? Yes you can. It's called a complex number and is necessary (gb please note) for the solution of many an advanced problem - including ones where the answer is an 'ordinary' real number.

Do the 'rules' apply to complex numbers? Yes they do. You have to be a bit more careful, especially with division - but you can do everything - plus, minus, times, divide, powers...

You can draw two lines on a piece of paper at right angles to each other (crossing at the centre of the paper - label that 'zero' on both lines), label one line from -10 to +10, label the other line from -10i to +10, and then every point on that sheet corresponds to a number.

Some examples would be: 5 3i (4 + i) (-7+5i) (10 - 10i)

Hey, wait a minute - isn't that just a graph? Yes.

It's blessed with the name Argand diagram, and the piece of paper could be renamed the complex plane, but it's just a graph.

Have I got to the ones which don't obey the 'rules'? No, not yet. But I wouldn't dream of trying to explain those without clarifying these first.

So - if you're still reading this, db - I didn't contradict myself; but obviously I didn't explain what I was doing clearly enough.

Hence this post.

Back later.

wow never though of numbers being anything but one dimentional - never considered having a further dimention before, yet spookily I have just clicked, this is what my daughter was doing in her homework last week  - is there a third dimention too ... or more

You're very astute, saddam!

Well done for spotting the next big question!

Were you any good at maths at school? (I'm betting the answer's 'no'!)

was that an insult or a compliment ... mmmmm

 I wasnt very good at anything to be honest - but am good at basic simple maths and working computers etc...

wow finrod. you are very good in maths. ur in college? what are you taking?

It was neither insult nor compliment - it's just that very few people take to maths naturally.

If you think the next step is to leap off the paper and fly through the third dimension, you'd be absolutely spot on. You'd be thinking like many 19th century mathematicians, especially one called -

But I'll tell his story later...

Feel free to read the profile, Ramong. It's all true. 

i think the proper way to say that

(a * b) = ( b * a)

is  

(a * b) = ( b * a), if and only if a and b is a real number. 

but Finrod already said that did'nt he? :p

p.s: Finrod, I would've used the notation. :)

Welcome, Ramong.

It's true of the real numbers, both positive and negative, yes. Since you join in halfway through this discourse, allow me to emphasise that it's also true of the imaginary and complex numbers which I have just been attempting (!) to describe - so a mathematician would quibble your use of 'if and only if', but you have essentially grasped what I'm getting at.

The Final Episode (?) of this story will follow soon.

Watch this space. Or not. I don't mind.

Oh I just read the entire discussion. I'm beginning to see your point, Finrod.

I think maybe I've done this before during my undergraduate years (Im still in my final year :p).

Anyway, I couldnt remember.

Please continue Finrod.

And for those who read my previous comment, please ignore it as it maybe wrong.

P.S: I would've edit the previous comment, but that would'nt make sense for the posts following it :)

anyways – it all goes to prove, the more you know about something the more you realise how little you know about it

Very true, saddam. At the end of the 19th century, someone made the (foolish, oh so foolish) comment that Physics was nearly complete - that there wasn't much left to discover.

Well, it wasn't long after his words had finished echoing when there came the discovery of the Ultraviolet Catastrophe - I'll put a suitable link in later - and the results of the Michelson-Morley experiment, which showed the speed of light (in a vacuum) to be the same whether you were stood still, moving towards the light source, or moving away from it.

Out of this mess, at the beginning of the 20th century, grew the disciplines of quantum mechanics and (special) relativity.

Ultraviolet Catastrophe - Michelson-Morley experiment, speed of light (in a vacuum) disciplines of quantum mechanics and relativity. ......

My head hurts - I think the closest I can get to understanding such things is to think about a fly flying around my car whilst I drive at 90kph

So the final part (?) of this leads us into the 19th century, where - as saddam very astutely points out - mathematicians worked like - well, mathematicians - to extend the idea of number into a third dimension - like taking off from the sheet of paper I mentioned earlier, and flying above or below it (which would be the equivalent of positive and negative).

Well, they struggled with it. And nobody got anywhere.

I'll now introduce William Rowan Hamilton, famous Irish man (I knew his name from the maths functions used in theoretical science long before I knew any of what I'm explaining now).

He hit upon the idea of throwing in the towel to three dimensions, and seeing if he could make it work in four. - which is not the sort of thing that academics tend to do in this sort of situation.

 But he made it work. He tested, he explored, he thought - and came up with - or, more accurately, discovered -  a set of numbers with four parts rather than the two of the 'complex' numbers I described earlier - which obeyed all the rules.

Well, nearly.

Hamilton called his three 'imaginary' parts i, j, and k. These units, together with the 'real' unit, the number 1, can be put on four lines at right angles to each other and  the 'rules' of plus, minus, times, divide etc. can go right ahead - with one odd exception.

 (i * j) is not equal to (j  * i).          The scheme only works if (i * j) = minus(j * i)

In full - seeing as I've got this far, I'll set it out completely - he found that:

    i squared = j squared = k squared  = minus 1

    (i * j) = minus(j * i) = k

    (j * k) = minus(k * j) = i 

    (k * i) = minus(i * k) = j

    And, bizarrely,

    (i * j * k) = minus 1

He named this number system quaternions. It's strange, it's weird - but it works, as long as you abandon the idea that (a times b) equals (b times a). Which is where we started.

To complete saddaam's point, does it work at any higher level? The answer is yes, you can make an eightfold system work - they are called octonions, and that is as much as I know about them.

So, in summary, you can get number systems to work with 1 ('real' numbers), 2 ('complex' numbers), 4 and  8 dimensions. Apparently it has been proved that no others are possible, which explains how they had such a hard time in the 19th century with 3 dimensions.

Wow you boys have been busy!

Thank you finrod for explaining it more clearly and finishing the coconut problem. And I remeber now that I learned that "i" is the squareroot of minus 1, and all about the Argand diagram. (its funny what you can remeber when someone brings it up again haha). Thank you so much for not going into quantum mechanics! All I have to say to quantum mechanics is YUCK.

And I'd  never heard of i,j, or k so thank you for enlightening me. but reading over the post it doesn't seem that complicated in the end, its just like you said , finrod, a person would just kind of have to forget what they know.

But 1 question still, did mathmaticians ever figure out a way to do the 3rd dimension simply?

Actually, quantum mechanics drives me insane too.

I don't understand your enquiry in your last line - do you mean about getting a 3-D (rather than 4-D) system of numbers? if so, read my last paragraph above. If not, I'm sorry, please could you rephrase the question?

ok so I re-read your post and your saying that to date there is no number system for 3-D?

Isn't it frustrating when you find the perfect Web page for your purposes and then lose track of it? I've just spent half an hour trying to rediscover a page I referred to before typing my last opus, and now I can't damn well find it!

I'm sorry, db, I'll try and express myself more clearly.

I've been looking for a maths page - not too technical - which I found this afternoon. It stated that you could formulate systems of 'dimensionality' 1 or 2 or 4 or 8. Only. Any other level is impossible. But I can't back that statement up with a link, as I write this. I remember the page said that any system higher than 8 'lacked certain fundamental properties' or some similar phrase, but until I find the cursed thing to link to it I can't tell you any more.

Incidentally, I also found that out today; and I can't help wondering how significant it is that if you start with the number 1 and keep doubling it... but that's part of the beauty (and I use the word carefully) of mathematics.

That reminds me... perhaps I'll write about Euler's ( German: pronounced 'oiler') formula next...

Tell me if you like this. Tell me if you don't. I'll keep rambling on if you like.

It's interesting because math is one of the most definate things in the world (next to death and taxes of course) and when i mean definate i mean that it's not at all if that makes any sense but compared to other subjects studied anywhere math is the most consistent (i guess thats a much better word to use here instead of definate) but please continue I find it interesting but I like math, (i know most people who read this will disagree with me and find me weird) but i like it a lot.

No, it's not weird; what could be more natural than investigating one's world? The world of the mind is just as real as the world of the body.

And when earlier I got it just right, saddam saw clearly what I was thinking. 

I'll shut up now. It's getting late here.

"A mathematician may say anything he pleases, but a physicist must be at least partially sane."

-The American Josiah Willard Gibbs (1839-1903), mathematical engineer, theoretical physicist and chemist.