Some interesting things about maths

Written by Dave

1. Intro

After reading articles in some emags and noticing that all articles were very similar in their contents, I had the idea of writing this (weird?) article. Yeah, I mean that the articles concerned the demoscene (design in intros, some news, opinions about the demoscene in a particular country, humour)...

Well, this is all what a guy in the demoscene wants to read in emags... Yeah, ok, that's true, of course, :) but, anyway, think for a moment and tell me: Why not read something that really relax you? Mmh... Now, the question is: What topic then? Ok, I can tell you: a simple article about jokes and interesting things in maths. :)

Now, relax and start reading this text... in the hope that, at least, it will entertain you... or... who knows, probably you'll learn some concepts which you don't know... or... who knows, probably you'll hate me for the rest of your life. ;)

2. 1=2.... eh? 1=2 ... How? Are you Mad?!

I'm sure this first thing will entertain some newbies at maths... Heheh, I'll demonstrate that 1=2!

Suppose that a!=0

Also, suppose that a=b, then a*b=b*b

(1) a*b = b*b

(2) a*b - a*a = b*b - a*a

(3) a*(b-a) = b^2-a^2

(4) a*(b-a) = (b+a)*(b-a)

(5) a = a+a

(6) a = 2a

(7) 1 = 2

Arghhhh!!!!! 1=2!!!!!!!!!!!

Now, HOW I CAN believe in MATH??? I always believed in maths because it's rigorous on axioms and proofs... but now, look... 1=2.

Hehe, of course, there's a tiny mistake in the 'demo'... Find it!

3. a^0 = 1

To begin with that, take a look here:

1^1 = 1
2^1 = 2
3^1 = 3
2^2 = 2*2 = 4
3^4 = 3*3*3*3 = 81
5^2 = 5*5 = 25

Now:

1^0 = ?

You know the result, every1 knows it. Are you thinking in the fact that '0' means nothing? Then, why does it equal to 1 when making the power?!

Is this really an axiom?

Now, I'll write the entire demonstration:

Take a number called 'a' where a!=0. (You cannot choose 0 because 0^0 is an indetermination.)

Take another number, er... 'b'.

Now, b=b (right, coz 0=0, 1=1, 2=2,...).

                          -------          -----------
                          | b=b |    <=>   | b-b = 0 |
                          -------          -----------

                                         a^b
        a^0 = a^(b-b) = a^b * a^(-b) =  ----- = 1
            |         |              |   a^b  |
            |         |              |        |
            |         |              |        ---> Our result.
            |         |              |
            |         |              |
            |         |              ------------> Leave the negative exp
            |         |                            down.
            |         |
            |         ---------------------------> Separating each
            |                                      exp. and mult. them.
            |
            -------------------------------------> Substitution.

4. 'e' number... and his alien origin

Every1 has used the 'e' number in a couple of problems at school and university. The 'e' number, like pi, is very important for many reasons (one could be that the derivative of 'e^x' is 'e^x', so the derivative of e^x is itself. IT makes e^x ideal when solving differential equs). I don't wanna bother you with math now... but, sometimes, it's nice to hear some ideas.

Now, look at this: e = 2.7182818 (approximately, because e, like pi, has infinite decimals). Now comes the interesting crappy thing (the comp. sci. students and others must know from WHERE the 'e' number comes, but 15, 16 years ppl probably don't).

        lim      [ 1 + 1/n ] ^ n   = 1 ^  = 1*1*1*1*1*1* ....*1 = 1
        n -> 

        Note: where  = infinity
        ----

WHAT!???? As you see, it cannot be possible... You know perfectly that e is different than 1... So take your calculator machine and do some calculus when n=1, n=2, n=3... :)

1 lim (1+1/2)^2 = 2.25 n->2 lim (1+1/3)^3 = 2.37 n->3 .....

As you see, it's increasing and it's also superior to 1... which leads us to the conclusion that the first limit (= 1) is false. Here's where 'e' appears.

Mmh.. As I see you're really bored with this shit, I won't explain the real demostration of this magic number... Instead of this, I'll tell you how one can obtain the 'e' number applying the Taylor Theorem.

Most CODERS know this theorem because in the past The Taylor Theorem was REALLY useful for computing sin/cos look-up tables in 4k intros without using the 'nice' FPU (read Imphobia Issues). :)

Yeah, in the past it was useful, but nowadays it's un-useful. Well, okay... then, why I'm talking about this theorem? To show you how obtain the 'e' number:

               n=
        e^x  =  -   (x^n) / n!
               n=0

        e^x = 1 + x + x^2 +  ....  + x^n
                      ---            ---
                       2!             n!

        Note: choose a big natural 'n' and use x=1 - the 'e' number appears :)
        ----

The real demonstration involves some steps and also some concepts like the Newton binomial formula.

'e' number appears in our reality in some experiments. This is like pi or the golden ratio. Making e^x and doing a substitution in 'x' we can obtain some nice & SMOOTH gfx. This is, like cos/sin functions, amazing because you can use the gfx for making textures in intros.

5. Some mysterious / special and also popular numbers

13,13, .....

Look: the number of this current Hugi issue is #13. The bad lucky number. Most people hate this number because they think it brings them a lot of bad luck. I REALLY can't understand the belief of this people... Probably it's the ancient culture of theirs... I can't see another reason for this... Let me think for a moment... Mmmh.. Is this a bad lucky number because it's prime? No? Well, then what's the reason... A special day in the history? No? Anyways, that's not the best number for supersticious people.

666,666,666, .....

Another nice number for some people is this: 666. Hehehe. Demon / Beast / Antichrist number or how you want call it. But then, it's only a number like any other... So what's the problem? Where's the problem? It's more repetetive than 13 because it's formed by three 6. Heheheh... But this number is also profoundly disturbing the peace of the mind of a certain people.

69,69,69,69, .....

Heck! Really pretty cool number: 69. It has been famous thorugh years only coz of the love posture... Ejem, I won't continue talking about this number...

18,18,18

The special one which gives us some more rights... :) Teenagers...

65536

In the past, we, the coders, hated it when developing programs... in special when playing with the VESA Hi-Rez modes... :) Shrinking a 640x480 8 bit image into 64kb segs is easy, but ugly too. However, nowadays this number continues disturbing coders...

2,10,16,..... bases

Number 2 is also nice. Two states, 0 and 1. Can you think for a momment? This whole computer world has been developed through this base: 2. Big numbers are horror in base 2. But however, this base number has some interesting properties...

- It's prime.
- It makes CPU processing with more security since it involves 0 and 1 only.
- Error detection is easier.
- Arit. operations like and/or/xor are quickly implemented.

As you know, this base 2 is ideal for computers, not for humans. We still prefer base 10, probably because we have 5 fingers in one hand and other 5 in the other hand. It's more natural for us counting with the fingers when we are child, and so we use the base 10.

Could you image what would have happened if we had only 2 fingers?
Would we count with the base 2?
Would we count with the base 4?
?

As I told in part 4, some numbers REALLY appear in our reality, in our life... In fact, Pythagoras and his friends thought that 'EVERYTHING HAS A NUMBER'. ;)

There are much more special numbers in the life which are contributing to our culture, some are taken from the past and some others are being created now. So, who knows... Maybe you can discover a special number too.

Last but not least, here you have a hidden message (reconstructing is easy... Coders know what I'm referring to... The date? Ah... 1677, Who sent it? Ah.. Isaac Newton):

6accdae13eff7i3l9n4o4qrr4s8t12ux

- Dave/PhyMosys