martes, 30 de octubre de 2012

M2C2A: Episode 14, What is 0^0? Let's unscramble this nonsense once in for all!

What is 00? And I'm not expecting some miracle answer or nothing, I'm expecting the truth. And actually, it's much more complex than you might think (and I'm not talking about i here).

My point is, powers with base 0 (0n) could be forbidden.

It is known that xm*xn=x(m+n).

But also that xm/xn=x(m-n).

So if you might call 02 as 0, or 03, you would be forced to say 03/02, that is 01, would be 0/0.

But there is something wrong about that.

Remember when we talked about 0/0? It could even be 0!

But we know that when we say 01, we're talking about a number that when you multiply 1 zero together (that is, leave it as it was), you're talking about 0! So 0/0 is 0 in this case.

But 00 has nothing to forbid it.

The only reason mathematicians called n0 as 1, was because n1 =(n)/n1, (that is n(1-1)=n0) was always 1... except for 0.

So the only thing that can define without barriers 0^0 is... Our number with infinite answers! 0/0! (or 0j, if you remember one of our last episodes)! But of course because almost everyone else believes 0/0 is undefined, 00 is as undefined as.

-The Roaring Thunder

M2C2A: Episode 13, ∞, what has always been wrong about it

Many say +1, 2, or -10^(10^(10^(10^10))), are simply , but no.

is like i, you (usually) can't simplify things with it. E.g.
If + is , then we would be obliged to say -=.

Is it?

We can use something I like to abbreviate as P.I.G (Patterns In Graphs, which can also refer to look for formulas), which is what I'll do.

If we graph x-x=y, we always get 0.

It's a truly linear equation.

We don't expect that when it gets to it will suddenly rise to too.
Same story with * and /.

What makes sense in both cases, is that neither + or * should be expressed as , but as 2, and 2, respectively.

So +1 should be written as it's written here, and so should 3/, 2+1, xy=yx, or x2+x+=0, etc.

...or is it?

This would work perfectly if it wasn't for the definition of , the biggest quantity that can be described, which doesn't have an end.
If +1>, then +1 should be infinity!

But if (+1)<+1 (+2), then +2 should be !
But if this is smaller than (totally infinite!), then SHOULD BE !!!
So my definition of is:


M2C2A: Episode 12, The ABACABA, the biggest word in the mathematical dictionary

To write this word is a total mess, but to write the first 63 characters, this you can do.

Write an "a" every 2 spaces or (21) spaces. This should look like this:

Then, from the first unoccupied space, write a "b", every 4 spaces (22) this time:


Repeat this last step, but with the "c", and now every 8 spaces (23) .


If you continue the pattern, this you should get:


Of course, if you continue this until the "z", you will get a word (227)-1 characters long, which is 134,217,727 letters long! 

And if you include all the character in the Unicode (the majority are Chinese characters and some are just representations as " ", but still, no discussing about that), the epic number of 109,449 you get:

5.40785978562894540705755027563290385780481607 × 1032947 letters!!! 

But what does this apply to?

The Sierpiński Gasket!
the sierpiński gasket

Every time you see a black square, write "a", if you see a little white triangle, write "b", a bigger white triangle, "c"... etc.

abacabadabacabaeabacabadabacabafabacabadabacabaeabacabadabacaba on sierpinski

And if all the "a"s in the triangle are one color, and the "b"s other... you get a colorful Sierpiński! (but that's another thing...)

And to really celebrate Sierpiński and the abacaba, I made the "rainbow gasket"!
Raińbow gasket

Cool, huh?
-The Roaring Thunder

lunes, 1 de octubre de 2012

M2C2A: Episode 11, The root of a number?!

A square root of a number y is by definition a number x in which x2 equals y. And a cube root of a number y is by definition a number x in which x3 equals y.
But we can also do n roots, in which the nth root of y is a number x in which which xn equals y. But there's a special property about roots:
They may have many answers.
For example, the square root of 4.
22 is 4, but also -22. So can 2 and -2 be the roots of 4?
Let's now take a different example:
The cube root of 8.
It may be true, but there are 2 more solutions.
First of all, you'll need to know complex numbers.
A complex number has a real part (2, 4, 7982) and an imaginary part, which can be any number times i.
But what is i?
i is the square root of -1, which doesn't exist, but can be used in many ways.
You can add i (23+i), subtract i (9-i), multiply i (23i) and divide i (i/5).
You can get i to the power of a number (i1=1, i2=-1, i3=-i, and i4=1. This pattern repeats, and in=in mod 4).
So the second answer of the cube root of 8 is
-1 + 1.7320508075688772i and the third answer is -1-1.7320508075688772i.
And fourth roots?
Let's say we have the fourth root of 81. There are 4 answers for this:
A) 3
B) -3
C) 3i
D) -3i

We know that 3*3*3*3 is 81, and that -3*-3*-3*-3 is 81, but why 3i and -3i?
3i*3i is -9, because if the square root of -1 is i, then i2 should be -1. And because 3*3 is 9, 9*-1 is -9, 3i2 is -9. But we want 3i4, not 3i2. So we square -9 to get 81. Same story with -3i.
So really, the root of a number is much more complex than you might think.
-The Roaring Thunder

M2C2A: Episode 10, Is 0 a prime???

Let's think of the definition: A number only divisible by 1 and itself. 0/1=0 ✓, but 0/0, itself is equal to...
Yes, calculators will tell you 0/0 is undefined, but if when we say x/y, you're looking for a number z that passes the test zy=x, so if x=0, and y=0, couldn't z be any number? 0×0=0, 0×1=0, 0×π=0, even 0×i=0!

So 0/0=R∪C, that is, the union of the real and complex numbers.

Any number in or out of the number line can be 0/0, but isn't a number divisible by another when the number is an integer?
So because 0/0 can or can't be an integer, it is only SOMETIMES divisible by itself. So 0/0 is NOT completely divisible by itself, doesn't this seem strange?
We're dealing 0/0 as if we weren't dealing with something comparable with tan(90) or ∞ .
So is it prime?
A little.