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 x^m*xⁿ=x^(m+n).

But also that x^m/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, x^y=y^∞x, 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:

nonsense.

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:
a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a_a

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

aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba_aba

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

abacaba_abacaba_abacaba_abacaba_abacaba_abacaba_abacaba_abacaba

If you continue the pattern, this you should get:

abacabadabacaba_abacabadabacaba_abacabadabacaba_abacabadabacaba
abacabadabacabaeabacabadabacaba_abacabadabacabaeabacabadabacaba
abacabadabacabaeabacabadabacabafabacabadabacabaeabacabadabacaba

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!


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



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"!


Cool, huh?
-The Roaring Thunder

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?
Yes!
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=i^{n 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!
YES!

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.

M2C2A: Episode 9, Goodbye Graham's number, you ain't big anymore.

What if we could make a number BIGGER than Graham's number, just for fun.
We'll call Graham's number G.
First of all, G+1.
Much more drastic, G*2.
Even greater, G^G. Or (G!)^(G!). 
But what if we invent some mathematical terms for a certainly bigger number, so, SO BIG, that by only writing the number of digits, of the NUMBER OF DIGITS, OF THE NUMBER OF DIGITS, YOU'D GET A NUMBER BIGGER THAN (((G!)!)!)!
How about we invent the "exponetorial"?
Let's write this as "¡".
This, instead of the factorial, which is 2*3*4*...n, goes 2^(3^(4^(5...n?))...)
So for example 2¡ would be 2,
3¡ would be 8,
4¡ would be 4,096,
5¡ would be 1,152,921,504,606,846,976, and
6¡ would be 2.3485425827738332278894805967893e+108!!! (the ! are exclamation signs).
7¡ would be 3.9408424552214162695348543183639e+758,
8¡ would be 5.8171811191842110297035069398346e+6068, and only
9¡ would be an overload for my 9.99999999999999999999999999999999e+9999 calculator!!!
How would we express G¡? And
(((((G¡)¡)¡)¡)¡)¡?!
Now, how about we try something definitively bigger than that.
Let's start with G¡. Now put G¡ "¡" after G (A.K.A G¡¡...¡¡¡, where the total number of "¡" is G¡). Let's call this G_1¡. 
Now let's make G_2¡ which will be G¡¡...¡¡¡, where the number of "¡" is now G_1¡. If you continue this to G_G¡, what will you get? An non-infinite number beyond what you could think possible!!!!! And G_G¡^G_G¡?! (Your brain can explode now).
-The Roaring Thunder

M2C2A: Episode 8, The magnitude of infinity.

The best way to see the magnitude of infinity, is to compare it with other big numbers and say "Infinity is bigger than that.". But with what big numbers?
In third place in the list of very big numbers, we have the googol (not to be confused with the Google searching machine), which is equivalent to 10¹⁰⁰ (A.K.A 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000), which is quite a big number.
In second place, we have the googolplex, equal to 10^googol (A.K.A "A number so big it couldn't be written because the observable universe isn't big enough").
But in first place, we have Graham's number, which is equal to this:
Suppose you want to write 3 cubed. You could say 3³, but you could also say 3↑3. But what if you wrote 3↑↑3? That would be 3 to the power of  3↑3. And 3↑↑↑3? This would be 3 to the power of 3↑↑3. But this is already 3^{7,625,597,484,987}! Why would someone need a number so ridiculously big?! Is this Graham's number?
Not even close.


We need 3↑↑↑↑3! And this isn't still Graham's number!!!
Now, let's call 3↑↑↑↑3 g_1. g_{1 is EXTREMELY BIG, but it is NOT Graham's number.}
Now, let's make g₂, where g₂ is equal to 3↑...↑3, and the number of arrows is g₁. Is this Graham's number?
Not by a googol of zeroes close (literally)!!!


Graham's number isn't achieved until g₆₄, and is so ridiculously large, that NOBODY KNOWS HOW MANY DIGITS IT HAS NOR IN WHICH DIGIT DOES IT START.
But this has nothing to do with infinity, does it? But this is the point I wanted to talk about:
Next time you hear "Infinity", remember this:
"BIGGER THAN GRAHAM'S NUMBER".

-The Roaring Thunder