+349 (don't worry this is not off-topic i have a question at the end)
We all know what infinity is, right? 1, 2, 3, 4, 5, etc. What makes counting/natural numbers "infinity" is the fact that there are infinite possibilities for a number. For example, one. The number one is one of the infinite possibilities for a counting number. So is 2, 3, 4, 5, and so on. But is it possible to square, or even cube, infinity? Let's find out.
Right now we are dealing with whole numbers. Simple ones, twos, and threes. What if we go deeper? What if we add decimals to the equation? Let's start with one decimal place. heve we have 0.0, 0.1, 0.2, 0.3, all the way to 0.9, then 1.0, etc. Now, every possibility for a counting number has 10 possibilities, making the amount of possibilities for counting numbers increse ten-fold. This would be ∞*10. But if we add the second decimal place, we get 10 more possibilities for each of the 10 possibilities of the infinite amount of possibilities of counting numbers, or ∞*10*10! That's alot of possibilities. If we keep going, we get ∞*10*10*10, then ∞*10*10*10*10, etc. until we reach ∞*∞, or ∞^2.
So we can square infinity, sure. But how do we cube infinity? It's simpler than you think, and it involves fractions. We know what fractions are, they are basically divisions or ratios, and they are composed of numerators and denominators. This means that fractions with decimals have (∞^2)*2 possibilities!
But this isn't infinity cubed, you might be thinking. But remember, we can stack fractions. Stacking fractions looks something like \(\frac{\frac{1}{2}}{\frac{3}{4}}\)
The fraction above has (∞^2)*2*2 possibilities. So if we keep stacking, over and over, just like squaring infinity, we get (∞^2)*∞, or ∞^3 possibilities.
So, we have cubed infinity. Something that can get a bit confusing, but still fun. What we actually did there was step up the level of infinity. We went from the infinity of natural numbers to the infinity of decimals to the infinity of decimal-fractions. But don't worry, you don't really have to know all this, just something fun I came up with.
Now my question is can we do ∞^4? Or even ∞^∞? If so, how? Is it the infinity of real numbers? Or could it involve "i" or imaginary numbers? Please leave answers.
And that's it. Thanks for reading 
Infinity is NOT a number but a CONCEPT or IDEA, such as space stretching into every direction FOREVER....etc. !!!!.
+349 I'm visualizing the concept of infinity as the theoretical "number" of possibilities for a number or decimal, or "numerical infinity".
But I agree with you 
By your post i can assume you're about 14 or 13 years old, exploring the wide imaginary world called math.
Other people told you you're wrong, "You cant multiply infinity" "infinity is not as number", the usual stuff. Then you defended your post because you are "visualizing the concept of infinity as the theoretical "number" of possibilities for a number or decimal, or "numerical infinity"".
But i think they are missing the point.
But that part will come later- first we need to go back.
30 minutes before writing this post, i came across your post. I started reading, and i got furious. "How could you write those thing?!" i thought to myself. "Is this guy out of his mind?! what is he doing?! you cant use infinity like that, you just cant!!"
you reminded me of a guy i came across on the internet- a fanatic guy that didnt finish highschool. That guy, seemed to appear in every math forum that mentioned his name. Whats interesting about him, is he had some interesting claims like the claim we can only infer things about math by trying those things in the real world- He came up with a really weird theory, using unprofessional, invented words to express some of his ideas- one of them, that was a direct result of his previous theorem- was that pie is not constant. The way he "proved" it was probably drawing circles in his office.
He had some other, weird ideas. But the most irritating thing about him, was he was never wrong. Every time someone tried to tell him why, He posted something he already posted, or explained why he's right using his broken logic.
I remember one of those people told him that math is not about measure, and that geometry is a mathematical thing that cant be proven using real life, but using math. The guy's response was "if this is the way of mathematics, this is not my way", and that mathematics and the "geometric reality" are different.
He wasnt a university student, but that didnt stop him from arguing with professors and genuises.
And you remind me of him in a certain way. the way you try to create your own math.
But i guess that's fine, because you are young and cant really start learning advanced mathematics. We all stumble sometimes, especially when we are thirsty for knowledge but cant get it.
Oh right, i almost forgot! i promised my part about why they're wrong-
What the other people that answered your question didnt understand is, the problem with your post is NOT the fact you cant multiply infinity, but another thing-
You think the wrole "combinations" thing applies to infinity as well. But it doesnt. And even if it did, you made some serious mistakes.
Here's an example: you said the number of "combinations" of numbers with all the decimal places up to the nth place is infinity*10n. But by that you just proved that the "infinity" you defined in your post, multiplied by any number, is infinity! why? what you said is, there are infinity combinations for the integer part of the numbers and 10 combinations for every decimal place, making infinity*10*10*10......(n times)*10=infinity*10n combinations. Here's another way to aproach this: Every number with n decimal places can be written as x/(10n) when x is an integer. in addition, every number that can be written as x/(10n) is a number that can be written with n decimal places. That means the number of combinations is the number of combinations for the x in my formula=infinity. Therefore infinity=infinity*10n.
Then your broken logic somehow concluded that 10*10*10.....=infinity. I know it looks "obvious", but you cant just say it without defining it, although the only way you defined infinity was "The number of combinations for an integer". You cant just say that. You have to prove it.
Another problem with your post is YOU DIDNT DEFINE INFINITY*INFINITY. Ok, So, somehow infinity*10*10*10.... is infinity squared (Suppose you actually proved it). What the h**l does it mean? what the h**l does "infinity squared" combinations mean? is it an infinite number of squares? is it the proof god exists? is it a banana?
I appreciate your interest in mathematics, i really do. But you have to understand that mathematics is a precise subject. You have to prove things strictly, and make sure you define EVERYTHING.
+118609 Hi :)
I always like to see people thinking about numbers the way you do but infintiy^2 has no real meaning. I mean it is no bigger than infinity. You can square huge numbers and get an even bigger number but infinity is not a huge number. It is a concept.
+349 I agree, but my visualization of infinity in this theory is the amount of possible numbers, or in this case, numbers, decimals, and fractions.
Thanks for your opinion!
You should read this article. Also, look up the mathematician Georg Cantor who spent a great deal of his life on such matters as infinity.
https://en.wikipedia.org/wiki/Infinity
+33616 There is an arithmetic of infinite numbers - search for "transfinite numbers".
The "smallest" infinite number is the number of natural numbers (1, 2, 3 etc), and is generally represented by the symbol \(\aleph_0\) .
Multiples and finite powers of this are still \(\aleph_0\) !
However, the number of Real numbers (including irrational) is a "bigger" infinity!
Also \(\aleph_0^{\aleph_0}\) is bigger!
Transfinite arithmetic is weird and wonderful!
Georg Cantor ranks among the greatest of mathematicians and, by implication, philosophers of the nineteenth and twentieth centuries. He gave us the first tangible dimensions for infinities. He moved the realm of infinity, and by extension, eternity, from the exclusive territory of metaphysics and religious philosophy to that of mathematicians and physicists. To him, finding and understanding infinities may have been the beginning to finding and understanding the essence of God. Most mathematicians use symbols based on the Greek alphabet, but Cantor used the first letter of the Hebrew alphabet, “Aleph,” to represent the “beginning” of his defined infinities. Since then, it’s the only commonly used Hebrew letter in mathematics.
Though infinities would seem to contain everything –they do not. Though each infinity continues indefinitely, they are missing elements—the infinites are unique. Each Aleph has a subscript to identify its uniqueness from the other infinites.
\(\aleph_0 \; \text {infinite set of integers}\\ \aleph_1 \; \text{the set of irrational numbers, including transcendental numbers.} \\ . \hspace {1em} \small \text {(There are more irrational numbers between two integers than there are integers).}\\ \aleph_2 \; \text{the set of lines that intersect a point. }\)
An amazing mind, for sure, but it seems that giving birth to great ideas requires a certain amount of travail.
Cantor had his critics and detractors. One of the most notable was Poincare, who declared Cantor’s ideas a "grave disease" infecting the discipline of mathematics, and Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan," a "renegade" and a "corrupter of youth." Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable. Source: (wikipedia.org/wiki/Georg_Cantor) and ( Joseph Dauben, (2004), "Georg Cantor and the Battle for Transfinite Set Theory")
Cantor seemed stoical about criticism, but he passed it on in kind –actually, he was much better at biting his contemporaries. Cantor opposed the theories of infinitesimals; describing them as both "an abomination" and "a cholera bacillus of mathematics" Cantor also published an erroneous "proof" of the inconsistency of infinitesimals. Apparently, Cantor could understand the infinitely large but not the infinitely small. Source: (wikipedia.org/wiki/Georg_Cantor) and (Nominalistic Tendencies in Contemporary Mathematics and its Historiography", Foundations of Science, 17 (1): 51–89)
I may have found another infinity after reading these technical mathematical publications. Though the information is finite, it might take me an infinite amount time to understand it.
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+9519 My rules of dealing with infinity:
1) ∞ + C = ∞, where C is any number.
2) C * ∞ = ∞ , same as above
3) ∞^C = ∞, same as above
Because there are nothing bigger than infinity, adding something to infinity is still infinity. Imagine a wall is blocking your way (because no bigger than infinity)and there are someone pushing behind you(the addition/multiplication/exponent), and that guy pushing behind you won't make you penetrate through the wall...XD
So that's why (I think) the 3 rules are true.
+9519 Anyone heard of this story?
There is an apartment where there are infinite numbers of rooms(and those are full for some reason...) and one person more wants to live here too, so everyone move to the next room(The guy who lives in room 1 moves to room 2, the guy who lives in room 2 moves to room 3,.... etc.). So that person also have a room! :)
And then there are a bunch of people whose number is same as the number of residents, they all wanted to live in the apartment. Then everyone move to the room of (number of original room * 2) (The guy who lives in room 2 moves to room 4, the guy who lives in room 3 moves to room 6... etc). So all people have a room :)
Conclusion: Infinity is awesome :)
What you mentioned is only a fraction of that story:
This story is called "hilbert's hotel" and it teaches, in a very sophisticated way, about cardinal number of different sets, and compares them to different cardinal numbers of different groups (talks only about infinite cardinal numbers, mostly about the smallest infinite one, aleph 0, the cardinal number of the set containing the natural numbers: 1,2,3,4...... also called "the countable infinity")
And again, before you say that "infinity plus/times something else is still infinity" you have to translate it to the "set theory" language and CLARIFY THE MEANING. otherwise, it means nothing.
