### Is there such thing as something with no end?

If I'm honest, I've never really thought about this very much. I know what you are thinking: "* What is this guy on about? He thinks he can write about maths but he hasn't even considered what is one of the biggest, literally, subjects in maths?*" and for you, I have no answer, It just never seemed me to be that interesting, nothing to prove wrong and calculations to make, infinity is infinity, a never ending number, and as brilliant that is, it has never had a real meaning to me...until yesterday:

As I sat down in maths yesterday, I expected the usual (if you need to know more about my maths class, look at my last post). Little was I to know that it would be my inspiration to start this post and, in fact, this blog as a whole. It was my friend who started the process, telling a maths trick. Here it is:

"I'm going to prove that x in this equation, x={^{1}⁄_{2}+^{1}⁄_{4}+^{1}⁄_{8}+...}, actually equals 1.

Ok so first what I want to do is divide both sides by 2, from that we get ^{1}⁄_{2}x={^{1}⁄_{4}+^{1}⁄_{8}+^{1}⁄_{16}+...}

Now if we add ^{1}⁄_{2} to both sides you get ^{1}⁄_{2}+^{1}⁄_{2}x={^{1}⁄_{2}+^{1}⁄_{4}+^{1}⁄_{8}+...}, and look, the right side in that last equation is the same as the right side in the first one, therefore ^{1}⁄_{2}+^{1}⁄_{2}x=x and x=1."

This is the beauty of ∞, that shouldn't work. Every logical part of you says that that couldn't work but no matter how many times you work over it you always receive the answer that x=1. See the problem is that when we add ^{1}⁄_{2} to the set, we expect there to be one more number in it, like if we had a set of {1,2,3} and then you added 4 to it, there would be four numbers in the set. But here is the magic, there are infinity numbers in that set and infinity is not a number, it is an idea and, as it is the highest number, theoretically, ∞ + 1 = ∞, this is how it works, the number of numbers in the set cannot increase because it is already the highest it can be. The maths is sound, we just don't have the brainpower to think naturally about infinity.

Good links to read up more on infinity:

Link 1 (Quite Complex)

Link 2 (Simpler)

Thanks for reading this, please leave your opinions in the comments.

surely when adding the half to each side you would add it outside of the brakets rather than inside..? thus rendering the theorem incorrect

ReplyDeleteBut it is the same thing in the end, it is like saying (4+3)+2 and (4+3+2), you always get the same answer.

ReplyDeleteWouldn`t taking `infinity` into consideration answer the many questions that `mathematics`cannot supply an answer to ? For example.....`Dark Matter`. I know this will upset many of your correspondents, but it seems obvious to me that the systems in place that are being used, have been tried and found wanting for millenia.............. Therefor it`s time to start thinking `outside the box`..........

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