A couple neat mathematical facts

[robly18]

So I learned something new today.
First of all, let me show off this kinda nice website I found: http://mathwithbaddrawings.com
It's basically the blog of a teacher and how he deals with his students, as well as a few random math ramblings. He also shows images in the form of drawings in a whiteboard, hence the name.
From there, I found this other neat thing: http://www.jamestanton.com/wp-content/u ... r-2012.pdf
It's basically a sort of story on the whole 0.999... = 1 thing. But it has a couple other conclusions.
Take an infinite string of nines. So ...9999
Infinite nines.
How much is that? Common sense would say it's infinity.
Let's see if this is true. Let's start by multiplying by ten.

Alright, now, let's see what happens if we add nine.

But... Wait, is that the number we started with? Wonderful, we can do algebra now.
So we get 10x + 9 = x. Solve for x.
Add x to both sides, subtract 9 to both sides and divide by 9 and you get x = -1
So, let's see what we've done thus far:
Take a natural number(9)
Add another natural number(90)
Repeat
The result: a negative number.
Yeah, that looks about right.
But the question is, is it self consistent?
Let's back away for a second and look at the big picture. By which I mean try other numbers.
How about ...9999.9999...?
Again, an infinite string of nines, stretching left and right. Proceed to multiply by ten. The result:
...9999.9999...
So, let's get this straight.
10x = x
Subtract x. Divide by nine.
x = 0
So, ...9999.9999... is equal to zero.
But what about ...999 + 0.999...? We get... -1 + 1 = 0?
So this is consistent!

Isn't it amazing how something that looks absolutely insane and wrong at first can actually be proven to be consistent?
Heck, from this we can even get some new never before seen numbers. Like ...99909.90999... So this is 0 -0.09 - 90 = -90.09
Wait, so a negative number... But this must mean..
There are ways to write negative numbers without the minus?
Let's try -3. is ...9999 = -1, subtract two and you get -3 = ...9997
To make matters even more interesting, mix that with infinitely repeating decimals. So -3 = -2.99... = ...9997 = ...9996.999...
So, let's get a recap:

0.999... = 1
...999 = -1
...999.999... = 0

And from this, we can represent any number (without infinitely repeating decimals) in four different ways.
For instance:
1 = 0.999... = -...9998 = -...9997.999...


Doesn't math make you feel all warm and fuzzy inside?

[A Random Player]

a + ar + ar^2 + ar^3 + ... = a / (1-r)
9 + 9*10 + 9*100 + 9*1000 + ... = 9 / (1 - 10)
= -1

[exfret]

So I learned something new today.
First of all, let me show off this kinda nice website I found: http://mathwithbaddrawings.com

This should be the new xkcd.

Take an infinite string of nines. So ...9999
Infinite nines.
How much is that? Common sense would say it's infinity.
Let's see if this is true. Let's start by multiplying by ten.

Alright, now, let's see what happens if we add nine.

But... Wait, is that the number we started with? Wonderful, we can do algebra now.
So we get 10x + 9 = x. Solve for x.
Add x to both sides, subtract 9 to both sides and divide by 9 and you get x = -1
So, let's see what we've done thus far:
Take a natural number(9)
Add another natural number(90)
Repeat
The result: a negative number.

Yes but, 10x-x could have multiple values. Since it's followed by infinite 9's, it could be infinity, in which case infinity times ten equals infinite, so 10x-x=x-x=0. But then, of course, 0≠-9, so you obviously can't do that in this case. This doesn't mean it isn't infinity, though. Anyways, since x=x+1 if x is infinity, you could plug than in and get x-(x+1), which is x-x-1, which is -1! Oh, wait, but then that's set equal to -9... Well, you get what I mean. I guess that the context of 10x and x means that when you subtract one from the other, you should get 9x without straying off and saying "Well, that could also be..." like I did. Also, this reminds me of the fact that the sum of all the powers of two greater than or equal to 1 is actually -1. Strange, right?

Yeah, that looks about right.
But the question is, is it self consistent?
Let's back away for a second and look at the big picture. By which I mean try other numbers.
How about ...9999.9999...?
Again, an infinite string of nines, stretching left and right. Proceed to multiply by ten. The result:
...9999.9999...
So, let's get this straight.
10x = x
Subtract x. Divide by nine.
x = 0
So, ...9999.9999... is equal to zero.
But what about ...999 + 0.999...? We get... -1 + 1 = 0?
So this is consistent!

Isn't it amazing how something that looks absolutely insane and wrong at first can actually be proven to be consistent?
Heck, from this we can even get some new never before seen numbers. Like ...99909.90999... So this is 0 -0.09 - 90 = -90.09
Wait, so a negative number... But this must mean..
There are ways to write negative numbers without the minus?
Let's try -3. is ...9999 = -1, subtract two and you get -3 = ...9997
To make matters even more interesting, mix that with infinitely repeating decimals. So -3 = -2.99... = ...9997 = ...9996.999...
So, let's get a recap:

0.999... = 1
...999 = -1
...999.999... = 0

And from this, we can represent any number (without infinitely repeating decimals) in four different ways.
For instance:
1 = 0.999... = -...9998 = -...9997.999...

Amazing! It's even cooler that it's self-consistent, just like my 1/0 thing. (I get to say that if you get to without a formal proof!)

Doesn't math make you feel all warm and fuzzy inside?

Um... Well, not exactly warm and fuzzy, but it's amazing how things twist and turn with a simple start. Simply put, math is just an abstract, logical version of chaos.

[exfret]

a + ar + ar^2 + ar^3 + ... = a / (1-r)
9 + 9*10 + 9*100 + 9*1000 + ... = 9 / (1 - 10)
= -1

SEE! Exactly like the fact that the sum of powers of two get you -1!!!

[testtubegames]

Cool stuff... boggles the mind. As fate would have it, I just stumbled across this video today.

1+2+3+4+5+.... = -1/12

Yup, the claim is the sum of all natural numbers is -1/12. Another one of those strange twists in math... where you go beyond your normal set of rules a bit, and find that, surprisingly enough, things are still self-consistent. If weird.

[robly18]

I'm subscribed to numberphile, and I saw that video as soon as it was uploaded
That video actually helped me a lot in terms of figuring out how infinite sums work.