Math proof that 2 = 1

tehsolace

tehsolace

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I found this pretty cool thing in my math book today. It explains whats wrong with the proof, but I certainly wouldn't be able to find the error on my own... see if you can (or if you can't, then I guess it must be true right?!)

Code: 
Lets say a and b are two positive integers.
a = b                       -> Given
aa = ab                     -> Multiply both sides by a
a^2 = ab                    -> aa is, of course, a^2
a^2 - b^2 = ab - b^2        -> Subtract both sides by b^2
(a - b)(a + b) = b(a - b)   -> Factor both sides
a + b = b                   -> Divide both sides by (a - b)
2b = b                      -> We stated before that a = b, so b + b is 2b
2 = 1                       -> Divide both sides by b.
 
edit: the fallacy is cancelling the (a^2 - ab) from both sides gives 1=2.

To "cancel" a quantity from both sides of an equation is to divide both sides of the equation by it. So, in this step of the proof you are attempting to divide both sides of the equation by a^2-ab.
However, division only makes sense when the number you are dividing by is non-zero. In this proof, a^2 - ab = 0 , because you assumed in step 1 that a=b!

Therefore, it is not legitimate to divide both sides of the equation by a^2-ab, because that would be division by zero, which does not make any sense.

In essence, this proof boils down to saying "1 times 0 equals 2 times 0, therefore 1 equals 2". The fallacy is that, just because two numbers give you the same answer (zero) after you multiply them each by zero, doesn't necessarily mean that the two numbers are the same, because anything when multiplied by zero gives zero.

This is also the reason division by zero does not make sense: there isn't just one unambiguously determined number q such that q*0 = 0 , so there isn't any number that we can uniquely and unambiguously define the quotient 0/0 to be.

If you tried to divide 1 (or some other non-zero number) by 0, you'd run into a different problem: in this case, there is no number q at all such that q*0 = 1, so there is nothing that we can define the quotient 1/0 to be.

That's why division by zero is undefined and not just because it's a rule somebody decided on!
 
The first line just hurt my head to the extent where I'm just going to go, "Yup!"
 
Where's your science now?!
Muahahahaha!
*gathers math, religion, mac vs pc, halo vs half life 2 debates
 
Hey, it's uriel.

I wonder what he's planning to do with all those arguments.
 
I'm making soup.

.......I'm living in Utah now, im gonna find you and steal your foodz, LOL

kthxbye
 
I got 20 bonus points on a math test in high school for figuring that out.
 
I FINALLY realised why .99999999... = 1 MATHAMATICALLY, although common sense is still bugging me about how it shouldn't.
 
I remember my 7th grade teacher posting this "proof" at the beginning of algebra class, she only got to the "/(a-b)" and I ruined her fun.

My younger brother and I were both evil when it came to catching teacher's mistakes or misunderstandings, especially in math.
 
I failed Math B. Twice. And that's after I struggled to get a B in Math A.



...but if anyone's interested, uncopyrightable is one out of the two longest english words without any repetition of letters. The other one is dermatoglyphics.
 
Jintor said:
I FINALLY realised why .99999999... = 1 MATHAMATICALLY, although common sense is still bugging me about how it shouldn't.
Click to expand...

No joke, It works in theory, but reality kicks it's ass :p

Anyway, yeah, I keep staring at that and it doesn't work. I can't figure it out. *Head 'splodes*
 
I hate maths
those greeks who invented maths deserve the death
 
I think most people understand the concept, but to think about it in a common sense way was what hurts the head.
 
<RJMC> said:
I hate maths
those greeks who invented maths deserve the death
Click to expand...

I am so with you on that, But Ikeorus, He does math for fun. Like hard math. Math from Physics class. During church, and school. He is a true Math Geek.
 
Idonno >_< said:
I am so with you on that, But Ikeorus, He does math for fun. Like hard math. Math from Physics class. During church, and school. He is a true Math Geek.
Click to expand...
;(
Say hello to him from me.
 
What was the conclusion from that "Does 0.999... = 1?" thread?
 
Stigmata said:
What was the conclusion from that "Does 0.999... = 1?" thread?
Click to expand...

Some people understand the concept and accept it, some understand it and reject it based on common sense, and some people don't understand it at all.

The End.
 
Stigmata said:
What was the conclusion from that "Does 0.999... = 1?" thread?
Click to expand...

On paper, and looking at it, yes it is true.

In reality, and being used, no, it isn't.

But here is how the thread virutally ended up...

"LOLOL IT DOES TO LAWL! *pic*"
"NO YOUR WRONG IM RIGHT FICKFACE!"
"closed!....
LAWL"
 
http://en.wikipedia.org/wiki/.999

Dr. Math said:
How can .999999.... equal 1?

Date: 03/21/2001 at 15:07:26
From: Emily F. and Jenny B.
Subject: .999999..... I still don't get it

Dr. Math,

In my math class in school, my math teacher always talks about how
whenever she has a problem she goes to your site and finds it or
writes to you. I have a problem.

I know .999999.... is supposed to equal 1. My teacher demonstrated
the subtracted thing and the other stuff you have on your site. I
still don't get it. If .99999999.... goes on forever, wouldn't it be
just a little below one? There would be just a tiny gap between it and
one. Please explain this to me.

Thanks,
Emily and Jenny


--------------------------------------------------------------------------------


Date: 03/21/2001 at 16:22:11
From: Doctor Ian
Subject: Re: .999999..... I still don't get it

Hi Emily and Jenny,

There's no doubt that this equality is one of the weirder things in
mathematics, and it _is_ intuitive to think: No matter how many 9's
you add, you'll never get all the way to 1.

But that's how it seems if you think about moving _toward_ 1. What if
you think about moving _away_ from 1?

That is, if you start at 1, and try to move away from 1 and toward
0.99999..., how far do you have to go to get to 0.99999... ? Any step
you try to take will be too far, so you can't really move at all -
which means that to move from 1 to 0.99999..., you have to stay at 1.

Which means they must be the same thing!
Here's another way to think about it. When you write something like

0.35

that's really the same as 35/100,

0.35 = 35 / 100

right? Well, you can turn that into a repeating decimal by dividing by
99 instead of 100:
__
0.35353535... = 0.35 = 35 / 99

Play around with some other fractions, like 2/9, 415/999, and so on,
to convince yourself that this is true. (A calculator would be
helpful.)

In general, when we have N repeating digits, the corresponding
fraction is

(the digits) / (10^N - 1)

Again, some examples can help make this clear:
_
0.1 = 1/9
__
0.12 = 12/99
___
0.123 = 123/999

and so on.

So, here's something to consider: What fraction corresponds to
_
0.9 = ?

It has to be something over 9, right?
_
0.9 = ? / 9

The _only_ thing it could possibly be is
_
0.9 = 9 / 9

right? But that's the same as 1.

Ultimately, though, this probably won't _really_ make sense until you
come to grips with what it means for a decimal to repeat _forever_,
instead of just for a r-e-a-l-l-y l-o-n-g t-i-m-e.

When you think of 0.999... as being 'a little below 1', it's because
in your mind, you've stopped expanding it; that is, instead of

0.999999...

you're _really_ thinking of

0.999...999

which is not the same thing. You're absolutely right that 0.999...999
is a little below 1, but 0.999999... doesn't fall short of 1 _until_
you stop expanding it. But you never stop expanding it, so it never
falls short of 1.

Suppose someone gives you $1000, but says: "Now, don't spend it all,
because I'm going to go off and find the largest integer, and after I
find it I'm going to want you to give me $1 back." How much money has
he really given you?

On the one hand, you might say: "He's given me $999, because he's
going to come back later and get $1."

But on the other hand, you might say: "He's given me $1000, because
he's _never_ going to come back!"

It's only when you realize that in this instance, 'later' is the same
as 'never', that you can see that you get to keep the whole $1000. In
the same way, it's only when you really understand that the expansion
of 0.999999... _never_ ends that you realize that it's not really 'a
little below 1' at all.

I hope this helps. Let me know if you'd like to talk about this some
more, or if you have any other questions.
Click to expand...

Best explanation I've seen.
 
hahaha

I'm of the opinion that 0.999... does equal 1, but that's neither here nor there. Onwards with statements relevant to the OP!

[edit] Ahh, very nice. The way I think of it is through this proof:

1/3 = 0.333...
3(1/3) = 3*0.333...
1 = 0.999...
 
Ahh, very nice. The way I think of it is through this proof:

1/3 = 0.333...
3(1/3) = 3*0.333...
1 = 0.999...
Click to expand...
That one is nice.
Although, I prefer
x = .999...
10x = 9.999...
10x - x = 9.999... - .999...
9x = 9
x = 1
 
Krynn72 said:
I still dont get it. I honestly cant see how how it wont just be a infinitesimally small amount less than 1.
Click to expand...
Think of it this way: 0.999... is a representation of 1. Like how there are multiple words that mean the same thing in English - those words are just different representations of the same idea.
 
Krynn72 said:
I still dont get it. I honestly cant see how how it wont just be a infinitesimally small amount less than 1.
Click to expand...

You can't really think of it that way.

"you're _really_ thinking of

0.999...999

which is not the same thing. You're absolutely right that 0.999...999
is a little below 1, but 0.999999... doesn't fall short of 1 _until_
you stop expanding it. But you never stop expanding it, so it never falls short of 1."

See? It HAS to be 1.
 
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