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# Why 1 = 2

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Assume x = y which means 1 = 1

x = y then multply both sides by x so...

x2 = xy then subtract y2 so...

x2 - y2 = xy - y2 then expand these quandratric equations to...

( x + y ) ( x - y ) = x ( x - y ) which is just another representation of the previous line.

Now, we currently have ( x - y ) on both sides, so lets divide both sides by ( x - y ) to get what is left...

(x+y)(x-y) = x(x-y) .

(x-y) . . . . (x-y)

That leaves us with the following

x + y = x

if we go back to the beginning and say that x equals y so that x = 1 and y = 1 then

1 + 1 = 1 OR

2 = 1

Logical except for the fact that you can not divide by x-y because if x=y then x-y=0 and you can not divide a number by nothing - ie, nothing goes into one an infinite number of times plus one.

Ohh..!!

I saw this before!

It was on my test (Geometry) as an extra credit (I didn't get it... )

I also saw a similar problem in my geometry book (another extra credit)

which described how 2X2=5 (something similar to that...)

All of these are just breaking only one rule, which is that you can't divide by zero.

However...I'm kind of confused on what people are talking about when it comes to things like .9999...

...Oh well...

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• 2 months later...

Actually, I've seen a similar mathematical fallacy stated as the following. I like it better because the error is hidden a little better and the proof looks a little cleaner. The proof is pretty much the same and is identical to the problem posted at the begging of the thread in that the algebra is correct but the proof violates a basic rule of mathematics.

Proof that any number 'a' is equal to a smaller number 'b':

a = b + c

a(a - b) = (b + c)(a - b)

a^2 - ab = ab - b^2 + ac - bc

a^2 - ab - ac = ab - b^2 - bc

a(a - b - c) = b(a - b - c)

a = b

Edited by 95yj
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• 4 weeks later...

x(x-2)=x-2

x(2-2)=2-2

0=0

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• 1 year later...

Alright, try this one on for size:

X=.9999999...

10x=9.99999...

(10x)-x=9

(9x)/9=1

x=1

.999999...=1

X=.9999999...

10x=9.99999...

(10x)-x=9

10x = 9+x

10x-9 = x

10-9 = x/x

1 = 1

so, it always results in 1=1

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• 1 month later...

Alright, try this one on for size:

X=.9999999...

10x=9.99999...

(10x)-x=9

(9x)/9=1

x=1

.999999...=1

There is no problem with this statement; it is true.

(1/3)=.333333.....

(2/3)=.666666.....

(1/3)+(2/3)=.999999....

or

(1/3)+(2/3)=(3/3)=1

The original question is wrong because he divides by (x-1) which would be division by 0.

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• 5 months later...

x(x-2) = x-2

because it had defined x=2 so （x-2） can't be reduced

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