2 Equals 1 - It's True 2=1

[electriic ink 1]

In light of the topic, 2+2=5 methought I'd create my own puzzle:

 

[hr=shade]Let x = y[/hr]

 

Therefore x? = xy

 

Therefore x? - y? = xy - y?

 

If we factorise these equations we end up with (x + y)(x - y) = y(x - y)

 

We now have a common term on both sides "(x - y)," which we will divide off to make our problem easier to solve

x + y = y

 

But x = y so y + y = y thus 2y = y

 

Then, if we divide off the common value "y," 2 = 1

 

[hr=shade]Simple! Where's the flaw?[/hr]

[husker 0]

I remember this from last year in my math class. It was in the step when you canceled (x-y) I believe, because (x-y) is 0, therefore you are multiplying by 0 which would be 0. You can't cancel (x-y). Good one!

Edited December 9, 2006 by husker (see edit history)

[Plenoptic 0]

Well like husker said (x-y) would equal zero and it's the fact that you can't divide by zero not multiply like husker said. So that would really mean that it ends up as 0 = 0 because if you plug in the 0 of (x - y) , that would multiply with the rest of the equation to make 0 because it is all multiplication, no addition or subtraction except between parentheses. There may be a way though so keep working at it.

[bhavesh 0]

As I have said in similar topic 2+2=5, I would again say that this type of questions are raised by people with little knowledge of Mathematics.As explained by Husker and Plenoptic as x and y are equal therefore x-y is zero and whenever zero comes it cant be cancelled from both sides, if this is possible then all numbers will be equal to each other, like this.0=0or, 0*a=0*bi.e. a=bwhere a and b are any number.therefore zeros from both sides cannot be cancelled.

[electriic ink 1]

Congrats! The answer was you can't divide by 0 or x - y. You can prove anything when you divide by 0.

[Terciel Silvi 0]

Here is another way of proving that 1 = 2

Everyone knows this:
-1/1= 1/-1

Now we will square root both sides:

√-1/1 = √1/-1

Now we break up the roots:

√-1 √1
--- = ---
√1 √-1

The square root of a negative 1 is i and the square root of 1 is 1. In other words:

i/1 = 1/i

Now we multiply the entire thing by 1/2:

i/2 = 1/2i

Now let's add 3/(2i) to this to make the math easier.

i/2 + 3/2i = 1/2i + 3/2i

Now we can multiply the entire thing by i:

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

So now we expand this beast:

1^2/2 + 3i/2i = i/2i + 3i/2i

We know that the square root of -1 is i, so i^2 must be -1.:

-1/2 + 3i/2i = i/2i + 3i/2i

Now we simplify the i's

-1/2 + 3/2 = 1/2 + 3/2

Let's calculate this thing:

2/2 = 4/2

And so

1 = 2

That does work as far as I can see. And with this proof you can prove that dividing by zero is possible, because

anygivennumber^0 = 1

Source

I know the source isnt reliable, but the 1-=2 thing I posted does work as far as I can see.
Edited December 20, 2006 by Terciel Silvi (see edit history)

[rsf 0]

1 will never equal 2 by definition. So I'm going to bother with the second proof because there's definately a flaw somewhere. If it was correct then there would be a flaw in our addition, multiplication, etc.Remember you can't divide by variables unless you can prove they aren't 0 like y = x^2 + 1 where x is a real number (no i's )

[Posicoln 0]

I noticed a flaw:We know that the square root of -1 is i, so i^2 must be -1.:anything squared will always be a positive number ... and i=1, as 1x1=1There are probably more flaws ... but those TWO I noticed ... making this very unbelieveable.

[linekill 1]

---

Let x = y
Therefore x˛ = xy

Therefore x˛ - y˛ = xy - y˛
If we factorise these equations we end up with (x + y)(x - y) = y(x - y)

We now have a common term on both sides "(x - y)," which we will divide off to make our problem easier to solve
x + y = y

But x = y so y + y = y thus 2y = y

Then, if we divide off the common value "y," 2 = 1

---

Simple! Where's the flaw?

Husker is right, the flaw is when you divide (x - y) on both sides. We have rules to follow, BODMAS, when solving equations. B - Brackets , O - Operations, D - Divisions, M - Multiplication, A - Addition and S - Subtraction.

When (x - y) was isolated from the equation, it means that we have to calculate it first. And since x = y, then x - y would give us 0. And down it goes the drain.

Nice one! Almost got me there.
Edited March 28, 2010 by linekill (see edit history)