More Maths

[Hysteresis]

A Mathematical Proof that 1 equals 2

Let X = Y
Multiply both sides by X gives X² = XY
Subtract Y² from both sides gives X² – Y² = XY – Y²
Factorise both sides gives (X + Y)(X – Y) = Y (X – Y)

Of course (X – Y) cancels to give (X + Y) = Y
As X = Y substitute Y for X (X + X) = X
Therefore 2X = X
Let X = any No., say 1, Therefore 2 = 1



It that division by Zero again !!!!!

 

[PrenticeBoyofDerry]

so 1=2 therefore 2=1, so do I have 1 or 4 bullux?

 

[empip]

Another for you - the earth is flat.

-0-

 

[EddieM]

X² – Y² = XY – Y²

just means 0 = 0. Nonsense.

 

[Hysteresis]

X² – Y² = XY – Y²

just means 0 = 0. Nonsense.

Exactly - that is the anomaly of the so called 'Proof'. But 0=0 is not Nonsense it is a fact. But once arrived at you can multiply the 0=0 equation by what ever you want and it will still be correct, but algebraically not quite so obvious.
The point is that the algebra appears to be correct if the observation of 0=0 or indeed the other one of Division by 0 ie (X-Y) is overlooked.
More complicated algebraic manipulations could use the same error but not so obvious as in this example.

 

[EddieM]

X² – Y² = XY – Y²

just means 0 = 0. Nonsense.

Exactly - that is the anomaly of the so called 'Proof'. But 0=0 is not Nonsense it is a fact. But once arrived at you can multiply the 0=0 equation by what ever you want and it will still be correct, but algebraically not quite so obvious.
The point is that the algebra appears to be correct if the observation of 0=0 or indeed the other one of Division by 0 ie (X-Y) is overlooked.
More complicated algebraic manipulations could use the same error but not so obvious as in this example.

sorry I, i didn't mean 0 = 0 is nonsense just the rest of it is is mathematical rubbish.

 

[EddieM]

the problem here is letting x and y be the same thing, fatal mistake. they have to be different.

 

[Hysteresis]

[

sorry I, i didn't mean 0 = 0 is nonsense just the rest of it is is mathematical rubbish.

Of course it is Mathematical rubbish, but on the face of it, algebraically correct.

I think it highlights the possibility in far more complicated algebraic operations, a zero could inadvertently form, but hidden in the complex expressions.

Arithmetically it is very easy to see a division or equating of zero, but algebraically, the anomaly can be missed don't you think?

 

[plugwash]

A Mathematical Proof that 1 equals 2

Or rather a demonstration that false assumptions can lead to false proofs.

Let X = Y
Multiply both sides by X gives X² = XY

This is where it starts to go off the rails. It is true that X = Y implies X² = XY but the reverse is not true. Like any quadratic X² = XY has two soloutions, X=Y and X=0.

Subtract Y² from both sides gives X² – Y² = XY – Y²
Factorise both sides gives (X + Y)(X – Y) = Y (X – Y)

These steps are fine.

Of course (X – Y) cancels to give (X + Y) = Y

This is where things REALLY go off the rails.

(X + Y)(X – Y) = Y (X – Y) has two soloutions, X=Y and X=0. By your cancellation process you throw out one of these soloutions and keep the other the one you throw out being different from the one you started with.

 

[Hysteresis]

Plugwash - We of course all agree with you, and we have said the Algebra is flawed with the zero anomaly.
What the point was that the algebra in more complicated manipulations could throw up unrecognised/detected zero's, and as such one has to be careful of such conditions.
'O' level stuff really, maybe a bit advanced for GCSE !!!!!!!!!

 

[gigz]

I tried to disprove this but couldn't find a pencil sharpener

 

[tony1851]

A Mathematical Proof that 1 equals 2

Let X = Y
Multiply both sides by X gives X² = XY
Subtract Y² from both sides gives X² – Y² = XY – Y²
Factorise both sides gives (X + Y)(X – Y) = Y (X – Y)

Of course (X – Y) cancels to give (X + Y) = Y
As X = Y substitute Y for X (X + X) = X
Therefore 2X = X
Let X = any No., say 1, Therefore 2 = 1



It that division by Zero again !!!!!

This goes wrong on the first line;

X cannot equal Y, because X and Y are different. X can only equal X.

 

[Hysteresis]

This goes wrong on the first line;

X cannot equal Y, because X and Y are different. X can only equal X.

Tony - You are not paying attention.

No it doesn't go wrong on the first line.
The declaration X=Y is not in error.
Say you have a bag marked X with 1 lb of sugar in it, and a second bag marked Y with 454gms of sugar in it, putting them on a balance would show that X=Y. So no problem there.
As has been pointed out, the error is the equating of Zero or and the division of Zero. Both operations being undefined.

 

[plugwash]

As has been pointed out, the error is the equating of Zero or and the division of Zero. Both operations being undefined.

Equating of zero is fine in the right context.

The error is multiplying or dividing through by something without ensuring that the "something" cannot be zero.

 

[Hysteresis]

Equating of zero is fine in the right context.

The error is multiplying or dividing through by something without ensuring that the "something" cannot be zero.

Don't Agree Plugwash.
Equating zero is never OK.
IF you equate zero then everything else is nonsense.
If 0 = 0 and it follows 12x0 = 0 is also correct.
If x=0 then 12x = x, so 12 = 1
Clearly silly.

I am not sure as to what context you find it is fine to equate zero. I don't know of one, but am interested to learn of your context.
You get similar meaningless answers by using Infinity in a mathematical operation.
The greeks did not have a number zero, I believe.
Very clever these ancient Greeks.