1 = 2

[Alea_iacta_est]

Found this highly amusing mathematical proof that concluded that 1 = 2, so without further ado, here it is.

1. Let A = B

2. Multiply both sides by A.

A^2 = AB

3. Add (A^2 - 2AB) to both sides

A^2 + A^2 - 2AB = AB + A^2 - 2AB

4. Factor the left side of the equation, and collect right terms on the right

2(A^2 - AB) = A^2 - AB

5. Divide both sides by (A^2 - AB)

2 = 1

 

[xisnotx]

i've seen it before.
xisnotx, get it?
it's the exact same proof.
you can define x as anything, even what it can't possibly be.
therefore, logic is limiting.

 

[nemo]

1. Let A = B

5. Divide both sides by (A^2 - AB)

This pains my soul.

 

[uumlau]

There are several variations on a theme of this proof. The error with all of them is division by zero (because A = B, the expression you factor out is zero, and 0*X = 0*Y is true for all values X and Y).

 

[ygolo]

This pains my soul.

Mine too.

I like this one a bit better:
1=sqrt(1)=sqrt(-1*-1)=sqrt(-1)*sqrt(-1)=i*i=-1.

 

[chubber]

Mine too.

I like this one a bit better:
1=sqrt(1)=sqrt(-1*-1)=sqrt(-1)*sqrt(-1)=i*i=-1.

That is indeed a conundrum.

 

[Fluffywolf]

Found this highly amusing mathematical proof that concluded that 1 = 2, so without further ado, here it is.

1. Let A = B

2. Multiply both sides by A.

A^2 = AB

3. Add (A^2 - 2AB) to both sides

A^2 + A^2 - 2AB = AB + A^2 - 2AB

4. Factor the left side of the equation, and collect right terms on the right

2(A^2 - AB) = A^2 - AB

5. Divide both sides by (A^2 - AB)

2 = 1

Wrong because A = B and you forgot that halfway.

Explanation:

1. Let A = B

2. Multiply both sides by A.

AA = AB

3. Add (AA - 2AB) to both sides

AA + AA - 2AB = AB + AA - 2AB



4. Factor the left side of the equation, and collect right terms on the right

Consider A=B

X = A or B

XX + XX - 2XX = XX + XX - 2XX
X = X

Just because of a misuse of the to the power symbol, does not mean something that is logic, suddenly isn't.

 

[Fluffywolf]

Mine too.

I like this one a bit better:
1=sqrt(1)=sqrt(-1*-1)=sqrt(-1)*sqrt(-1)=i*i=-1.

Just "-1 * -1 = 1" is enough to see the logical conundrum, but then, so is -1 or any negative number in fact. It is impossible to have -1 of something after all.

 

[netzealot]

5. Divide both sides by (A^2 - AB)

2 = 1

If A=B, then A^2 = AA = AB, and anything minus itself is zero. You can't divide by zero.

null = null... noting new about that.

 

[Mal12345]

Mine too.

I like this one a bit better:
1=sqrt(1)=sqrt(-1*-1)=sqrt(-1)*sqrt(-1)=i*i=-1.

Are you sure that 1=sqrt(1)?

 

[uumlau]

Are you sure that 1=sqrt(1)?

While it is true that (-1)^2 = 1, sqrt(1) is defined as the positive square root of 1, so that is not a technical problem with the proof, but it is a clue as to the real problem.

Because the "proof" invokes complex numbers, the real difficulty is the sqrt(1) = sqrt(-1*-1).

 

[Mal12345]

sqrt(-1*-1) is undefined.

 

[ygolo]

While it is true that (-1)^2 = 1, sqrt(1) is defined as the positive square root of 1, so that is not a technical problem with the proof, but it is a clue as to the real problem.

Because the "proof" invokes complex numbers, the real difficulty is the sqrt(1) = sqrt(-1*-1).

sqrt(-1*-1) is undefined.

I would actually say the problem is with this step:

sqrt(-1*-1)=sqrt(-1)*sqrt(-1)

-1*-1=1

So any function acting on both sides over the whole side is still valid.

So sqrt(-1*-1)=sqrt(1)

However, the square root function, in order to be a well defined function, can only have one unique answer. There is a degree of arbitrariness to it, but you need to pick the region of the complex plane for the range to which the square root maps its domain.

Suppose we pick the angles from the real axis of the range to be [0,pi).

Then sqrt(-1)=i, and sqrt(-1*-1)=1. But clearly sqrt(-1)*sqrt(-1) does not equal 1.

So the error was extrapolating a property of square roots when thinking about reals to one when thinking about complex numbers.

Specifically, that sqrt(a*b)=sqrt(a)*sqrt(b). This is not true anymore when we allow square roots of negative numbers and a ranges (co-domains) that are complex.

 

[uumlau]

sqrt(-1*-1) is undefined.

Or rather, insufficiently defined, because even sqrt(1) is undefined for complex numbers without further clarification.

Using e^(i*theta) notation, for -1, theta = pi (or rather pi + 2*pi * n for all integers n). Without loss of generality, let n = 0.

So sqrt(1) = (e^(i*pi*0))^(1/2) = 1
while sqrt(-1*-1) = (e^(i*pi)*e^(i*pi))^(1/2) = e^(i*pi) = -1

for n=0 (or more generally, n = even integers; it reverses sign for both expressions for n = odd integers)

In general, for complex numbers, the results are defined as long as you keep track of the number of windings around the origin. The "proof" of 1 = -1 deliberately obfuscates of the number of windings.

 

[ygolo]

Or rather, insufficiently defined, because even sqrt(1) is undefined for complex numbers without further clarification.

Using e^(i*theta) notation, for -1, theta = pi (or rather pi + 2*pi * n for all integers n). Without loss of generality, let n = 0.

So sqrt(1) = (e^(i*pi*0))^(1/2) = 1
while sqrt(-1*-1) = (e^(i*pi)*e^(i*pi))^(1/2) = e^(i*pi) = -1

for n=0 (or more generally, n = even integers; it reverses sign for both expressions for n = odd integers)

In general, for complex numbers, the results are defined as long as you keep track of the number of windings around the origin. The "proof" of 1 = -1 deliberately obfuscates of the number of windings.

Indeed, we could make one with cube roots or fourth roots or what have you and obfuscate all sorts of angles in the complex plane with each other.

Edit: another such "proof" would be:
1=1^(1/4)=(i*i*i*i)^(1/4)=i^(1/4)*i^(1/4)*i^(1/4)*i^(1/4)=i^(4/4)=i

Edit 2: More generally:
Suppose a^n=1. There are generally n solutions to this equation, only one of which is 1.
Now the "proof" would be:
1=1^(1/n)=(a^n)^(1/n)=(a^(1/n))^n=a^(n/n)=a

 

[Mal12345]

Or rather, insufficiently defined, because even sqrt(1) is undefined for complex numbers without further clarification.

Using e^(i*theta) notation, for -1, theta = pi (or rather pi + 2*pi * n for all integers n). Without loss of generality, let n = 0.

So sqrt(1) = (e^(i*pi*0))^(1/2) = 1
while sqrt(-1*-1) = (e^(i*pi)*e^(i*pi))^(1/2) = e^(i*pi) = -1

for n=0 (or more generally, n = even integers; it reverses sign for both expressions for n = odd integers)

In general, for complex numbers, the results are defined as long as you keep track of the number of windings around the origin. The "proof" of 1 = -1 deliberately obfuscates of the number of windings.

Are you talking about an Argand diagram?

 

[uumlau]

Are you talking about an Argand diagram?

Not specifically, no. I'm talking about complex numbers in general.

An Argand diagram in terms of polar coordinates would perhaps be the most useful for interpreting/visualizing my statements, where

z = |z|e^(i*theta) = |z| * (x*cos(theta) + i*y*sin(theta))​

If we let theta increase/decrease indefinitely, then each time it passes an integer multiple of 2*pi is a single "winding". In the case of this problem, |z| = 1, so we only need deal with the number of windings, and not the magnitude of z.

 

[Mal12345]

Not specifically, no. I'm talking about complex numbers in general.

An Argand diagram in terms of polar coordinates would perhaps be the most useful for interpreting/visualizing my statements, where

z = |z|e^(i*theta) = |z| * (x*cos(theta) + i*y*sin(theta))​

If we let theta increase/decrease indefinitely, then each time it passes an integer multiple of 2*pi is a single "winding". In the case of this problem, |z| = 1, so we only need deal with the number of windings, and not the magnitude of z.

In pure mathematics, but not in QM.

 

[RaptorWizard]

You folks here are speaking a language I have yet to comprehend, but I can still appreciate the elegance and complexity contained within mathematics. My criticism against it though is that it operates around well-defined laws, assumptions of sorts that limit us from the start.

I have a theory that if we agree to play by the Game's "rules", then we have to understand that contract first in order to break it. In essence, with higher awareness of our world system (ruled largely by mathematics), we can come to command it better and perhaps 1 day even transcend it all.

Just "going with the flow" of the Game will take us by the initial conditions under their set laws to a fixed destination, but by the power of choice, we can use greater amounts of free-will and pretty much become the "X-factors". There's a lot of the significance I think behind asking "what-if" questions.

Risks are often calculated by math as well. The "probabilities" encourage us to stay away from the choices deemed less likely to succeed - but they key here is: risk is the chance for failure!

Failure is in reference to progress not made or even lost when risks bring us down and stunt our development (within the context of the system). The escape is that these risks are just the choices as presented to us by the Game.

Why is that significant? It's because the "risks" are made through choosing from the initial choices, where you will trail behind the Creators (game designers) in the lead who programmed them into the system, but truth be told, we can actually select from options outside the Game! Basically, we can open the "wish-box" from inside to let new "presents" in and, at sufficiently high evolutionary levels, we may even spring out from the box and into the beyond!

The top of the box can be seen as the mountaintop, but then you must the faith needed for leaping out into the unknown territories and flying away with the wings of freedom. We from the room beyond the box can even lift it up in the air, come to "move the mountain"!

So if we can lay the right outlines towards the accomplishment of this seemingly impossible task, then it shall be conquered by first believing and changing the world as the variables rather than the results when our sum-total equations are all taken together into the master schematics.

The final step is the unlimited expansion possible from this path - ultimate power; let God's Force of Will be unleashed and spring us free! Then truly, when we ascend with angels all the way into the Heavens, we can then turn the world on its head and hold it within our hands.

Love needs no reason (hence no restrictive mathematical projections); Love is the reason. Let Love lay the foundation and point the way! Love can revolve around forever in eternally whirling spirals if we're open to its input, receive its calls and transmit our responses back out into the horizons as we chase our Eternal Love ever onwards towards our ultimate victory; it's quite simple - if you don't believe it, then that's why you fail!

On a novel closing note, if you want to see sexy Angels "topless", then you must have the faith to make it through victory road, breaking away from this mundane place and soaring across into the other side with the Divine. It's there where we can jump into the classroom action, listening to our lessons and witnessing them "unfolding" from our Teacher. Together we shall pass the Test with triumphal success!

 

[Jonny]

[MENTION=20385]Alea_iacta_est[/MENTION] regarding 1=2

When you divide by zero, anything is possible.


[MENTION=825]ygolo[/MENTION] regarding 1=-1

I agree for the most part.

1 = sqrt(1) if and only if f(x)=sqrt(x) is well-defined on the positive real plane. The issue is that the proof interchanges well-defined functions that are actually each distinct subsets of the broader general mapping f(x) = sqrt(x). If we consider the general function, then:

sqrt(1) = {-1,1}

sqrt(1) = sqrt( -1 x -1)

sqrt(1) = sqrt(-1) x sqrt(-1)

sqrt(1) = {-i, i} x {-i, i}

sqrt(1) = {-i x -i, -i x i, i x -i, i x i}

sqrt(1) = {-1, 1, 1, -1}

sqrt(1) = {-1, 1}