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).
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Thread: 1 = 2

01012014, 08:48 PM #11An argument is two people sharing their ignorance.
A discussion is two people sharing their understanding, even when they disagree.

01012014, 08:54 PM #12
sqrt(1*1) is undefined.
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01012014, 09:28 PM #13
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

01012014, 09:30 PM #14
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 (n) around the origin. The "proof" of 1 = 1 deliberately obfuscates of the number of windings.An argument is two people sharing their ignorance.
A discussion is two people sharing their understanding, even when they disagree.

01012014, 09:51 PM #15
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
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

01022014, 09:56 AM #16

01022014, 10:14 AM #17
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 = ze^(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.An argument is two people sharing their ignorance.
A discussion is two people sharing their understanding, even when they disagree.

01022014, 11:38 AM #18

01022014, 12:15 PM #19
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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 welldefined 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 freewill and pretty much become the "Xfactors". There's a lot of the significance I think behind asking "whatif" 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 "wishbox" 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 sumtotal 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!

01022014, 03:48 PM #20
@Alea_iacta_est regarding 1=2
When you divide by zero, anything is possible.
@ygolo regarding 1=1
I agree for the most part.
1 = sqrt(1) if and only if f(x)=sqrt(x) is welldefined on the positive real plane. The issue is that the proof interchanges welldefined 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}[SIGPIC][/SIGPIC]