Is it Possible that 1 Equals -1: A Mathematical Inquiry?

[olek1991]

I think that I have proof of 1 being -1 and I can't find any flaw in it.
Could you please take a look?

-1=i² =>
(-1)²=(i²)² =>
1 = i^4 => take the square root both sides
1 = i²

i² = -1 v i² = 1

Thus proving
1 = -1

 

[Cyosis]

To confuse you a little more can you find the mistake: 2=\sqrt{4}=\sqrt{(-2)^2}=-2.

 

[olek1991]

Yea I know those too xD
Does that mean that it is correct? (but just not used since it's crazy)

Edit: the √(-2)² is not -2, but 2 btw :P
You probably meant (-2)² = √4 = 2²

 

[Cyosis]

No I meant exactly what I wrote, the root cancelling the square. We can't do this because we have defined taking the square root of a real number to be a positive value. This is called the principal square root. For the complex numbers this principal root is defined as \sqrt{z}=\sqrt{|z|}e^{i \pi/2}. In general for complex numbers it is not even true that \sqrt{zw}=\sqrt{z}\sqrt{w}.

 

[Gigasoft]

a^{bc}=\left(a^b\right)^c is not generally true. For example \left(\left(-1\right)^2\right)^{\frac 1 2}\neq-1. You should be careful with this rule when the base is not a positive real number and the exponent is not an integer.

 

[olek1991]

No I meant exactly what I wrote, the root cancelling the square. We can't do this because we have defined taking the square root of a real number to be a positive value. This is called the principal square root. For the complex numbers this principal root is defined as \sqrt{z}=\sqrt{|z|}e^{i \pi/2}. In general for complex numbers it is not even true that \sqrt{zw}=\sqrt{z}\sqrt{w}.

I really don't get that
Could you dumb it down a little? (I'm a collage student)

 

[Cyosis]

We can write every complex number z in the form z=|z|e^{i \theta} with |z| the distance between z and the origin and \theta the angle between the x-axis and |z| (polar coordinates). If you have had some complex numbers you should know this representation of a complex number. From this it follows that i=e^{i \pi/2} and i^4=e^{2 \pi i}. Now taking the square root of i^4 we get \sqrt{i^4}=e^{i \pi}=-1.

 

[wisvuze]

Yea I know those too xD
Does that mean that it is correct? (but just not used since it's crazy)

Edit: the √(-2)² is not -2, but 2 btw :P
You probably meant (-2)² = √4 = 2²

No that's not what it means, all of our mathematical foundations would be bogus if we ever said "it's true, but it's too crazy.. so it's pretty much false".
Sqrt(x) is a function (input/output relationships are unique), so given a number (perhaps 9), Sqrt(9) will map to 3.. never -3. If Sqrt(9) could be either -3 OR 3, it wouldn't be a function. Even though (-3)^2 = 9 = (3)^2, the root function is defined to take positive values and produce positive values.


Edit: the √(-2)² is not -2, but 2 btw :P
You probably meant (-2)² = √4 = 2²

This is exactly what you kind of said.. sqrt( (-2)^2 ) is indeed 2 since (-2)^2 gives us 4, and by the definition of the function, we will get the positive possible "root" only.

"You probably meant (-2)² = √4 = 2²" You probably made some typing mistake here.. (-2)^2 = sqrt(4) = 2^2?? 4 = 2 = 4? I don't know