I Explore Math & Theories Behind 2=-2

[fahraynk]

Does this relationship come out anywhere interesting in math? Are there any anything interesting theories built with this at its foundations?
$$2=2*1=2*\sqrt{1}=2*\sqrt{(-1*-1)}=2*i*i=2*i^2=-2$$

 

[fresh_42]

Does this relationship come out anywhere interesting in math? Are there any anything interesting theories built with this at its foundations?
$$2=2*1=2*\sqrt{1}=2*\sqrt{(-1*-1)}=2*i*i=2*i^2=-2$$

No, because it is wrong. You applied rules outside their area of validation. See:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/

 

[e-pie]

First of all square root does not come out of nowhere.

If y^2 = 1 then y=±(1)^(1/2). That means y is +1 or - 1.

Next i is a special complex unit in the form of (0,1) where as 1 is a real number. A REAL NUMBER CANNOT BE REPRESENTED AS PURELY COMPLEX.

<Edited>

A real number system is a subset of Complex number system but the converse isn't true.I assume that this was a random post from a popular social media. The most of them are baseless, only written to attain popularity. Don't waste time on them.

 

[jbriggs444]

-3=sqrt(3) x i^2 FALSE

$-\sqrt{3}$ is indeed equal to $\sqrt{3} \times i^2$

Edit: I'd failed to notice that the claimed inequivalence could rest in part on the obvious fact that 3 <> $\sqrt{3}$. I've repaired that oversight and hope that I've now rendered the intended claim properly.

What is not true is that $\sqrt{-1} \times \sqrt{-1}$ is equal to $\sqrt{-1 \times -1}$.

 

[mathman]

square roots are two valued functions. The roots are always of opposite sign. In deriving an expression, make sure you have the right sign. Otherwise you get silly things like $4=4$, $\sqrt{4}=\sqrt{4}$, therefore $2=-2$.

 

[Stephen Tashi]

square roots are two valued functions.

Yikes! How many threads are there where sudents are lectured that $y = \sqrt{x}$ is a function ? (i.e. not a "multi-valued" function).

"Square root" is an example of ambiguous terminology in mathematics. "$y$ is equal to the square root of $x$" has one definition as a function. If $x^2 = y$ then $x$ is a square root of $y$ has a different definition, which describes a property of $x$.

In complex analysis no one blinks at speaking of "the n-th roots of unity". or even "multi-valued" functions.

 

[mathman]

Yikes! How many threads are there where sudents are lectured that $y = \sqrt{x}$ is a function ? (i.e. not a "multi-valued" function).

"Square root" is an example of ambiguous terminology in mathematics. "$y$ is equal to the square root of $x$" has one definition as a function. If $x^2 = y$ then $x$ is a square root of $y$ has a different definition, which describes a property of $x$.

In complex analysis no one blinks at speaking of "the n-th roots of unity". or even "multi-valued" functions.

You can argue about terminology, but the fact remains 4 has two square roots, 2 and -2. This is the source of many silly proofs, such as in the original post!

 

[Mark44]

You can argue about terminology, but the fact remains 4 has two square roots, 2 and -2.

Yes, no one disputes that, but by common agreement, the symbol $\sqrt 4$ evaluates to a single number, + 2.

The OP's question has been answered, and there are multiple threads here about this "paradox" and similar ones, so I'm closing this thread.