# Relationship : 2=-2

• I

## Main Question or Discussion Point

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

## Answers and Replies

fresh_42
Mentor
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.

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jbriggs444
Homework Helper
2019 Award
-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 $\sqrt{-1 \times -1}$.

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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
Mentor
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.