Imaginary paradox

The imaginary paradox is a false paradox that arises due to the wrong course of reasoning.

To resolve the contradictions of the imaginary paradox, it is enough to use the means of formal logic .

An Example of an Imaginary Paradox

Consider an example of proof that the number 1 is −1:

${\begin{aligned}{\color {Blue}i}^{2}&=-1\\{\color {Blue}i}~&={\sqrt {-1}}~={\sqrt {1 \over -1}}~={1 \over {\sqrt {-1}}}~={1 \over {\color {Blue}i}}\\{\color {Blue}i}^{2}&={\color {Blue}i}\cdot {\color {Blue}i}~={\color {Blue}i}\cdot {1 \over {\color {Blue}i}}~={{\color {Blue}i} \over {\color {Blue}i}}~=1\\&\Rightarrow 1=-1\end{aligned}}$ ${\ displaystyle {\ begin {aligned} {\ color {Blue} i} ^ {2} & = - 1 \\ {\ color {Blue} i} ~ & = {\ sqrt {-1}} ~ = {\ sqrt {1 \ over -1}} ~ = {1 \ over {\ sqrt {-1}}} ~ = {1 \ over {\ color {Blue} i}} \\ {\ color {Blue} i} ^ {2} & = {\ color {Blue} i} \ cdot {\ color {Blue} i} ~ = {\ color {Blue} i} \ cdot {1 \ over {\ color {Blue} i}} ~ = {{\ color {Blue} i} \ over {\ color {Blue} i}} ~ = 1 \\ & \ Rightarrow 1 = -1 \ end {aligned}}}$

The reason for the erroneous conclusion is the use of incorrect equality: $i={\sqrt {-1}}.$ ${\ displaystyle i = {\ sqrt {-1}}.}$ The error is caused by the fact that the square root of the complex number has 2 values, in this case $i;-i$ ${\ displaystyle i; -i}$ . Therefore, this equality does not hold.

Other examples of imaginary paradoxes

The Paradox of Chorus
The paradox of Lorentz contraction
Twins paradox
Paradox of stairs
The Curry Paradox
The problem of two envelopes

Imaginary paradox

An Example of an Imaginary Paradox

Other examples of imaginary paradoxes

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