What is [math]x[/math] if [math]x+\left(\dfrac{1}{x}\right) =0[/math]?

To solve the equation ( x + \left(\dfrac{1}{x}\right) = 0 ), we can follow these steps:

1. Multiply by ( x ) to Clear the Fraction:
   Multiplying both sides of the equation by ( x ) gives us: [ x^2 + 1 = 0 ]

2. Rearrange and Solve for ( x ):
   Rearranging the equation to set it equal to zero gives us: [ x^2 + 1 = 0 ] This is a quadratic equation. However, it does not have real solutions because ( x^2 ) is always non-negative (or zero), and adding 1 means the left side will never equal zero for any real value of ( x ).

3. Complex Solutions:
   If we allow complex numbers, we can solve for ( x ) using the formula for the square root of a negative number: [ x^2 = -1 ] [ x = \pm \sqrt{-1} ] Since ( \sqrt{-1} = i ) (where ( i ) is the imaginary unit), we have: [ x = \pm i ]

Therefore, the solutions to the equation ( x + \left(\dfrac{1}{x}\right) = 0 ) are ( x = i ) and ( x = -i ), where ( i ) is the imaginary unit.