Imaginary unit

square root of negative one, used to define extra-dimensional complex numbers

In math, imaginary unit, or , is a number that can be represented by equations, but refers to a value that can only exist outside of real numbers. The mathematical definition of the imaginary unit is (i.e., the principal root of ), where satisfies the property .[1][2]

The reason why was created was to answer a polynomial equation, , which normally has no solution (as the value of would have to equal ). Though the problem is solvable, the square root of could hardly be represented by a physical quantity of objects in real life.

Square root of iEdit

It is sometimes assumed that one must create another number to show the square roots of  , but that is not needed. The square roots of   can be written as:  , a result which can be shown as follows:


Powers of iEdit

The powers of   follow a predictable pattern:


This can be shown with the following pattern (where   is any integer):


Related pagesEdit


  1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-10.
  2. Weisstein, Eric W. "Imaginary Unit". Retrieved 2020-08-10.