The imaginary unit, denoted by the lowercase letter i, is a value that represents the square root of -1, and is typically defined by the solutions of this quadratic equation$${\displaystyle x^2+1=0}$$This equation can be solved using algebraic manipulation to find that ${\displaystyle x=\pm {\sqrt {-1}}}$. However, restricting our domain to real numbers, these numbers does not exist. Because no real number squared can result in a negative solution. However, despite these numbers not existing in the real domain, mathematicians have found it immensely useful to 'declare' that these numbers exist, as a completely separate class of number, known as Imaginary Numbers, which extend our domain of the square root function to all real numbers.

The Imaginary Numbers revolve around the imaginary unit, just how the real numbers could be thought of as a multiple of 1. The imaginary numbers revolve around a real multiple of the imaginary unit. The Imaginary unit is written as the lowercase letter 'i', however the lowercase greek letter 'Iota' (ι) and occasionally the lowercase letter 'j'. But the lowercase letter 'i' is by far the most common notation for the imaginary unit.$${\displaystyle i=\pm {\sqrt {-1}}}$$While one may think that defining i in its positive or negative form would result in different properties, the behavior of imaginary numbers, and Complex Numbers are unaffected by the sign of the square root in the definition of the imaginary unit. So whichever sign is used is a matter of mere preference.