Why do positive numbers have negative roots

The imaginary number i

Squaring a negative or positive number always gives you a positive number. So it applies


The square root of a positive number is therefore always a negative or a positive number. However, you cannot extract a real root from a negative number. Often, however, as mentioned in the introduction, one comes across this case when solving simple, quadratic equations. For this reason the set of complex numbers was introduced. The unit i of the complex numbers is defined as follows:
DEFINITION: The number i has the property:


i is the unit of the imaginary numbers. This is comparable to the 1 for real numbers. (* Any imaginary number can be constructed from the imaginary unit i and a real number y: .
Negative roots can now be written in the following way:


According to the definition above, the root of a negative number is an imaginary number. It can be helpful when simplifying terms if you note that even powers of i always result in +/- 1:
 , ,.
(* In computer science, physics or electrical engineering, the letter "j" is used instead of the "i", since "i" is traditionally used in programming as a running variable and the current is also referred to as i.)