Square Roots and Complex Numbers

I watched a short video on YouTube and reflected on it, and maybe many of you want to know about it. You need to be cautious when applying square roots to complex numbers, as it's different.

TREMENDOUS EQUATION THAT ALMOST EVERYONE FALLS FOR! Basic Mathematics.

An irrational equation that causes a lot of confusion among students. Step-by-step explanation of what not to do...

Understanding Square Roots

By definition, square root values are always positive, they cannot result in negative values. This is established to avoid unnecessary complications through notation.

Current Definition of Square Root (with Real Numbers)

But if we start from x²=4 we know there are two possible solutions:

• An equation with two solutions.
• Then, when we refer to both solutions, we artificially add a 'plus or minus', but be aware! This is different and separate from the square value definition, don't get confused!

Ideally, we would write two different equations, one for 'plus' and another for 'minus', but for simplicity, we use this 'plus or minus' symbol for both equations.

An equation can have two (or more) solutions, but a square root can only have one outcome.

By 'branching' (which might not be the best word), I mean that we want a notation to signify a value, a single value, and not two possibilities since this could lead to confusion, which one or maybe both? By definition, whenever we find a square root value, it means its value is positive, that's why we work square roots with absolute value.

The Complex Numbers Problem

This is a problem for complex numbers because we want to work across the entire plane and not have an operator like the square root with its inherent absolute value that only works, producing positive results…

The main issue with this is that the absolute value operator breaks continuity, limiting the entire complex plane to only positive results.

The Solution

How to solve this? It is already solved, just undo or rewrite the definition to one that allows not only positive values but any other answer, it might be negative or even imaginary (or complex number).

As previously mentioned, we want to avoid the definition that causes 'branching' or at least curb it. If we can solve this, then we do not need to use a restrictive definition as the absolute value square root does.

Yes, we can resolve branching and maintain continuity in its operator, not like the absolute value does.