High School Understanding the Inconsistencies of Imaginary Numbers

[Tian En]

I ran into such problem. Not sure if some one can help.

$$\sqrt{-i^2}=\sqrt{-1\times i^2}=\sqrt{-1\times -1}=\sqrt{1}=1$$

I also have

$$\sqrt{-i^2}=\sqrt{-1}\times \sqrt{i^2}=\sqrt{-1}\times i=i\times i=-1$$

Can anyone explain to me the inconsistencies?

 

[NFuller]

My interpretation is that both of these equations are correct since a square root yields both a positive and a negative answer.

 

[fresh_42]

You can find a short essay about complex numbers in this insight article of ours:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/
It starts with your example.

 

[Mark44]

I ran into such problem. Not sure if some one can help.

$$\sqrt{-i^2}=\sqrt{-1\times i^2}=\sqrt{-1\times -1}=\sqrt{1}=1$$

I also have

$$\sqrt{-i^2}=\sqrt{-1}\times \sqrt{i^2}=\sqrt{-1}\times i=i\times i=-1$$

Can anyone explain to me the inconsistencies?

My interpretation is that both of these equations are correct since a square root yields both a positive and a negative answer.

I disagree. The first equation is find, because the radical on the left is essentially $\sqrt 1$, which is 1.
The second equation is not correct, because the property that $\sqrt a \sqrt b = \sqrt {ab}$ is applicable only if both a and b are nonnegative. This is pointed out in the Insights article that @fresh_42 cited.

In addition, the real square root of a nonnegative number represents a single number, so it's not correct to say that, for example, $\sqrt 4 = \pm 2$.

 

[FactChecker]

Either answer is incomplete. The best answer is ±1. Once complex numbers are allowed, you need to be aware that the square root function always has two possible answers.
Using the '√' notation together with its assumption of a positive answer is treacherous in a context where complex numbers have been introduced. The '√' radical notation implies that the positive square root of a positive number will be used. Use '±√' if you want both to be considered. But once complex numbers are involved, those conventions do not apply.

 

[Mark44]

Either answer is incomplete. The best answer is ±1. Once complex numbers are allowed, you need to be aware that the square root function always has two possible answers.

You make a good point, but the first equation starts off with $\sqrt{-i^2}$. Since -i² = -(-1) = 1, we are taking the square root of a positive number, and complex numbers are not involved.

Using the '√' notation together with its assumption of a positive answer is treacherous in a context where complex numbers have been introduced. The '√' radical notation implies that the positive square root of a positive number will be used. Use '±√' if you want both to be considered. But once complex numbers are involved, those conventions do not apply.

 

[Tian En]

Thank you.