Intro to imaginary numbers: If $i$ = $\sqrt{-1}$ and $i^2 = -1$, then when do you use $i^2$ and when $-1$?

Question

I just am learning about complex numbers from a high school math textbook (yep, I'm playing catchup from a long time ago, as a programmer I feel that I have amazing biceps but the rest of my body is normal, time to change that).

I have watched enough YouTube videos to believe that complex numbers are real. As Eddie Woo explains: sometimes you need to dive into deeper parts of the number system in order to get solutions to 'normal problems'. Which he showed quite simply by using the quadratic formula.

But what I don't get is the following.

If $i^2 = -1$, then if I have any equation, when should I use $-1$ and when should I use $i^2$?

I suppose they're the same thing, so I should just imagine both at the same time? Just like $\sqrt-1$ and $i$ are the same thing.

Is that what you do? Imagine them both at the same time, when reading equations?

Answer 1

No, $\sqrt{-1}$ and $i$ are not the same thing. Actually, since there are two square roots of $-1$, it is not a good idea to use the expression $\sqrt{-1}$, unless you defined it as something more that “square root of $-1$”.

On the other hand, I suggest that you use $i^2$ instead of $-1$ whenever that is useful. Such as when solving the equation $z^2=-1$:$$z^2=-1\iff z^2=i^2\iff z=i\vee z=-i.$$

Answer 2

I would hesitate to say $\sqrt{-1}$ and $i$ are the same thing. While we know that $\sqrt 3$ is the unique positive real number whose square is $3$, there's no reason to say $\sqrt{-1}$ means $i$ instead of $-i$.

However, any time two expressions represent the exact same thing you can substitute one for the other. For example, $2$ and $\sqrt4$ are completely interchangeable because they are the same number. Similarly, any place you want to use $-1$ you could use $i^2$, and vice versa.