Can we say that $i = -\sqrt{-1}$ because $i^2 =- 1$?

Question

It is defined as $i^2 = -1$ then we can say $i=\pm \sqrt{-1}$

Then is true that $i$ can be equal to $ -\sqrt{-1}$

Answer 1

I've found that a lot of mathematicians get irked by saying $i = \sqrt{-1}$ for various reasons pertaining to the algebra. (I'm not kidding, I had a professor rant about this one day.) For example, here's just one consequence of that definition:

$$i^2 = (\sqrt{-1})(\sqrt{-1}) = \sqrt{-1 \cdot -1} = \sqrt{1} = 1 \neq -1$$

Needless to say, bad idea.

I honestly don't know how to really think about it other than the following: that we just say $i$ is some constant with the property $i^2 = -1$. We don't philosophize about what $i$ itself is, we just know it has this property, and go from there.

$i$ and complex analysis wouldn't even be the weirdest case of such constants. Personally, until I adapted this philosophy, I found the dual numbers and split-complex numbers even worse to wrap my head around. These are systems similar to complex numbers, except $i$ is defined differently: in the dual numbers, $i^2 = 0$, and in the split-complex numbers $i^2 = 1$. Imagine how that begins to break if we try to solve for $i$: it starts looking like "why are we even doing this?"

And of course there are even other systems with more complicated units. For example, the quarternions use units $i, j, k$ satisfying

$$i^2 = j^2 = k^2 = ijk = -1$$

which is also weird to wrap your head around in terms of trying to find $i, j, k$. There are even more systems beyond the quaternions. If you think about the powers of $2$, there is a number system for each that is basically $2^n$-dimensional:

• $2^0 = 1 \Rightarrow$ real numbers ($\mathbb{R}$)
• $2^1 = 2 \Rightarrow$ complex numbers ($\mathbb{C}$)
• $2^2 = 4 \Rightarrow$ quaternions
• $2^3 = 8 \Rightarrow$ octonions
• etc.

Philosophizing about what the unit itself actually is gets all of the more screwy in these systems - be they alternate definitions of $i$ that produce alternatives to complex numbers, or systems with even more units than just $i$.

Honestly, all I really think we would use the $i = \sqrt{-1}$ definition for is simplifying radicals and such, but even then you circumvent that by saying the following... Let $x > 0$. Then

$$\sqrt{-x} = i \sqrt{x}$$

You don't even bother with the notion of "square root of a negative" or "square root of negative one" in this way, you just say "this is how you find the square root of a negative by definition."

I'm not sure how this notion of $i = \sqrt{-1}$ really got started. Maybe we actually started with that in defining complex analysis, and people ignored its contradictions and such? I feel like at least in the modern era that it does have the use in our middle/high school algebra classes insofar as making the simplification of radicals easier to understand. (Since we're taught that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$) early on, saying $\sqrt{-x} = \sqrt{-1} \cdot \sqrt{x} = i \cdot \sqrt{x}$ makes more intuitive sense, especially to a teeanger, a novice, right?)

So perhaps the issue is inaccurate teaching maybe? Or perhaps just the human desire to want to know what $i$ itself is, and taking the square root of each side of its definition ($i^2 = -1$) makes for the simplest conclusion? I'm not really sure. I think overall, though, it's just easier to accept $i^2 = -1$ - that whatever $i$ really is, we don't know, but it has that property.

Of course this might sound dumb but this is just my take on the issue. Though I suppose I can understand any ire I might get over "well, it's dumb and unscientific to just take it on faith that $i^2 = -1$ without knowing what $i$ is." It's just the best way I've been able to come to terms with this.