Ambiguity textbook exercise involving $\sqrt{-144}$

Question

Consider the following questions, the whole exercise is dedicated to determining the square root of negative numbers, after introducing the complex numbers. Eg.

$$\sqrt{-144}$$

Solutions for the whole exercise only give one solution for each problem, the above being $12i$.

However, this is providing to be rather confusing.

From my understanding, we know that $i$ is a number that when squared $i^2=-1$. However, this does not mean that $\sqrt{-1}=i$, as $-1$ has 2 roots that are indistinguishable. Defining $\sqrt{-1}=i$ goes bad really quickly.

Hence for the problem above, the answer should be $\pm12i$ to avoid confusion, we cannot say that it is $12i$, unlike for positive real numbers, where the principal square root is defined, there is no such counterpart in the complex plane. Hence when we write $\sqrt{-144}$, it must be $\pm12i$.
So are the answers incorrect?

By defining $i^2=-1$, how can we extend this, such that we can use it for such problems?

Answer 1

There is always one principal root. The radical "$\sqrt{}$" always indicates the positive root if the roots are real. e.g. $\sqrt {144} = 12.$ It does not equal $-12.$ Yet, $(-12)^2 = 144$

Or, we could say $x^2 = 144$ has two solutions $x = \pm 12.$

How does this work with principal roots? If we take the square roots of both sides:

$\sqrt {x^2} = \sqrt {144}\\ |x| = 12\\ x = \pm 12$

This concept of principal roots does not create a contradiction.

So, moving on to imaginary numbers... $i$ is the principal root. $\sqrt {-1} = i$ However, $-i$ is also a root, just not the principal root.

$x^2 = -1 \implies x = \pm i$

In the problem at hand

$-12i$ is a solution to $x^2 + 144 = 0$

$\sqrt {-144} = 12i$ has one solution... the principal root.

Answer 2

If you are asked to find the value of x such that $$x^2=-144$$ $$\implies x^2-(12i)^2=0$$ $$\implies (x-12i)(x+12i)=0$$ $$\implies x= (+12i), (-12i)$$ similar case for the equation $x^2=k^2,\; k\in \mathbb R$

but if you are asked to find $\sqrt {k^2}$ it is equal to $|k|$ (by definition) but it is for $k \in \mathbb R$. Your textbook author might be considering this theory for complexnumbers also. Always remember $\sqrt 4 = 2$ not $\pm 2$

Answer 3

In complex world, square root (or even power function) is actually multi-valued, so we need to define a branch cut on it. In most usual convention (without otherwise specified), we take $\sqrt{\cdot}$ to give the half argument complex number. That is $$\sqrt{re^{i\theta}}=\sqrt{r}e^{i\frac{\theta}{2}}$$ This prevents the ambiguous problem you mentioned. And this is also why we call $i=\sqrt{-1}$, because $-1=e^{i\pi}$ and $i=e^{i\frac{\pi}{2}}$.

It seems like we use $i$ to define $i$, a bit circular argument. But in fact there is other way to first define the $i$ without specifying $\sqrt{-1}$, by like stating $i$ is the root of $x^2+1=0$ with smaller argument, or by constructing the field of $\mathbb{R}[X]/\left<X^2+1\right>$ and call the $\bar{x}$ ($x$-coset) $i$.

Answer 4

OP line of thinking is good to high-light the Issue.
Naturally , there must be a flaw in either OP argument or in textbook.

The CORE flaw is that "Principal Square Root has no such counterpart in the complex plane"

SUMMARY : "Principal Square Root" is that value of the root which has the smaller angle when viewed in the Polar form.

Expand the Domain to all Complex Numbers ( rather than just $\pm i$ ) & the Issues get resolved.
Yes , we have no Positiveness & Negativeness among Complex Numbers , hence that is not the Basis with which we extend the Principal Square Root Concept.

In Complex Domain , $\sqrt{C}$ & $\sqrt{i}$ could be 2 values.
With that , we then get two values $X$ & $Y$ for $\sqrt{C}$ like in the Image :

Here , $X$ has smaller angle , hence that is the Principal Square Root.

When we initially arbitrarily chose one $i$ & represent it with Positive angle $+90^\circ$ , while the other was at larger angle $+270^\circ$ , we had made the same choice.
More-over , when we chose Positive Side for $\sqrt{144}=+12$ , we were choosing angle $0^\circ$ versus $180^\circ$ : Still Principal Square Root was having smaller angle.
Hence this way to Extend "Principal Square Root" is totally Consistent.

SUMMING UP :

"Principal Square Root" is that value of the root which has the smaller angle when viewed in the Polar form.
$\sqrt{-144}=12i$ is the Correct Consistent Principal Square Root.

Consistently applicable to Positive numbers , Negative numbers , Imaginary numbers & Complex numbers , Definition will work with $\sqrt[3]{\cdots}$ , $\sqrt[4]{\cdots}$ , $\sqrt[5]{\cdots}$ & higher radicals too.