What will change if we admit a different definition of $\sqrt a$

Question

We know that $\sqrt a$ is the non negative solution of the equation $x^2=a$ with $a\geq 0$.
So if we want to solve the equation $x^2=a$, we say that $x=\pm\sqrt a$.
How will mathematics be affected if we define as $\sqrt a$ any (not necessarily non negative) solution of $x^2=a$?
For example what if we say that $\sqrt 4= \pm 2$?
The solution of $x^2=4$ will be then $x=\sqrt 4=\pm 2$ and it seems that there is no problem.
Will there be another problem (somewhere) in maths?
Thanks in advance!

Answer 1

This is one of those "beating-a-dead-horse" questions.

A function is something that takes an input and produces one output. The squareroot function $f(x)=\sqrt{x}$ with $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ is a function because for every $x$ it gives one output. $f(4)=2$. If $f(4)=\{-2,2\}$, then it is not a function.

Ultimately you can make up all kinds of notation you want but if you want to speak of $\sqrt{x}$ as a bona-fide function, then you have to decide on a sign, either positive or negative. By convention (in the mathematics community), we assign it the positive root if nothing else is mentioned. If you're not assigning it a positive root, then just say so: "We define $\sqrt{x}$ to be the negative root." Mathematics won't break, you'll just have to convert between conventions every once in a while. This actually happens often in complex analysis, where one sometimes needs to assign a particular branch to a square-root for easier calculation. The polite thing to do is mention you are doing this.

If you define $\sqrt{4}=\{-2,2\}$, great, that's fine. You can write this as $\sqrt{x}:=\{y: y^2=x\}$ and that's also fine. However this is a definition. You have not fundamentally changed anything, just taken a symbol and redefined it to something that's not a function. That's fine too but it goes against convention, which is important when communicating results between two parties.

Answer 2

Nothing will change. The $\sqrt{a}$ as we know it would still exist and all the mathematics will be the same again.

We need not care about this definition of square root, because it would be defining something else other than the actual square root, which exists whether you define it or not.

Answer 3

I think this is a nice question because it forces us to reflect on the notation $\pm$. Indeed, this notation is a little unusual in math. Usually in math, strings of symbols represent one and only one number, but $\pm$ is an exception. Generally, $\pm$ is somewhat sloppy notation that means something along the lines of "there are two solutions to [the problem we're talking about], one with a $+$ sign here and one with a $-$ sign." Since this situation arises so much in math, and since the reader usually knows what [the problem we're talking about] means, it's worth the slight ambiguity. But even $\pm$ can lead to trouble.

For example, what is $\pm \sqrt{3} \pm \sqrt{2}$? In the absence of clarification, this is talking about $4$ numbers, and maybe that's what we mean; for example, all 4 of those are solutions to $(x^2 - 5)^2 = 6$. But, on the other hand, only two of those numbers are solutions to $x^2 = 5 -\sqrt{6}$, and only two of those numbers are solutions to $x^2 = 5 + \sqrt{6}$. In the former case we could write $\pm \sqrt{3} \mp \sqrt{2}$, or alternatively $\pm(\sqrt{3} -\sqrt{2})$ to get rid of the ambiguity; something like the second solution works in the latter case although there seems to be no way to write it without parentheses.

Now let's consider your proposal. To keep from getting confused with the normal meaning of the symbol, I will use $\circ$ instead of the radical sign to mean $\pm \sqrt{}$. That is, I will write $\circ(x)$ for $\pm \sqrt{x}$.

What are the solutions to $(x^2-5)^2 = 6$? That's no problem; they are $\circ (3) + \circ (2)$. Now what are the solutions to $x^2 = 5 - \sqrt{6}$? Can't write them in $\circ$ notation. Is this a huge problem? No. We could still adopt the $\circ$ notation. But I contend that we wouldn't want to. It's very nice for symbols to stand for one and only one number when we do calculations. We allow ourselves indulgences for things like the $\pm$ sign because they abbreviate a very common situation, but even these run us into trouble.

As others have said, a change in notation wouldn't change the underlying mathematics (for those who argue that this notation would be strictly illogical; I disagree -- it's no more illogical than the $\pm$ notation, it's just not as flexible).

Answer 4

I try to write another explanation.

Let's look at it as the function,which is the domain is time and range is place:when you write at follows "$$f\left( x \right) =\pm \sqrt { x } $$ it means for instance at $4$ o'clock you can be in two different places$\left\{ -2,2 \right\} $ which is imposible,and it is contradiction with the definition of function.

Answer 5

Actually, when dealing with complex numbers your proposal is essentially what is used: for a complex number $z$ there is no single number that is called $\sqrt{z}$, rather we say $z$ has two square roots, and $\sqrt{z}$ is a "multivalued function". Sometimes we designate one of the two as the "principal branch", but this is not always done.