What would happen if $\sqrt{}$ instead meant the negative square root?

Question

I read that it is only by convention that $\sqrt{}$ means “the positive square root of”. The downvotes and user 'Thomas' 's answer compel me to clarify that I asked this hypothetical question only because of curiosity, not because of any desire to desecrate $\sqrt{}$.

What if $\sqrt{}$ meant “the NEGATIVE square root of”? Why might convention not have caused this?

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I already understand the following and already read: 2013/10/23, 2013/11/16, 2014/5/26, 2015/5/12, 2015/9/24.

Source: Page A7, Appendix A, Calculus Early Transcendentals (6th ed.; 2008) by James Stewart:

Recall that the symbol $\sqrt{}$ means “the positive square root of.”
Thus $\sqrt{r} = s$ means $s^2 = r$ and $s \ge 0$.
Therefore, $\color{darkred} { \text { the equation $\sqrt{a^2} = a$ is not always true. It is true only when $a \ge 0$ } } $.
If $a < 0$, then $ -a > 0$, so we have $\sqrt{a^2} = -a$.
[...] we then have [...] $\sqrt{a^2} = |a|$.

Answer 1

You are free to change notation. The only thing that is going to happen is that it will confuse and annoy a lot of people. Also, if you use notation that is contrary to established notation without giving a good reason, people aren't going to take your serious as a mathematician.

As mentioned in the comments above, if you define $\sqrt{a}$ to be the negative number $b$ such that $b^2 = a$, then we would just have to put a minus in front of all square roots appearing in the literature. It would (as also mentioned in the comments above) cause the problem that $$ \sqrt{ab} \neq \sqrt{a}\sqrt{b} $$ And then you have have to make sense of stuff like $x^{1/3}$. How is this now defined? There are good reasons for picking the notation we have.

So basically, you have to ask yourself: why would you want to do that?

Answer 2

So imagine that convention caused $\sqrt{}$ to mean “the NEGATIVE square root of”. Then what would change or differ?

We would get negative signs everywhere instead of positive ones. This will happen every-time we solve an equation with even powers.

Would mathematics be fundamentally changed?

Good question, but I don't think it would, only the signs will change.

Answer 3

As commenters said, mathematics would be the same up to the substitution $\sqrt{\ } \mapsto -\sqrt{\ }$.

Actually, a similar phenomenon already happens in the context of differential geometry and PDEs. The same symbol $\Delta$ means $-\frac{\partial^2}{\partial x_1^2} -\ldots -\frac{\partial^2}{\partial x_n^2}$ in the geometers's literature and $\frac{\partial^2}{\partial x_1^2} +\ldots +\frac{\partial^2}{\partial x_n^2}$ in the PDEs literature. It's a bit confusing but people are used to it.