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?
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 [...] $\sqrt{a^2} = |a|$.