Why is $\pm x = \sqrt y$ incorrect and $x=\pm \sqrt y$ correct?

Question

Let $x^2=y$, then $x=\pm\sqrt y$. But why can't it be $\pm x=\sqrt y$? I started thinking about this when i encounter this answer to a question but I didn't really understand it. In the answer, it says that there are two possible definitions for the notation $√$:

1. For any positive real number $a$, $\sqrt a$ is defined as the square roots of $a$
2. For any positive real number $a$, $\sqrt a$ is defined as the positive square root of $a$

By the first definition, $\sqrt {16}= \pm 4$. It also says that if I use the first definition,then I will encounter some problems in the future and that’s why we use the second definition. In the answer it says that solving for $x$ in $x^2 - \pi =0$ for $x>0$, by doing $x=\sqrt \pi$, would be incorrect. I still can’t understand why $x=\sqrt \pi$ would be incorrect if I use the first definition.

For me, $\pm x = \sqrt y$ and $x=\pm \sqrt y$ looks like they mean the exact same thing, but I suppose they don’t. Why is that?

Answer 1

Neither is incorrect. $x =±\sqrt{y}$ is the same as $\sqrt{y}=±x$.

However, you'd usually want to have the variable on one side (usually left) and its solution(s) on the other side (usually right).

Thus, one would normally interpret $x=±\sqrt{y}$ as $x$ is the variable and $±\sqrt{y}$ are its solutions while $\sqrt{y} = ±x$ would be interpreted as $\sqrt{y}$ is the variable and its solution is $±x$. Both are of course, equivalent ways of saying the same thing.

Answer 2

$\sqrt {x}$ is the "principal root." That means that $\sqrt x \ge 0.$ (Or, at least it is until you learn about complex numbers.)

$x^2 = 4$ is multivalued and has two solutions. That is $x = \pm \sqrt 4 = \pm 2$

When you write $\pm x = \sqrt y$ this requires that it be possible that $\sqrt y$ be negative, which it can't.

$x = \pm \sqrt y$ says $\sqrt y$ is always positive, but $x$ can be negative.

Answer 3

Let $y\geq0$. If $\sqrt{y}$ is defined as the set of solutions to the equation $y=x^2$ and $x$ is one solution then I would say that $\sqrt{y}=\pm x$ is perfectly sensible notation (since $x$ and $-x$ are the only solutions). More commonly, however, $\sqrt{y}$ is defined as the unique non-negative solution to $y=x^2$. In this case, if $x^2=y$ then $x=\sqrt{y}$ or $x=-\sqrt{y}$. In my opinion one should avoid writing either of the expressions "$x=\pm\sqrt{y}$" and "$\sqrt{y}=\pm x$". My point: $x$ is some number, but what is $\pm\sqrt{y}$, and similarly, $\sqrt{y}$ is some number but what is $\pm x$?

Answer 4

There has been a lot of nonsense posted here. The facts are these:

The notation

$$a=\pm b$$ is simply a universally accepted shorthand for $$a=b\text{ or }a=-b$$

The notation $$\pm a=b$$ has no universally recognised meaning, and should never be used.