Why is $-n^2$ seen as $-(n^2)$?

Question

I understand order of operations plays a large role, but I am still confused. I figured out:

If $-8^2 = -(8^2)$ then $-8^2 = -64$. Now the can do the inverse. To do the inverse we square root the entire expression. So $\sqrt{-64} = \pm8i$, or we can separate $-64$ into $-1 \cdot 64$ and we also get $\sqrt{-1}\cdot\sqrt{64} = i \cdot ±8 = \pm8i$. This means that we end up with different value to what we started with, meaning the inverse does not work. Now where $-8^2 = (8\cdot-1)^2$, we can say that $-8^2 = 64$, and doing the inverse we get $\sqrt{64} = \pm8$, which works.

So firstly, is there anything wrong with this proof. And secondly, if the proof above is valid, why is $-n^2$ seen as $-(n^2)$?

Answer 1

Because $-8^2 = -(8^2)$ is the way that needs the fewest parentheses in the long run.

Order-of-operations is a man-made concept. We could very well do math entirely without it, or with a different order, and no objective part of math would be any different. The only thing that would change is writability and readability. We humans have decided that that's what it should mean because it's the most convenient, so that's what it means.

Answer 2

The inverse does work, but you're using the inverse of the wrong function. If your function is $x \mapsto -x^2$, the inverse relation is $x \mapsto \pm\sqrt{-x}$: in other words, $y = -x^2 \iff x = \pm\sqrt{-y}$. You can check that with $x = 8$, there is no contradiction.

Answer 3

$-a$ is generally an abbreviation for the solution of $x+a=0.$ This means, that $-a$ stands for the inverse element of an additive written group. It is a bit of a sloppy notation since the minus sign abbreviates what would be $ a^{-1}$ in the multiplicative case.

Thus, $-n^2$ is a solution for $x+n^2=0.$ The question, whether it should represent $(-n)^2$ instead doesn't even occur. BEDMAS and alike has no say here either. $-a$ is short for $+(-a)$ regardless of what $a$ is or whether it is used in an equation or as a standalone.

Answer 4

The order of operations is just a convention, and over the years there have been attempts to introduce other conventions. Polish notation is an example.

The order of operations attempts to model the spoken language symbolically. With $-n^2$ being the negative of $n$ squared rather than the square of negative $n.$ In part this is because the $-$ symbol has double duty. It could indicate that we are negating a number or it could mean that we are subtracting a term. Since we need to do both, which interpretation will be more useful? If we want to read $-n^2$ as $(-n)^2$ then if we are subtracting the term $n^2$ we will need to write $-(n^2).$

We have decided to read the exponentiation as happening before the subtraction or negation. But, again this is a convention. One that seems to be useful most of the time.

As for your algebra. If we want to invert the function $y = -n^2$ then $n = \pm \sqrt {-y}$