I am trying to make a very clear explanation of why -4^2 is -16 and not 16.

You all know that in mathematics an expression must always be solved in the right order. You don't always go from left to right. Everyone knows that in expression:
2 * 3 is evaluated before the 1 + 2.

Similarly everyone knows that in expression:
5 ^ 2 is evaluated before 2 * 5.

This is called order of operations. I've referred to it as "operator precedence" many times in my posts before, which is a term more common to programming languages, but it means the same thing. Order of operations.

Now, the reason why people miscalculate the -4 ^ 2 is because they don't know that there's an operation "hidden" in the -4. Yes, -4 is a number on its own (a real number), but the -4 is also an operation. This operation is called negation. Everyone knows this as well. Everyone knows that a minus sign can be used to negate a number, or an expression, but what they don't know is that it's also got its place in the order of operations. And furthermore they don't know that it's not on top of the list (and this is not necessarily their fault, it's surprising in how many schools they don't apparently teach this at all). Exponentiation has higher precedence in order of operations than the negation (which I've referred to as "unary -" in my earlier posts, which is also more of a programming terminology). That is why -4^2 is calculated as -(4^2). For the very same reason why 1 + 2 * 3 is calculated as 1 + (2 * 3).

Hopefully that clears it up a bit.

It's -16, since -4^2 is the same thing as in something like 17 - 4^2, which is 17 - 16 = 1 and not 17 + 16 = 33

And for people who think -4^2 is 16, because "-4" is a number, think it like this:
-(2+2)^2

Clearly the - here is not part of the number. It's used to negate an expression. This shows you that subtraction of two numbers, and negative numbers are not the only things the - is for. It's also for an operation called negation, which is shown in the expression above.

are there any other "hidden" operations btw.

Not that I know of!


Also Ragnar is right, it's always a good idea to use parenthesis to clarify an expression. Although I don't really think that simple expressions like 1 + 2 * 3 need them.

I have never heard everyone as a synonym of I before

Sorry, English isn't my native language. (assuming you're talking about me)

no rami, everyone DOESN'T know that, this is the point. just because something is perfectly logical to you it does not mean that everyone else knows it at all. your explanations are constantly "everyone knows this" "everyone knows that" but they DON'T. i am pretty sure everything you are saying here is always 100% true, but it's made confusing by (as ase said) you're constantly using technical and programming terminology instead of basic maths terms everyone can understand, and your explanations are just consistently confusing.

Ok, excluding people under age 10, you seriously claim that people don't know that multiplication is evaluated before addition in an expression? What a fine level of education they must have in the UK...