I just read an article on MSN about a tricky mathematical problem that apparently most people get wrong. The premise of the article is that, while (-9)^(2) \+ 81 would solve as 162, the lack of braces causes us to do the exponentiation first (PEMDAS) resulting in -81 + 81 = 0, which seems very incorrect as -9 isn’t an operation, it’s a literal. (PEMDAS doesn’t talk about a number’s sign.)
Stating a term of -9 doesn’t mean “take 9 and negate it”, it means “take the literal which is 9 less than 0”. Unless there’s some historical shorthand I’m missing here or something? Does -9 represent a shorthand for 0 – 9? Most programming languages treat a negative literal as a literal and not an operand. (In languages where you can override the – operand, specifying a number with a – in front of it still takes the literal without calling the operand override function.)
So, I guess my question is this: Does -9^(2) evaluate as (-9)^(2) = 81 or as -(9^(2)) = -81, and why?
Here’s the article in question.
[https://www.msn.com/en-us/lifestyle/parenting/tricky-equation-confuses-public-mathematical-misconception-leads-to-widespread-error/ar-BB1hD1mJ?rc=1&ocid=winp1taskbar&cvid=4a3ad03d63b84eb1d76097a4d6c147d3&ei=22](https://www.msn.com/en-us/lifestyle/parenting/tricky-equation-confuses-public-mathematical-misconception-leads-to-widespread-error/ar-BB1hD1mJ?rc=1&ocid=winp1taskbar&cvid=4a3ad03d63b84eb1d76097a4d6c147d3&ei=22)
Who’s right (me or the article) and why?
In: Mathematics
-9^2 = -(9*9) = -81
(-9)^2 = (-9 * -9) = 81
And my god this is weird because I literally just asked my math professor about this, exact numbers and all. It wasn’t related to that article either, just a random homework question. But that was his response.
When you encounter exponentials, you always take only the first thing before it. So -9^2 indeed means “take nine, raise it to the power of two, then negate”, as number and its negation are two things.
If you want to raise to the power whole -9, then you have to write it (-9)^2. In this case, negation is also raised.
This is actually popular problem and many matematicians are educated wrong way. But simple equation is enough to explain:
a – b^2 = 0
By logic, a has to be greater than zero, as b^2 cannot be negative (ignoring complex numbers). If we use 9 as b and raise whole -9 to the power of two, we get:
a + 81 = 0 => a = -81.
Contradiction with previous statement, thus we made an error.
Other example is when we rearrange equation to a = b^2. That way we get rid of negation and we see that 9 and -81 are not solutions.
Its a notation thing, if you let -9^2 be -81, you can do (a-b)(a+b) and it works out to a^2 -b^2, which you can write as a^2 -b^2 instead of having to write a^2 -(b^2 )
And in equations, you would always just drop the – if you were squaring it and make it a positive, so they went with the notation that is easier for equations.
But I do agree with you -9^2 by its self should be 81. Unfortunately, the common notation says it is not.
The trick is that we don’t typically write expressions that inefficiently. The expression would almost always be written as
81 – 9^2
and when you see that expression you don’t think twice about the answer being 0. Someone has rearranged the terms with, I presume, intentional misdirection.
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