Just to make everyone aware, to finally put this question to rest, there are two pitfalls in this question that will lead you to the wrong answer.
1) Because it's written as PEMDAS, that implies that multiplication has higher precedence than division. But in actuality, they have the same precedence.
2) 1/2x = 1/(2x). This is actually not true because as state above, multiplication and division have the same precedence, and thus must be done in order from left to right. Therefore, 1/2x = (1/2)x = 0.5x.
Originally Posted by mathman26
1/2n = 1/(2n) This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). It was simply 1/2pi, or 1/2e^2.
Now consider the Identity Law:
a = 1a = 1(a)
We know there is ALWAYS an 'invisible' 1 as a ceofficient of a variable if no other number is there. Therefore:
a/a = 1, and if a is also 1a, then a/1a = 1. Blindly using 'pemdas', some folks would do this:
a/1a = a/1*a = a*a = a^2. I hope this drives home the silliness of this calculation.
Your understanding of math does seem to be quite good. And your arguments hold true under the assumption of 1/2x = 1/(2x). Unfortunately this is not true and is a common misconception due to the limitation of typing. I'm sure if it was written out on paper that many people here would not be falling into these pitfalls because it would be much more obvious because you can actually use fractions lines as well as showing actual numerators and denominators. Maybe this is what the OP meant when he wrote the equation and forgot to add the parenthesis, or maybe they left out in order to create this pitfall. The reason Wolfram or any graphing calculator gives the answer of 9 is because they follow the order of operations down to the individual character. By adding the parenthesis to the (2x), you are not using the associative property of multiplication, you are actually altering the problem. If you were to change the equation to a pure multiplication question using the inverse property real number field under multiplication, the problem 1/2x would change to 1*(0.5)*x . This makes it much easier to see that the x is multiplied in the numerator due to both the commutative and associative property of multiplication.
You're example using the identity property of a/1a equals a^2 under the standard order operations. However a/(1a) does equal 1 as long as a =/= 0. But once again, this relies on the condition mentioned above holding true. As for the cases with pi and e, this is commonly accepted due to laziness, but a real stickler of a professor like myself would take off points for that
It is interesting that wolfram contradicts itself like that. Should raise that with their programmers. If I type 2x/2x on my ti-89, I get x^2
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