I'm on Khan Academy, learning how to multiply decimals, and in this video (around the 1:10 mark) Sal shows that if you're multiplying a decimal by a decimal, first you convert both into natural numbers divided by exponents of 10 and then multiply, ending up with this:
2.91 x 3.2 = 291 / 100 x 32 /10
So far, so good. But then he goes and reorders the expression to the right of the equals sign into this:
2.91 x 3.2 = 291 / 100 x 32 / 10
= 291 x 32 / 100 / 10
This is the point where I get lost, because in a much earlier video about the Order of Operations around 5:40 (and more prominently around 7:40), he explains that when there's multiple operations on the same level (which multiplication and division are), you have to complete those operations from left to right. So by that logic, the above then violates Order of Operations, right?
After that, he then further seems to violate it by clearly performing the operations out of order, by doing the following:
2.91 x 3.2 = 291 / 100 x 32 / 10
= 291 x 32 / 100 / 10
= 291 x 32 / 1000
He divided 100 by 10, which is at the far right of the expression! How does that work, if you're supposed to perform your operations from left to right?
Now, I did both calculations (the way he did it and just taking the original expression and resolving it from left to right – 291 / 100 x 32 / 10 ) on my calculator and came to the exact same answer of 9.312, which tells me that the Order of Operations rule is wrong and you actually don't have to perform them from left to right?
So to sum up my question: Do you have to perform operations on the same level, like multiplication and division, from left to right, or not?
UPDATE 1: I've now done some more experiments on the Order of Operations' left to right rule, which I think might have given me more clarity on exactly how exactly it works. I tried the expressions 1 + 3 - 5 and 2 x 3 / 4 (and a couple others, unimportant) and I have found that I can re-arrange the numbers and wind up with the same answer, so long as I move their symbol (+/-) with them. For example:
1+3-5 = 1-5+3 = -5+1+3 = -1
2x3/4 = 2/4x3 = 3x2/4 = 1.5 // although I'm not sure how I would move "/4" to the left-most position of this expression.
Based on that, I want to update the conclusion of my question: Can you rearrange the numbers in an expression, so long as you move the numbers' symbols with them, thus not breaking the left to right rule?
UPDATE 2: It seems that my problem was caused when I confused the Order of Operations' left to right rule with moving numbers around in an expression (which was my suspicion in UPDATE 1). Thanks to everyone that helped me come to this realization!
Best Answer
The PEMDAS rule should be understood as PE(MD)(AS) or even better PE(DM)(AS), where the operations inside brackets have equal priorities and should be performed from left to right.
For example: $$16\cdot 8:4=(16\cdot 8):4=16\cdot (8:4)=32.$$ However: $$8=16:8\cdot 4=(16:8)\cdot 4\ne 16:(8\cdot 4)=\frac12.$$
One way to avoid confusion (or violation of PEDMAS) is to consider "multiplication" and "division" as mutually inverse operations and remember commutativity of multiplication. For example: $$16\cdot 8:4=16\cdot 8\cdot \frac14=16\cdot \left(8\cdot \frac14\right)=16\cdot \frac14\cdot 8=16:4\cdot 8;\\ 16:8\cdot 4=16\cdot \frac18\cdot 4=16\cdot 4\cdot \frac18=16\cdot 4:8.$$