Please Excuse My Dear Aunt Sally

When I was in elementary school, I remember (sort of) being taught this mnemonic device to remember my Order of Operations.

Please Excuse My Dear Aunt Sally

Wtf?

This stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Okay, now that we've finished the review for the old people, I'll elaborate on how I was taught this and why I think I had some major problems doing math in Physics.

When I was taught the Order of Operations, we were told that Multiplication and Division can be done in any order, you'd still get the right answer.

We were also told that Addition and Subtraction can be done in any order, you'd still get the right answer.

Well, yeah, with Addition and Subtraction, you can do that.  But not Multiplication and Division.

It's easy with adding and subtracting:

9 -2 + 6

7 + 6 = 13

or

9 - 2 + 6

9 + 4 = 13

See how I got there in both cases?  First I did the 9 -2 subtraction to get 7, plus 6 is 13.  In the second version, I did -2 + 6 first (remember, the - is attached to the 2, making it a negative number) and got +4.  So 9 + 4 also equals 13.

Try to do that with division and multiplication and you'll get two different answers, but not with every equation.

Here's one floating around the interwebs right now that'll net you two different answers:

48 / 2 (9 + 3)

I saw this and thought oh how easy.

48 / 2 (12)

48 / 24

2

Nope.  Chuck Testa.

That's wrong.  You apparently have to do multiplication and division, if they're the only operations remaining, from left to right.

48 / 2 (9 + 3)

48 / 2 (12)

24 * 12

288

Doesn't need to get more complicated than that.  You don't even need to explain it such that 9 + 3 is a variable x and you're being given a value to put in to x.  Straight up, I was edumacated wrong.

This type of crap came up a shit ton, not in my math classes (I seemed to do perfectly well in those classes) but in my Physics class in high school.  I managed to get so much math wrong that my teacher wanted me in remedial math classes to review elementary level shit because that's what I was fucking up on.

But whenever I was in calculus, I'd get it right.  Somehow.

Because of shit like this

9 * 2 / 3

You can do this in any fucking order and get 6.

18 / 3

6

or

9 * 2/3 (2 divided by 3 becomes the fraction 2/3)

9/1 * 2/3 (put 1 under 9 to represent whole number fraction)

3/1 * 2/1 (reduce numerator 9 to 3 and denominator 3 to 1)

6 (multiple straight across to get 6/1 which is 6)

Yes, technically the first method is the "correct" method because it is less work.  But this shows that our teachers thought that there was no case in which you could get two different answers out of an equation in which only the operators multiplication and division remained.

I'm almost positive this is why I nearly failed high school Physics.

When I think it matters most is when the division is in the front of the equation.  Let's take the example above and mix it up.

9 / 2 * 6

This equation will net you two different answers.

4.5 * 6

27

Okay now ...

9 / 2 * 6

9/ 12

1/4

I've been testing this out a bunch this morning, and it still seems like the only case in which you must do multiplication and division in the order in which they show from left to right, is when division precedes multiplication.

Let's make it a bit more complex.

8 / 4 * 3 * 5 / 6

So far, I've gotten 6 different answers.  I won't go through them all but you can see that if you have several of the two operations, it makes for some confusion when you've been taught you can start anywhere in the problem and go to town.

From left to right (the correct way):

8 / 4 * 3 * 5 / 6

2 * 3 * 5 / 6

6 * 5 / 6

30 / 6

5

Weee.

Now let's get rid of that first division, see if we can get one answer no matter where we start in the equation.

4 * 3 * 5 / 6

4 * 15 / 6

60 / 6

10

or

12 * 5 / 6

60 / 6

10

or

4 * 3 * 5/6

12 * 5/6

12 * .8333~

10

Yup.

Looks like it just so happens that if you have a division operation at the beginning of your equation, you must do it first, then you can do the equation in just about any order you want.

8 / 4 * 3 * 5 / 6

2 * 15 / 6

2 * 2.5

5

or

2 * 3 * 5 / 6

6 * .8333~

5

Math is fun!