What is the order of operations in math?

Please excuse my dear Aunt Sally

Parentheses, exponents, multiplication and division, addition and subtraction.

Pemdas

BIDMAS is what my 9 year old has learned
Brackets Integers Division Multiplication Addition Subtraction

I don’t know what Integers are

@Stinley From context, I would think it’s some Common Core euphemism for exponents.

Do the most powerful operations first. Exponentiation is equivalent to several multiplications, which in turn is equivalent to many additions. Multiplication and division are on the same level as are addition and subtraction. The order is 1. exponentiation, 2. multiplication and division, 3. addition and subtraction.

Also, to clarify: you evaluate everything in the innermost parenthesis according to the full rule set of the order of operations until that thing in the parentheses is just one number. Now replace the parenthesis and the stuff in it with that number you calculated. Now work through the next innermost set of parenthesis using the order of operations, and replace that blob with a single number. Repeat this process until there are no more parenthesis. Now evaluate the rest according to the the order of operations.

I hope that makes sense and isn’t MORE confusing.

PEMDAS is close, but not quite true.
Multiplication and Division are on the same “level”.
Addition and Subtraction are also on the same “level”.
What I mean to say is that they are the same operation.*
When it comes down to these two operations in an expression**, solve from left to right. This is something that isn’t clear by the simple “PEMDAS” acronym.

Example:

4–16/4+3*6–2*7 >(multiply and divide)> 4–4+18–14 >(add/subtract, left to right)> 0+18–14 >> 4

Now go back to 4–4+18–14 and add first, as PEMDAS suggests. Did you get -32? That answer is wrong. PEMDAS is wrong.

Likewise, here’s an analogous example, but with multiplication and division:

24/3*2

What’d you get? 4 is wrong. The correct answer is 16.

I love @LostInParadise‘s explanation of the order of operations.

*For the pedantic people: They are inverse operations, with the inverse of multiplication not being defined for r/0, where r is a non-zero real number.
**I don’t exactly know what an expression is, but I’ll say it’s a collection of numbers with operations like multiplication, addition, etc. In this quip, the bolded things are expressions.

@dxs “What I mean to say is that they are the same operation. When it comes down to these two operations in an expression, solve from left to right. ”

Why left to right, particularly? Doing math in your head can go a lot faster if you first handle recognizably easy combinations of numbers within a multiplication/division or addition/subtraction group.

Left to right is convention, and if you break it, then your answer will not be consistent with such established standards. The examples I had given directly explained this. The only way to truly solve them in any order you want is to instead form inverses from the inverse operations. That is, instead of 4 – 4 + 18 – 14, you have 4 + (-4) + 18 + (-14). In the M/D example, it becomes 24 * ⅓ * 2. Now it really looks as though there’s only one operation. I find it more difficult to do this in my head, but if that’s easier for you, then have at it with a bib and a fork.

I entered all of the answers into Wolfram Alpha to be sure, and lo and behold my answers are all consistent the Great Math Lord.

@dxs Pfft. Well, okay. If you want to ignore the fact that -4 + 18 is in no way the same as -(4 + 18), then it could be confusing. Some basic understanding of math has to be assumed.

OK. When my kids were small I always gave them the anatomically correct name for their privates, but if they wanted to change it up after that it was ok with me.

My daughter changed hers to “China.” My son changed his to “Pema.” SO PEMA is just like PEMDAS, (mentioned above) except I tossed out the D for division (because it’s done at the same time as the multiplication just which ever comes first) and I threw out the S, subtraction, (because it’s done at the same time as the addition just which ever comes first.)
So, PEMA is is. Think you’ll remember now? :D

@dappled_leaves I have no idea what your point is.

-4 + 18 is the same as subtracting 4 from 18, which = 14. Order of operation applies.

-(4+18) is the same as adding 4+18 (doing the operation in the parenthesis first), then making it all negative, -22. Order of operation applies.

Another way to look at it -(4+18) is the same as -4 + -18= -22. The order of operations still applies.

@Dutchess_III Were you trying to explain to me @dappled_leaves‘s point?

No. I really didn’t understand what her point was either. I got the impression that she was suggesting that the order of operations was some how didn’t apply between the equations of -4 + 18 and -(4 + 18.) I was just showing that it still did apply.