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I'm having trouble remembering PEMDAS (in regards to precedence in mathematical equations), ie:

  • Parentheses
  • Exponentiation
  • Multiplication & Division
  • Addition & Subtraction

I understand what all of the above mean, but I am having trouble keeping this in my head. Can you recommend any tricks or tips you use to remember this.

Thanks

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  • $\begingroup$ This was incorrectly tagged as being related to functional equations. $\endgroup$ – Simon Hayward Dec 9 '12 at 12:49
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    $\begingroup$ you may want to try: xkcd.com/992 $\endgroup$ – Dave Dec 14 '12 at 13:12
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    $\begingroup$ Don't remember PEMDAS. $\endgroup$ – Pedro Tamaroff Sep 9 '14 at 0:40
  • $\begingroup$ Maybe you find BEDMAS easier to remember? Brackets, Exponentiation, Division, Multiplication, Addition, Subtraction. (I changed that from BODMAS [Brackets Off ...] to contain your inclusion of Exponentiation) $\endgroup$ – Pixel Oct 20 '15 at 21:02
  • $\begingroup$ Don't remember it. Tell your teachers and those who use infix notation to either use prefix notation for terms, use postfix notation for terms, or completely parenthesize terms in infix notation. $\endgroup$ – Doug Spoonwood Oct 21 '15 at 15:34

13 Answers 13

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PEMDAS is an acronym to help you remember. try different forms of mnemonic devices, like acrostics: Please Excuse My Dear Aunt Sally; Pancake Explosion Many Deaths Are Suspected; Purple Egglants Make Dinner Alot Sickening; Pink Elephants March, Dance, And Sing; Pizza ended my donuts addiction Saturday

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    $\begingroup$ +1: Did you make up these hilarious mnemonics yourself or did you find them in a book? $\endgroup$ – Dominik Dec 9 '12 at 13:28
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    $\begingroup$ lol, i heard them over time from teachers and classmates $\endgroup$ – lias Dec 9 '12 at 17:48
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I think it makes most sense to remember it in four steps:

  1. Parentheses trump everything else -- because that's what they are for.

  2. Addition and subtraction have the same priority, because they are each other's inverse.

  3. Similarly, multiplication and division have the same priority. But that's rarely relevant, because proper mathematics prefers writing division with a faction bar, which delimits its arguments unambiguously without referring to rules.

  4. The ordering between the various groups of operation is chosen such that polynomials work. What "polynomials work" means is that we can use the laws of arithmetic to rewrite everything into something that doesn't need parentheses:

    • Exponentiation must come before multiplication, because then we can rewrite $(2x)(3x)$ into $6x^2$. If multiplication had higher priority than exponentiation we'd need either parentheses as $6(x\text{ to the power of }2)$, or introducing square roots to make $\sqrt 6 \cdot x\text{ to the power of }2$, both of which are inconvenient.

    • Multiplication must come before addition and subtraction, because the distributive law allows us to rewrite an arbitrary product of sums into a sum of products, but not the other way around: $$ (5+x)\cdot y = 5y+xy $$ but there's no product of sums that correspond to $3x+5y$, so it would be inconvenient if we couldn't write that without parentheses but had to write it as $(3\text{ times }x)+(5\text{ times }y)$.

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    $\begingroup$ Writing division with a fraction bar is a way of hiding brackets: $$\frac{\color{red}{a}}{\color{blue}{b}} \space\equiv\space \left(\color{red}{a} \div \color{blue}{b}\right)$$ $\endgroup$ – Nick Sep 9 '14 at 5:39
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Well, I suggest you remember the word 'pemdas'. Another way:

  • brackets were designed to be evaluated first, so they go first
  • the other operations go from advanced to simple
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    $\begingroup$ Yup, not sure you can get much simpler than a one word acronym. Otherwise there's nothing so good as practice! $\endgroup$ – Simon Hayward Dec 9 '12 at 12:50
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please eat mom's delicious apple strudel

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    $\begingroup$ Why it is downvoted? It is a proper answer for a question about memorization. $\endgroup$ – Przemysław Scherwentke Nov 4 '14 at 0:48
  • $\begingroup$ Not the downvoter (to be exact, can't). While it's a proper answer, a short explanation might help, like "Here is how I memorize it". Also, using correct formatting might give a hint to each letter. Else, it looks like a (somewhat) nonsense answer. $\endgroup$ – Andrew T. Nov 4 '14 at 1:59
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I recommend thinking about precedence a different way.

  • What did you learn first? Addition and subtraction.
  • What did you learn after that? Multiplication and Division.
  • What after that? Exponentiation.

If you perform these operations in the opposite order that you learned them, you don't have to remember about anyone's Dear Aunt Sally. When these rules are not enough to avoid ambiguity, use parentheses.

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Daan's suggestion of remembering that parenthesis go first, then advanced to simple is good because there is some meaning in it instead of just having a random word. But if you want to remember a word, and you remember that multiplication and division go together (they're inverses of each other), and addition and subtraction go to gether (again, they're inverses), you just have to remember PEMA or PEDS or whatever, and then fill in the gaps.

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I always taught students to remember this mnemonic as PE(MD)(AS) since multiplication and division are performed left-to-right in the order they appear, just like addition and subtraction.

I'm sure many have seen this viral Facebook question: $$6 \div 2 \ ( 1 + 2 ) = \ ?$$

Is is 9 or 1? Here's a video that shows you how it's done.

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  • $\begingroup$ division and subtraction aren't commutative operations, when using them you should always employ parentheses. $\endgroup$ – Zackkenyon Jun 30 '13 at 18:40
  • $\begingroup$ I think that we should get rid of the dreadful $\div$. The only time I have used it is when I teach it. $\endgroup$ – Baby Dragon Oct 25 '13 at 5:18
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I think a better acronym would be

$P$arenthesis $E$xponents $F$actors $T$erms.

Students often think that multiplication has higher priority than division because $M$ is before $D$ in $PEMDAS$.

However, if you want a funny way to remember $PEMDAS$, then

$P$lease $E$xcuse $M$y $D$ope $A$ss $S$wag.

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Aunt Sally (PEMDAS) has been banished from mathematics.

Read this very short article to understand why: https://blogs.edweek.org/teachers/coach_gs_teaching_tips/2011/01/math_teachers_please_excuse_dear_aunt_sally--forever.html

PEMA is the new mnemonic: Purple Elephants Munching Apples ???

  1. Parentheses (and other grouping symbols).
  2. Exponents.
  3. Multiplication and Division from left to right.
  4. Addition and Subtraction from left to right.

The importance of from left to right is absent in PEMDAS.

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I am a step by step person. (remembering always left to right) 1. Parenthesis 2. exponents 3. multiply/divide 4. add/subtract

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  • Please Excuse My Dreadful Art Skills
  • Please Eat My Delicious Anaconda Snakes
  • Please Excuse My Dear Aunt Sally

The first $2$ I came up with. The last is a commonly used phrase to remember PEMDAS.

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I Think a good P.E.M.D.A.S Is

Parrots eat up magic dassies and snoozed off

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Please Excuse My Dear Aunt Sally.

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  • $\begingroup$ This is mentioned in the upvoted and accepted answer. $\endgroup$ – pjs36 Oct 20 '15 at 20:56

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