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Let me blindly follow for a second BODMAS. $$ \begin{align*} 1-3+2 &=1-(3+2)\\ &=1-5\\ &=-4 \end{align*} $$ (The brackets are just to make the error clear - I wouldn't write them in "real life".) This is correct according to BODMAS, because BODMAS doesn't talk about positive or negative numbers. It just says that addition should come before subtraction.

What am I missing here? Is it like "i before e except after c", and then you realise their is a word and the whole rhyme is as follows:

i before e,

Except after c,

Or when sounded as "a,"

As in neighbour and weigh.

So, am I missing some subtlety? Is the rhyme incomplete? Is it not true that subtraction has greater precedence than addition, and so BODMSA is actually correct and BODMAS is incorrect?

NOTE: This issue is true for all other mnemonics I know of, such as BIDMAS, PEMDAS, PIDMAS...

user1729
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  • $1−3+2=1−(3+2)$????? – barak manos Aug 06 '14 at 09:54
  • @barakmanos Yes. No. That's my point! – user1729 Aug 06 '14 at 09:55
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    I always though $+$ and $-$ had the same priority and the rule was to do the operations from left to right... :p – Daniel R Aug 06 '14 at 09:57
  • $1-3+2=-4$????? lol – Abraham Zhang Aug 06 '14 at 09:59
  • @EpicGuy: Yes. In OP's set of rules, $+$ is performed before $-$. But there were something like three errors besides that in the original post... – barak manos Aug 06 '14 at 10:00
  • Which are all now fixed! – user1729 Aug 06 '14 at 10:01
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    I just felt like joining in – Abraham Zhang Aug 06 '14 at 10:02
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    The "i before e" rhyme is nonsense. Think about "science". Think about "being". – Gerry Myerson Aug 06 '14 at 13:24
  • @Gerry I fell that "i before e, except (sometimes) after c, or when sounded as ahh, with some exceptions listed below..." is slightly less snappy... – user1729 Aug 06 '14 at 13:29
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    I don't know about BODMAS and I don't care about such acronyms, having seen that they do more harm than good. Just think to $a-b$ as a shorthand for $a+(-b)$, where $-$ is a unary operator. Leave subtraction out of your considerations. The same for $/$. – egreg Jan 04 '16 at 14:31
  • @egreg That is basically what I tell my students: subtraction doesn't exist - it is just a figment of their school teachers imagination! (For division, I find it interesting that a/b/c looks "odder" to my eyes than a-b-c, and I wonder if this is true of everyone. But yes, it is exactly the same issue.) – user1729 Jan 12 '16 at 12:29

4 Answers4

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In the Danish educational system we are taught the order of operations in the following hierarchy:

  1. Brackets
  2. Roots and powers
  3. Multiplication and division
  4. Addition and subtraction

along with the rule of carrying out operations of equal order in the reading direction. In effect we have a BO(D,M)(A,S) system where (D,M) and (A,S) are pairs of equal order. This yields the same results for any calculation as all international standards prescribe, so no need to worry. The funny thing is that in our system you are meant to calculate as follows

$$ a\cdot b/c=(a\cdot b)/c $$ whereas the English BODM(A,S) prescribes $$ a\cdot b/c=a\cdot(b/c) $$ Of course those two expressions are equal to one another, so it produces no controversies about Danish and international order of operation standards.

My best guess is that BODMAS was ordered that way in order to form a pronouncable word rather than the tongue twisting BODMSA knowing that A and S are equal as long as you apply the rule of the reading direction too.

NOTE: Actually PEMDAS would cause problems too unless we interpreted it as PE(M,D)(A,S). For instance $a/b\cdot c$ would equal $a/(b\cdot c)$ if multiplication really came before division. But in reality what is meant is reading from the left to the right performing operations of equal order, namely $a/b\cdot c=(a/b)\cdot c$.

String
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In the case involving only addition and substraction, follow the left to right rule. (This also applies for operations involving division and multiplication only.) Do not add brackets when untold in question. So

$1-3+2=-2+2=0$

user1729
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Tosh
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I figure that I should answer my own question, as I am not fond of the other answers and long ago settled this in my head.

Subtraction and division do not exist. There are only addition and multiplication. (Subtraction is addition of negative numbers, division by $x$ is multiplication by $1/x$.)

So BODMAS should be BOMA.

For example, if subtraction "existed" then it would be associative, so $$-4=(1-2)-3=1-2-3=1-(2-3)=2.$$ This is clearly wrong, so subtraction cannot exist. Rather, $1-2-3$ is shorthand for $1+(-2)+(-3)$ (this is what egreg says in the comments) - it is short hand for addition of negative numbers.

Note that BODMSA would not differentiate from either of the above "answers".

I am uncomfortable with saying "we read left to right", as addition and multiplication of read numbers is commutative. Stipulating an order looses this strong feature!

user1729
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-1

It must be BODMSA at all times. BODMAS is fake. Don't try to defend it and reject the impeccable BODMSA on the grounds of pronunciation. Can the 'nice' pronunciation right the wrong?
Someone suggests we teach our students to consider, for example: 12-5+4 as 12+(-5)+4 without regarding the minus sign. I ask my fellow teacher, "what about 12-(-5)+4? Is the minus sign still insignificant? In fact, we must admit to give priority to subtraction over addition in the order since it is correct at all times, whether it be at the left or the right. BODMAS says 12-9÷3+5=4; also 12-(-5)+4=13 while BODMSA gives correct answers of 14 and 21 respectively. It's true doing the operations from left to right is correct doesn't mean we should embrace what is not true.

John Amanor Samuel
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