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When I learned about BODMAS / PEMDAS rules in mathematics. I was told to follow certain rules.

For example if I have this expression: $3 − 2 + 4 − 1$ and to solve it I was told to always go from left to right because addition and subtraction in this case have the same rank or weight.

Or, some rule that goes like "collect positive terms first and then negative term ..."

But I found that it doesn't really matter in which order I perform or solve it as I would get a same answer.

  • $3 + 4 = 7 - 2 = 5 - 1 = 4$
  • $3 - 1 = 2 - 2 = 0 + 4 = 4$
  • $3 - 2 = 1 - 1 = 0 + 4 = 4$
  • $-2 + 4 = 2 + 3 = 5 - 1 = 4$
  • $-1 + 4 = 3 + 3 = 6 - 2 = 4$
  • $4 - 2 = 2 + 3 = 5 - 1 = 4$

Am I right or wrong?

user963241
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    If I don’t go from left to right I can add 2 with 4 first. You then get the equation $3-6-1=-4$. – Tommiie Oct 18 '18 at 06:12
  • Of course. Addition is commutative and associative. You can rearrange any finite sum as you wish. – Berci Oct 18 '18 at 06:13
  • @Tom: But you forgot to notice that 2 is a negative number. – user963241 Oct 18 '18 at 06:14
  • @Tom: Last I checked, 2 was a positive number... –  Oct 18 '18 at 06:15
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    If you interpret the expression $3-2+4-1$ as $(3)+(-2)+(4)+(-1)$ then certainly it doesn't matter what order you do the additions in, because addition is commutative. However, if you interpret it as it is written, as $(3)-(2)+(4)-(1)$, then it does matter what order you do the additions and subtractions in. –  Oct 18 '18 at 06:18
  • I know. Thus that would require the student to see that subtracting 2 from a number actually means adding a negative number, thus it’s not 2+4 but -2 + 4. I’m sure this is not an easy thing to grasp for young students struggling with math, hence the easy rule to go from left to right. – Tommiie Oct 18 '18 at 06:18
  • @Rahul: +1 for comment, because I was just beginning to type something like what you did when your comment appeared! – Dave L. Renfro Oct 18 '18 at 06:19

3 Answers3

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You may rearrange the terms how you wish before actually adding and subtracting without changing the value (assuming you do it the right way). That's not what "go from left to right" is about.

Subtraction is not an associative operation, and it doesn't associate with addition either. That means that given an expression with subtraction (and possibly addition), inserting parentheses in different ways may change the outcome. For instance, in your example, we can do $$ ((3-2)+4)-1=4 $$ Or we could do $$ 3-(2+(4-1))=-1 $$ or any of a number of different orders.

Going left-to-right means that the top one is the correct one. You evaluate the leftmost binary operation (which is $3-2=1$) first, then the next one (which becomes $1+4=5$) and finally the rightmost one ($5-1=4$).

So left-to-right isn't about the terms, it's about the operations. You even follow this yourself since you in your calculations always put the next operation to the right of the previous one (abusing the $=$ sign in ways I don't really approve of, but that's another story).

You seem to have internalised the fact that in an expression like $3-2+4$, the $-$ in some sense "belongs" to the $2$, and only the $2$. That's not a bad thing in general, but in this specific case I think it made you confused about what the whole left-to-right thing is about. I personally prefer the "belongs to" interpretation over the "go from left to right" imperative, but that's not usually what's taught in schools, so it's a realisation most of us have to come to on our own.

Arthur
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  • Does the same also go for multiplication and division? E.g. I can perform them in any order. For example: $6 / 3 * 5$. I can perform multiplication here first i.e. $6 * 5$ to get 30 and then division with $3$ to get $10$. Or, I can perform division here $6 / 3 = 2$ and then $2 * 5 = 10$. – user963241 Oct 18 '18 at 06:57
  • @user963241 I know many people say multiplication and division also goes strictly from left to right, but it's not as ubiquitous. Thus when you have an expression with either both division and multiplication or multiple divisions, my advice is to always use parentheses or fractions to clarify your own expressions (write either $(6/3)\cdot 5$ or $\frac63\cdot 5$, or go for $6\cdot 5/3$, which cannot be misunderstood), and to consider any expression like $6/3\cdot5$ as ambiguous. – Arthur Oct 18 '18 at 07:04
  • But I thought $6/35$ is just $\frac{6}{3} 5$ Is it really not? – user963241 Oct 18 '18 at 07:12
  • @user963241 Very often it is. But you cannot be as certain of that as you can about $6-3+5=8$. – Arthur Oct 18 '18 at 07:22
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Depends on how it is taught, the relationship between minus and negative might not be obvious to beginners. The notion of negative usually comes much later after subtraction. That is $$a-b = a+(-b)$$

Hence upon giving $a-b+c$, some might misinterpret it as $a-(b+c)$. Stating a clear rule for people to follow reduces such mistakes.

The result is true by commutative and associative law for finite number of terms for the addition operations. That is you are dealing with only addition operation on the set of real numbers.

As we go on to discuss infinitely many terms, rearrangement might not be that trivial.

remark about your examples:

Be careful about your equations, I know what you are doing but $3 + 4 \ne 7 - 2 $

Siong Thye Goh
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You are correct. For any numbers $a,b$, we have $a+b=b+a$ (this works whether or not $a$ or $b$ is positive or negative), so you can permute the terms of a sum any way you want and you will still get the same result.

Albert
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