What's the fuss about performing operations from left to right?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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 $