# Order Of Operation- Does the order matter..

The BODMAS rule states that while solving any given mathematical expression we first solve it in this manner-

(i) Brackets

(ii) Orders or powers

(iii) Division

(iv) Multiplication

and at last we solve (vi) subtraction

I get that division and multiplication should be carried before addition and subtraction but I don't think the manner in which addition and subtraction are carried out matter..

For example consider this expression- $$1 + 2 - 3$$ Now, it doesn't matter if we add 1 and 2 first and then subtract 3 from it, or subtract 3 from 2 first and then add 1. It WILL give 0 either way.

So does the order of addition and subtraction really matter?

Similarly does the order of multiplication and division matter. Shouldn't the rule be like-

First multiply/divide(doesn't matter which one you do first as division is multiplication in a way) and then do addition/subtraction(doesn't matter which one you do first as subtraction is another form of addition, where you add a negative number)

Am I right?

Also SIDE QUESTION- Does BODMAS also apply on eqations or is it just used to solve expressions to derive an equation?

• Try $5-4+3$, and see whether you still think the order of operations doesn't matter. – Gerry Myerson Oct 22 '16 at 5:30
• 5+3=8;8-4=4. 5-4=1; 1+3=4. -4+3=-1;-1+5=4. Nope, doesn't seem to matter. Was that the point you wanted to make? – fleablood Oct 22 '16 at 5:35
• @fleablood $(5-4)+3=4$ vs. $5-(4+3)=-2$. – dxiv Oct 22 '16 at 6:46
• That's if you interpret -4 + 3 as -(4+3) instead of (-4) + 3. Which .... If you do shows that BODMAS is WRONG. 5-4+3 Brackets, no, orders of power, no, division, no, multiplication, no, addition, yes 4+3 = 7 so 5-4+3 = 5 -7. And finally vi) multiplication???? I think to OP made a typo, so subtraction 5 - 4+3 = 5-7 = -2. Which is WRONG. – fleablood Oct 22 '16 at 7:07
• Frankly, I learned before BODMAS and BODMAS just confuses and irritates me. Now, get off my lawn!...."Shouldn't the rule be like- First multiply/divide(doesn't matter which one you do first as division is multiplication in a way) and then do addition/subtraction(doesn't matter which one you do first as subtraction is another form of addition, where you add a negative number) Am I right?" Well, I think so! But others wont.... sigh... – fleablood Oct 22 '16 at 7:44

BOMDAS is just an acronym - Subtracting a number is the same as adding a negative number, while division is just multiplying by a number's multiplicative inverse. So it doesn't matter what order you do addition/subtraction on.

I don't understand your side question - but I think that you mean if BOMDAS should apply to both $3+7-6/2$ and $3\times(1+2)+2^2+2x = x$, which it does. The first expression simplifies to $7$, while the equation simplifies to $3(3)+2^2+2x = x\implies 3(3)+4+2x=x\implies 9 + 4+2x=x\implies13+2x=x\implies x=-13$

Addition and subtraction are interchangeable, as are multiplication and division - the only reason the acronym states each in order is because you can't make an acronym that's ambiguous about order.

As for your side question: There is absolutely no difference, in any situation, in any mathematically relevant respect, between expressions that involve variables and expressions that do not. Every single rule you know that works for numbers works for variables - for example, $(x + y) + z = x + (y + z)$, just like $(1 + 2) + 3 = 1 + (2 + 3)$. Variables are just stand-ins for numbers; you should think of them the same way.

We can make up any rule we want. As long as we are consistent about it.

So what is 5+4x3+2?

We could make up a rule that: 1) You always do it strictly left to right

So 5+4x3+2 = 9x3+2=27+2=29.

Or we could make up a rule that: 2) You always do addition first

So 5+4x3+2 = 9x5 = 45.

Or we could make up a rule that: 3) You always do multiplication first

So 5+4x3+2 = 5+12+2 = 19.

Or we could make up a rule that: 4) You always go right to left

So 5+4x3+2 = 5+4x5=5+20=25.

So which rule is best?

Well, for many reason 3 is best and 1 and 4 are the worst. But really we could get by with any one provided once we pick one we stick with it.

For many reasons "Multiplication first; Addition second".

What about brackets and parenthesis? Well, the entire reason we have brackets and parenthesis is to tell us to do things first. The are precisely used when the normal rules are not what we want to do so we put them in to indicate something must be done first.

Seriously, if we had a rule that we had to do parenthesis last you can see that that wouldn't work. How could we express "3 times the result 4 plus 5" if we can have any way to say "add the 4 and 5 first". "Add the 4 and 5 first" is what 3x(4+5) means.

So why do we do multiplication first, then addition? Or for that matter powers first, then multiplication, and addition?

Well, I think it's because of "grouping". When we add things we are grouping but sets of units. 3 + 5 is really "3 ones grouped with 5 ones makes 8 ones". When we multiple we are grouping by big factors rather than small units. 3x4 + 5x6 means "we have a set of 3 fours and a set of 5 sixes; that combines and we have 12 and 30 and we combine them by the units to make 32". I don't know. That seems to me the most natural way to do it. In my opinion anyway....

So 3x(4+5) means "okay, first we specifically group the 4 and 5 and then we take a set of 3 of the results of 9. Three 9s is 27".

And powers are in even larger grouping.

Okay... so what about subtraction and division.

Well, addition/subtraction are inverses. 5 - 3 means ? + 3 = 5. Or more algebraically 5 + [-3] where [-3] is the number that takes 3 away. Basically subtraction and addition are the same level of grouping. It doesn't really matter which you do first. I think the BODMAS mnemonic fails as with 3 -4 +5 you definitely dont want to add the 4+5 to get 3-4+5 = 3 - 7 before you subtract. You really want to consider subtraction is adding negative numbers 3 - 4 + 5 is 3 + [-4] + 5 and now it's just addition in any order.

And likewise division, $8\div 4$, is the inverse of multiplication. $8 \div 4 = 8 \times \frac 14$.

So... no. The distinction between addition and subtraction is not as important as all that. BUT do be careful. If you get cavilier mistakes will happen.

Anyway, BODMAS is just a memory aid. It's not actually a mathematical rule.