Is Bodmas a convention or a fundamental feature of mathematics?

[jake-the-peg]

One of the regular "order of operations" questions came up in How it works the other day and this thought struck me.

Why is the order of operations the way it is? Is it just a general convention or is there a subtle underlying reason for it?

Would the world be different if everybody decided that 3+6/3 was 3 rather than 4?

Would planes fall out of the sky if engineered under this assumption? is this truely fundamental or just a convention?

[rojash]

"rather than 4? "

?!?!?!?!?

[factor-fiction]

A convention. Otherwise we'd get different answers to the same problem.

3+6/3 is neither 3 nor 4: it's 5

[jomifl]

doesn't it just define the order of operations?

[jake-the-peg]

Duh ! still getting my first coffee!

So we think mathematics would still be internally consistant if this order were changed - a different civilisation might decide on a different one?

I'd instinctively feel the same but I'm not sure how you'd demonstrate that was true - I don't know where our order of precidence came from

OK so if a car accelerates at 2 m/s² from 3 m/s for 2 seconds

V=U+at

It is doing 3+2*2 =7 m/s

If we changed the order of operation it is doing 10 m/s

Obviously inorrect so I guess many physical formulae would have to acquire brackets so that the correct answer is achieved

V=U+at becomes V=U+(at)

[factor-fiction]

You can put brackets if you want, and I often use them when they aren't necessary under BODMAS, especially in spreadsheets. But the BODMAS convention means we don't need them. I am not sure why we omit the brackets and use BODMAS- I suppose that it just looks messy with lots of brackets.

But even if we use brackets as you suggest there still has to be a convention that operations in brackets are dealt with first.

[Thatcherite]

it's a general convention decided on at some point. Maths by it's very nature must be consistant. It is the only universal language. Even aliens from another world would agree that 2+2=4, they'd no doubt have different symbols and terminology. As long as we all agreed on the order of operations I can't see it makes any difference mathmatically. The convention is really just to avoid having to code parenthesis all over the place but if you wanted 3+6/3 to be 3 and often you do, then in computing at least, we code (3+6)/3

[Jonathan-Joe]

BODMAS is just a convention. When evaluating an expresion, it makes sense to apply it only if you are confident that the expression was formulated with BODMAS in mind. Some other convention (eg taking terms and symbols in order) could be used instead, provided all dealing with the expression were agreed on it.

[factor-fiction]

Yes, and to expand on J-J's point- they don't use Bodmas on Countdown

[factor-fiction]

Or in those calculations you see on some puzzlepages in newspapers

[beso]

It is a syntactical convention.

What we write is only a representation of a mathematical concept. We could chose an entirely different convention and so long as everyone used it we would understand the same mathematical concept.

In effect bodmas is a default abbreviation of a more explicitly represented expression. We could choose to include all the understood brackets to transcend the bodmas convention entirely without losing meaning.

a+b*c = a+(b*c)

[chakka35]

As I said in the How it Works discussion that jake mentions, if you can get two values to the same expression according to whether you use BODMAS or not, then the expression is sloppily written.

Arithmetic should be absolutely precise, not dependent on some arbitrary convention, otherwise we could all invent conventions allowing us to get any any answer we liked!

[factor-fiction]

I'm not sure I follow that. How would you tell a calculator/spreadsheet to add 2 and 3 and and then multiply the result by the sum of 3 and 4?
if you didn't use BODMAS you'd use brackets. Isn't that just another convention?

[zebo]

Countdown also irritates the life out of me by things like 3 x 9 = 27 + 4 = 31. Thats sloppy, the first equality is untrue, 3 x 9 isn't the same as 27 + 4, if it's supposed to be 31, then it should be written 3 x 9 = 27, 27 + 4 = 31. It may sound picky, but a lot of children watch Countdown and shouldn't be seeing incorrect Mathematics.

[factor-fiction]

I agree Zebo. I doubt Countdown is major cause of the problem but as a maths teacher it's a real battle to get students to set calculations out properly. Most set them out the way your Countdown example is set out and because they understand their workings they don't accept that it needs to be set out properly.

[vascop]

chakka
What you say is just not right. The whole of mathematics is based on the Bodmas convention and brackets are normally omitted unless needed since they just make the expression more difficult to understand.
There are other conventions, such as reverse Polish. Read about it here:
http://en.wikipedia.o...verse_Polish_notation

[vascop]

I think what they do on countdown is OK as long as you redefine what you mean by '=', and call it 'gives', and remember that the first number after 'gives' is the result of what you did on the left.
It's just a new convention that's more convenient when time is at a premium since it avoids repetition. Let's call it the Vorderman-Riley convention - a very attractive convention indeed!

3+4 gives 7 x 2 gives 14 etc
Perhaps even writing it like this:
3+4-->7 x 2-->14

[vascop]

Also when people are calculating out loud they use this same system:
"Three plus four is seven, times 2 is 14, minus 6 is 8, divided by 2 is 4".
Notice how the commas here are essential for understanding.
So punctuation is important after all!

[chakka35]

vascop, BODMAS (which I had never heard of when doing maths at grammar school), when practised properly, merely performs those operations which normal arithmetic demands anyway, without giving them a fancy name.

The example in the How it Works thread with which jake started this discussion gave two possible results, 18 and 20. One could be justified by using BODMAS, the other by taking another equally valid route.
Which meant that the question was badly formulated in the first place.

But those using BODMAS would not have recognised this because they would be working to a formula instead of spotting the sloppiness in the construction.

The expression was ambiguous and therefore invalid, but BODMAS did not reveal this.

[factor-fiction]

I'm not sure whether I disagree with you chakka or I don't understand your point. If you went to Grammar School you will certainly have looked at the order of operations( and I think you have just forgotten that the term BODMAS was used).

[vascop]

You are wrong Chakka, 3+6/3 in mathematics always equals 5. Every mathematician in the world would agree to that. It's all down to the convention.