The BODMAS mathematical rule.

[XFool]

The same faulty logic that leads to an answer of 1 would require 6-3+(2+1) to be 0 rather than 6.

Eh?

It would undoubtedly be faulty logic to conclude 6-3+(2+1) equals 0 rather than 6. But why would anyone conclude that? It would only be 6 if it was written such as: 6-(3+(2+1))

In this case these two examples are impossible to confuse - no faulty logic anywhere! Not so with the OP.

OK, I will try and clarify what I meant in the final sentence of the post you quoted.

The faulty logic in the original expression ("6/2(2+1)") was to give precedence to the (implied) multiplication operator over the division operator. Multiplication and division have equal precedence and cases of equal operator precedence are resolved in mathematical expressions by applying operators in a left-to-right order. The faulty logic applies them incorrectly in a right-to-left order yielding an incorrect result of 1. Applying them in a left-to-right order yields a correct result of 9.

But the original expression as given in the OP was:

6 / 2(2+1) = ?

As written, this is clearly ambiguous. If taken as a straight line only expression: 6 / 2 multiplied by (2+1) the 'answer' obviously is 9. But, if taken as 6 divided by 2(2+1) it would be one. From the spacing given in the OP, this seems an entirely reasonable interpretation. A previous post points out that the expression was originally given online with a ÷ sign and different spacing which may clarify the interpretation as an expression all on one line.

In the expression "6-3+(2+1)" the sub-expression in the brackets is evaluated first to yield 3 (just as in "6/2(2+1)") so the expression is equivalent to "6-3+3". The two operators of addition and subtraction have equal precedence. Applying them in a right-to-left order (the "faulty logic" way) yields a result of zero whereas applying them in the (correct) left-to-right order yields the correct result of 6.

That's all I meant by "same faulty logic that leads to an answer of 1 would require 6-3+(2+1) to be 0 rather than 6." Both incorrect results flow from the incorrect use of right-to-left ordering of operations of equal precedence.

But with both those expressions you give, using only [ + - ( ) numerals ], there can be no ambiguity. They both can only be an expression on one line.

[modellingman]

6 / 2(2+1) = ?

As written, this is clearly ambiguous. If taken as a straight line only expression: 6 / 2 multiplied by (2+1) the 'answer' obviously is 9. But, if taken as 6 divided by 2(2+1) it would be one. From the spacing given in the OP, this seems an entirely reasonable interpretation. A previous post points out that the expression was originally given online with a ÷ sign and different spacing which may clarify the interpretation as an expression all on one line.

Embolding added.

This is clearly the nub of it and I disagree. There's no ambiguity and it is not a reasonable interpretation to separate the "2(2+1)" and evaluate it before the division operation. Further, "/" is simply an alternative to "÷" and the meaning of an expression does not change if one is substituted for the other. The expression is all on one line because its author placed all its characters on the same line.

Spaces have no significance in mathematical expressions. If they did, as is being asserted, then for an expression with spaces to differ from one without there would need to be a convention that spaces before or after a symbol such as "/" somehow insert a pair of implied brackets to the left or right, respectively, of that symbol. Whilst it might be clear where the opening bracket of a right-inserted pair goes what does such a convention say about the position of the closing bracket? Does it go at the end of the expression, to the left of the first subsequent space after the spaced symbol or what? Similarly, for a left inserted pair, where does the convention place the opening bracket? And do double spaces have precedence over single spaces? Etc, etc.

In terms of the OP, how would this "implied brackets" convention resolve an expression such as

6 / 2 × (2+1)

(ie spaces around both the "/" and "×" symbols)? How does the expression change if a space is omitted or doubled up?

Some posters have gone further than this. For at least one, spaces around a "/" mean the expression is split vertically with a horizontal divisor operation line inserted at the split point (so no longer "an expression all on one line"). To such posters there are similar questions. Under such a splitting convention, how much of the expression before and after the split is placed in the upper and lower components respectively? Do other spaces in the expression limit the scope of the split?

I look forward to seeing from posters the many references from mathematical texts that fully define these conventions, which somehow escaped my mathematical education. Until then, I shall maintain (along with my fellow graduates of mathematics) the much simpler convention that spaces have no significance, that "/" is simply another character for "÷" and that an expression where all the characters are physically on the same line is precisely that, ie "an expression all on one line".

[Alaric]

Until then, I shall maintain (along with my fellow graduates of mathematics)

Did your studies include non-commutable algebra? That's where order matters.

If the intent is that the answer is 9, it would be much better to write the expression as a(b+c)/d. If the answer is intended to be 1, write it as (a/d)(1/(b+c)) or a/(d(b+c))

a=6
b=2
c=1
d=2

[modellingman]

Try reversing/inverting the formula.....

(1+2)2 / 6 = ??

Is that 1 or 1/9?

Paul

Nice try but, I'm afraid, no cigar if you are trying to prove that 6 / 2(2+1) = 1.

You are asserting that if a is 6 / 2(2+1) then 1/a is (1+2)2 / 6. And if (your version of) 1/a is 1 (and I agree that (1+2)2 / 6 is 1) then a must be 1 (I also agree, in general, that 1/a = 1 implies a=1) , so 6 / 2(2+1) is therefore 1 (I disagree with that bit because I disagree that (1+2)2 / 6 is the inverse of 6 / 2(2+1) ).

Let's take the idea of inversion one step further. I hope we can agree that the inverse of an inverse is the same as the original, ie in mathematical terms for any non-zero number a

1/(1/a) = a - Equation 1.

So, I'm going to invert your version of 1/a and show that it is not equal to the original version of a. This will then be in contradiction of Equation 1. The only resolution of the contradiction is that the starting assumption, (ie that you version of 1/a is correct) is false. Mathematical fules kno that this is called a proof by contradiction. Its a handy and well used tool of the mathematical armoury and generally covered in 101 courses in maths degrees.

First, lets deal with the pesky spaces, I hope we can also agree that (1+2)2 / 6 (with spaces) is the same as (1+2)2/6 (no spaces). If so, then it should be uncontroversial that your version of 1/a can be written as (1+2)2/6. And, perhaps more controversially, using the same process that lead you from 6 / 2(1+2) to (1+2)2 / 6, say that this process also leads back from (1+2)/2 / 6 to 6 / 2(2+1) and, therefore, the process will lead from (2+1)2/6 to 6/2(2+1). This is just a long way of saying that if we can, uncontroversially, drop the spaces from (1+2)2 /6 then the same should apply through your processes of reversing/inverting to dropping them from 6 / 2(2+1). Doing this means a can be written without spaces as 6/2(2+1).


Inverting your version of 1/a leads to

1/((1+2)2/6)
= 6/(6(1+2)2/6) [multiplying numerator and denominator of the leftmost division operation by 6]
= 6/(1(1+2)2/1) [dividing numerator and denominator of rightmost division operation by 6]
= 6/((1+2)2) [simplifying by removing a multiplication by 1 and a division by 1]
= 6/(3x2)
= 6/6
= 1

So 1/(1/a) is 1

But, according to the conventions underlying BODMAS, evaluating a leads to

6/2(2+1)
= 6/2x3
= 3x3 [because / and x have equal precedence and so because the / is to the left of the x it takes priority under the left-to-right convention]
= 9

So a=9.

So 1/(1/a]) ≠ a.

This contradicts Equation 1 and the only resolution of this is that the assumption 1/a = (1+2)2/6 is false. In other words, your version of 1/a is incorrect. The assumption that (1+2)2/6 is the inverse of 6/2(2+1) is incorrect. No cigar. Sorry.

It is the reversing bit that is the Achille's heel of your argument. Proof left as an exercise!

[modellingman]

Until then, I shall maintain (along with my fellow graduates of mathematics)

Did your studies include non-commutable algebra? That's where order matters.

If the intent is that the answer is 9, it would be much better to write the expression as a(b+c)/d. If the answer is intended to be 1, write it as (a/d)(1/(b+c)) or a/(d(b+c))

a=6
b=2
c=1
d=2

Yes, though my preference was for applied stuff, particularly mathematical statistics. I became an operational research practitioner. Hence, the moniker.

I agree with your comment about expressions. Brackets and ordering can add clarity, so what's not to like about them. That there have demonstrably been different interpretations placed on the original expression indicates its scope for ambiguity, even though the conventions underlying BODMAS should lead to a clear conclusion.

[XFool]

6 / 2(2+1) = ?

As written, this is clearly ambiguous. If taken as a straight line only expression: 6 / 2 multiplied by (2+1) the 'answer' obviously is 9. But, if taken as 6 divided by 2(2+1) it would be one. From the spacing given in the OP, this seems an entirely reasonable interpretation. A previous post points out that the expression was originally given online with a ÷ sign and different spacing which may clarify the interpretation as an expression all on one line.

Embolding added.

This is clearly the nub of it and I disagree. There's no ambiguity and it is not a reasonable interpretation to separate the "2(2+1)" and evaluate it before the division operation. Further, "/" is simply an alternative to "÷" and the meaning of an expression does not change if one is substituted for the other. The expression is all on one line because its author placed all its characters on the same line.

I still disagree! Why?

I am not a mathematician, as a self described mathematician you describe all this purely in terms of a process solely involving a set of formal rules. In other words, as purely mathematical! Fair enough, but is that really adequate here?

Mathematics (as I understand it) is a set of purely formal rules, so the expression (if well formed) should be correctly evaluated by any correct mathematical 'machine'. Interestingly, the only two mathematical machines I have tried this on both give 'wrong' answers.

My Windows Scientific Calculator evaluates on the fly and so gives two separate answers of 3. Firstly, 3 for 6/2 (correct as per your description) and then a completely separate answer of 3 for (2+1). But, without an explicit symbol to multiply the first part of the expression by the second part of the expression it just treats them as two separate sums. My hand calculator takes in the whole expression (no spaces) and evaluates it to 1. With an explicit multiply symbol added it evaluates to 9.

You may well say they are both wrong, in which case you will have to take it up with their programmers. An expression evaluated by a human containing a digit '7' may be correct, but what if it had been given to a mathematical machine, and the '7' had been written in the continental style; perhaps the machine wouldn't be able to evaluate it according to the rules? What if Roman numerals had been used?

What is going on here involves a person interpreting what another person intended by the given arrangement of symbols in the expression in the OP.

You say that / is the same as ÷ (formally it is) and that the OP was "all on one line", which it literally was. BUT, without typesetting tools and using only normal keyboard characters, all such expressions typically are literally "all on one line" even though, in some cases, mathematically they are not. Of course, it that case, the onus is on the poster to form the expression in an unambiguous manner. Which, in the expression in the OP it was easily possible to do in more than one way.

In this case the OP was itself something of a 'translation' from the original source, which muddied the water even more.

The point is not that the answer 9 is wrong and the answer 1 is correct, rather that as it isn't possible (for an average human) to be sure what was meant. So either you have to go with how YOU interpret it, based on experience and judgement, or ask what actually WAS intended, which rather defeats the point of the question...

Spaces have no significance in mathematical expressions.

No. But they have significance in how humans interpret the meaning of text put in front of them.

In terms of the OP, how would this "implied brackets" convention resolve an expression such as

6 / 2 × (2+1)

Unambiguously. Because it is unambiguous.

Some posters have gone further than this. For at least one, spaces around a "/" mean the expression is split vertically with a horizontal divisor operation line inserted at the split point (so no longer "an expression all on one line").

That's right - along with the absence of spaces between the 2 and the (2+1), implicitly tying them together.

To such posters there are similar questions. Under such a splitting convention, how much of the expression before and after the split is placed in the upper and lower components respectively? Do other spaces in the expression limit the scope of the split?

The question should be: "Why such a misleadingly formed expression (bearing in mind the limitation of the character set being used)? It is simplicity itself to format it unambiguously."

I look forward to seeing from posters the many references from mathematical texts that fully define these conventions, which somehow escaped my mathematical education. Until then, I shall maintain (along with my fellow graduates of mathematics) the much simpler convention that spaces have no significance, that "/" is simply another character for "÷" and that an expression where all the characters are physically on the same line is precisely that, ie "an expression all on one line".

Again, you may be formally correct but, for most it was firstly a question of interpreting what the poster intended. (Which could, of course, be something else.)

I don't think we can learn anything very significant from the OP, other than to understand the importance of clarity when writing mathematical expressions using normal keyboard characters.

[XFool]

This has reminded me of a Maths class, following an exam, when I was at school. I and some others had had difficulty with one question. The very patient maths teacher explained at the blackboard. Some now understood, I and a few others didn't. He explained again. The same people failed to understand. He explained yet again, silence. The teacher was now baffled and interested: "It's always the same people..." Then somebody asked a particular question.

Teacher: "You mean you think this is solid? Oh no, it isn't, it's flat. It's causing enough trouble without it being three dimensional!"

Bong!

[ten0rman]

I have been following this thread with some interest, not because of any great insights, just pure interest.

When I was at school, a long, long time ago, similar to someone else, I was taught BODMAS, and always understood that the O stood for "of", as in 50% of 28, ie a synonym for multiplication. I must admit though, that "Order" as for indices does make more sense.

I have just had a word with my 24 year old grandson. he knew both BODMAS and BIDMAS, and said that he learnt BODMAS in about Year 6 or 7, and then BIDMAS in Year 10 or so. He could not remember what "O" stood for, but came out almost immediately with "Indices" for the "I" of BIDMAS. He also tells me that his 11 year old sister knows both.

All of which tells me that BIDMAS is much more correct than BODMAS, and that I shall have to change my thinking.

Strangely, their mother (my daughter), when asked the same question as my grandson was unable to tell me either word! But without being nasty, she did have to have two attempts at 'O' level maths, indeed she had to attend the local Technical College where a different syllabus was taught, so it's perhaps not too surprising that she could not remember either word.

Regards,

ten0rman

[XFool]

I have been following this thread with some interest, not because of any great insights, just pure interest.

When I was at school, a long, long time ago, similar to someone else, I was taught BODMAS, and always understood that the O stood for "of", as in 50% of 28, ie a synonym for multiplication. I must admit though, that "Order" as for indices does make more sense.

I have just had a word with my 24 year old grandson. he knew both BODMAS and BIDMAS, and said that he learnt BODMAS in about Year 6 or 7, and then BIDMAS in Year 10 or so. He could not remember what "O" stood for, but came out almost immediately with "Indices" for the "I" of BIDMAS.

All of which tells me that BIDMAS is much more correct than BODMAS, and that I shall have to change my thinking.

I believe O in BODMAS stands for 'orders' (i.e. indices), possibly a somewhat dated term in this context hence the change to BIDMAS. Though the 'order' of an equation is still used in 'proper' maths.

Strangely, their mother (my daughter), when asked the same question as my grandson was unable to tell me either word! But without being nasty, she did have to have two attempts at 'O' level maths, indeed she had to attend the local Technical College where a different syllabus was taught, so it's perhaps not too surprising that she could not remember either word.

Neither can I, from school.

[stevensfo]

This has reminded me of a Maths class, following an exam, when I was at school. I and some others had had difficulty with one question. The very patient maths teacher explained at the blackboard. Some now understood, I and a few others didn't. He explained again. The same people failed to understand. He explained yet again, silence. The teacher was now baffled and interested: "It's always the same people..." Then somebody asked a particular question.

Teacher: "You mean you think this is solid? Oh no, it isn't, it's flat. It's causing enough trouble without it being three dimensional!"

Bong!

It's strange how we assume we must be thick when in fact, the idea just hasn't been taught in the right way. Our Maths teacher at school was dreadful and just wrote very fast equations on the board without going into depth about what they meant. I was having hydrotherapy sessions following an accident and mentioned to the physio about differentiation being impossible to understand. She laughed and proceeded to explain it in a totally different way that made perfect sense. It was the lightbulb moment!

Steve

[XFool]

It's strange how we assume we must be thick when in fact, the idea just hasn't been taught in the right way. Our Maths teacher at school was dreadful and just wrote very fast equations on the board without going into depth about what they meant. I was having hydrotherapy sessions following an accident and mentioned to the physio about differentiation being impossible to understand. She laughed and proceeded to explain it in a totally different way that made perfect sense. It was the lightbulb moment!

Oh, don't get me started on this! It's a pet subject of mine.

Take 'the'average' (really: the arithmetic mean), I remember being taught this in primary school. It was simple and easy as long as you got the 'sums' right. And yet... then came secondary school... and it came to be far, far more bothersome to me.

In fact, I only really sorted it out in my own mind in my fifties. And that's just the arithmetic mean! (I contend it's trickier than it looks, or is taught.)

[AndyPandy]

This equation challenge is always circling the Interweb somewhere. It's clearly been set up as a deliberately ambiguous equation to polarise opinion and create discussion.

Within current Mathematical rules it cannot be interpreted in just one way and therefore, as previous posters have stated or alluded to, the correct answer is "rewrite it to remove the ambiguity".


Time for a quick scone before my bath. Can anyone just remind me of the pronunciation of those last two nouns and jog my memory - cream or jam first? Cottonwool for brains here - sheesh.

[UncleEbenezer]

Time for a quick scone before my bath. Can anyone just remind me of the pronunciation of those last two nouns and jog my memory - cream or jam first? Cottonwool for brains here - sheesh.

Crucial difference: those issues really don't matter. Nothing bad happens if your choice differs from mine.

Pass the port, please.

[XFool]

...possibly. OTOH, they do make you think and can sometimes be illuminating.

P.S. It's 'scones' and 'bath'. Obviously!

[melonfool]

Anyone who says anything else is I am sorry to say talking nonsense. BODMAS =Brackets, Orders, Division, Multiplication, Addition and Subtraction. I have not strictly followed it because I never have but you could have done 6/2*(2+1) =6/2*3=3/3 =1 in strict accord with the BODMAS rules.

You do not need to be a mathematician to know that, just a reasonably competent school boy or girl.

Dod

I started reading this thread out of interest as I have never really understood how this stuff works and actually I am going to do my maths GCSE next year.

But comments like this 'talking nonsense' (where in fact there are several pages so clearly more than one view, plus the mod box on the first post suggests that same) simply put me off reading any further.

Mel

[modellingman]

Anyone who says anything else is I am sorry to say talking nonsense. BODMAS =Brackets, Orders, Division, Multiplication, Addition and Subtraction. I have not strictly followed it because I never have but you could have done 6/2*(2+1) =6/2*3=3/3 =1 in strict accord with the BODMAS rules.

You do not need to be a mathematician to know that, just a reasonably competent school boy or girl.

Dod

I started reading this thread out of interest as I have never really understood how this stuff works and actually I am going to do my maths GCSE next year.

But comments like this 'talking nonsense' (where in fact there are several pages so clearly more than one view, plus the mod box on the first post suggests that same) simply put me off reading any further.

Mel

Unfortunately, Mel, the snippet you quoted contains a bit of a "howler" in Dod101's equation, so the snippet is really only useful for supporting learning as an exercise spotting the mistake.

I agree with the remaining sentiments you express. Hope the studying goes well.


Spoiler alert (following the conventions adopted in viewforum.php?f=73 , select text to view.)


Its the 6/2*3 = 3/3 bit that is wrong.

6/2*3 = 3*3 if you follow BODMAS.

6/2*3 = 6/6 if you do not follow BODMAS rules, but instead do the multiplication before the division.

So a result of 9 under BODMAS and 1 if multiplication given priority over division.

Under BODMAS division and multiplication have equal precedence. And when two operators (a symbol such as *, /, + or -) have equal precedence in an expression the one that is used first is the one that is to the left of the other. So the division is undertaken before the multiplication in 6/2*3, but multiplication before division in, say, 6*2/3.

[Dod101]

Anyone who says anything else is I am sorry to say talking nonsense. BODMAS =Brackets, Orders, Division, Multiplication, Addition and Subtraction. I have not strictly followed it because I never have but you could have done 6/2*(2+1) =6/2*3=3/3 =1 in strict accord with the BODMAS rules.

You do not need to be a mathematician to know that, just a reasonably competent school boy or girl.

Dod

I started reading this thread out of interest as I have never really understood how this stuff works and actually I am going to do my maths GCSE next year.

But comments like this 'talking nonsense' (where in fact there are several pages so clearly more than one view, plus the mod box on the first post suggests that same) simply put me off reading any further.

Mel

Unfortunately, Mel, the snippet you quoted contains a bit of a "howler" in Dod101's equation, so the snippet is really only useful for supporting learning as an exercise spotting the mistake.

I agree with the remaining sentiments you express. Hope the studying goes well.


Spoiler alert (following the conventions adopted in viewforum.php?f=73 , select text to view.)


Its the 6/2*3 = 3/3 bit that is wrong.

I do not see much wrong with my answer or logic. It is the / sign as I said later that is ambiguous and I will concede that either of the answers would be correct depending on how the / is interpreted.

Dod

6/2*3 = 3*3 if you follow BODMAS.

6/2*3 = 6/6 if you do not follow BODMAS rules, but instead do the multiplication before the division.

So a result of 9 under BODMAS and 1 if multiplication given priority over division.

Under BODMAS division and multiplication have equal precedence. And when two operators (a symbol such as *, /, + or -) have equal precedence in an expression the one that is used first is the one that is to the left of the other. So the division is undertaken before the multiplication in 6/2*3, but multiplication before division in, say, 6*2/3.

[chas49]

I understand the reasons for using coloured text to hide spoilers in the Games and Puzzles board. I'm not convinced they help here. And Dod's last response is unreadable (to me at least) as a result of that.

I don't moderate on this board so can I just ask as a normal Fool, please be careful about using tags like this and don't do it when it's not really necessary.

[XFool]

And Dod's last response is unreadable (to me at least) as a result of that.

Highlight it with the mouse.

[PinkDalek]

And Dod's last response is unreadable (to me at least) as a result of that.

Highlight it with the mouse.

Try that with both posts, at the same time, on a mobile.