A Simple Calculation

If the red house has blue shutters and the green house has red shutters, what's this section for?

6 / 2 (2+1) = x

9
2
67%
1
1
33%
Both of the above are correct
0
No votes
None of the above are correct
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No votes
 
Total votes: 3

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A Simple Calculation

Postby djembeweaver » Wed Nov 11, 2015 3:30 pm

6 / 2 (2+1) = x

Find the value of x...what could be simpler?

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Re: A Simple Calculation

Postby Poodle » Wed Nov 11, 2015 4:00 pm

Well - the typographical confusion over that denominator may cause a problem or two.

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Re: A Simple Calculation

Postby djembeweaver » Wed Nov 11, 2015 4:16 pm

Poodle wrote:Well - the typographical confusion over that denominator may cause a problem or two.


Eh? Is the space causing confusion?

6/2(2+1)=x

Does that help?

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Re: A Simple Calculation

Postby djembeweaver » Wed Nov 11, 2015 4:18 pm

Obviously equations don't require the multiplication sign before brackets...

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Re: A Simple Calculation

Postby Monster » Wed Nov 11, 2015 4:42 pm

I voted for nueve.
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Re: A Simple Calculation

Postby xouper » Wed Nov 11, 2015 5:10 pm

djembeweaver wrote:6 / 2 (2+1) = x

Find the value of x...what could be simpler?

Spoiler:
The answer depends on which operator precedence one chooses to use.

The most common interpretation converts the implied multiplication to an explicit multiplication:

6 / 2 * (2+1) = x

and then performs the multiplications and divisions left to right.

(6 / 2) * (2+1) = x

(3) * (3) = x = 9

This is what most computer programming languages and pocket calculators do, including the google calculator.

https://www.google.com/search?q=6+/+2+%282%2B1%29

This is what most mathematicians and programmers expect to happen.

My preference as a professional programmer is not to rely on the compiler's precedence rules. It is safer to add explicit parentheses to force the precedence I want. Then there can be no ambiguity and it is clear to anyone reading the code what I had in mind.

However, some academic journals treat implied multiplication as having higher precedence than division, as follows:

6 / (2 (2+1)) = x = 1

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Re: A Simple Calculation

Postby bobbo_the_Pragmatist » Wed Nov 11, 2015 6:38 pm

xoup: why do professional programmers violate simple universal rules we all got in the fifth grade?
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Re: A Simple Calculation

Postby xouper » Wed Nov 11, 2015 7:10 pm

bobbo_the_Pragmatist wrote:xoup: why do professional programmers violate simple universal rules we all got in the fifth grade?

I'll do my best to answer your question, but before I can do that, I need to know what specific rules you are referring to.

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Re: A Simple Calculation

Postby Poodle » Wed Nov 11, 2015 7:16 pm

Told you so.

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Re: A Simple Calculation

Postby bobbo_the_Pragmatist » Wed Nov 11, 2015 7:19 pm

xouper wrote:
bobbo_the_Pragmatist wrote:xoup: why do professional programmers violate simple universal rules we all got in the fifth grade?

I'll do my best to answer your question, but before I can do that, I need to know what specific rules you are referring to.

Ummmm....the very subject of this thread? "My preference as a professional programmer is not to rely on the compiler's precedence rules."===and I asked, why not? More an ironic comment than a real question. Yes.....how can we LOCK DOWN what is meant...even with maths.

I still enjoy that space probe that crashed into the planet instead of going into orbit because one team used the metric instead of the (English/American--?) system of notation. Given this one big example of confusion, you'd think some LOCK DOWN expression of what 100 was referring to would be right in the equation???

Confusion always arises like this. A universal definition is being used and some little group wants to do something different......and things go downhill from there. A poor use of expertise?
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Re: A Simple Calculation

Postby xouper » Wed Nov 11, 2015 7:41 pm

bobbo_the_Pragmatist wrote:
xouper wrote:
bobbo_the_Pragmatist wrote:xoup: why do professional programmers violate simple universal rules we all got in the fifth grade?

I'll do my best to answer your question, but before I can do that, I need to know what specific rules you are referring to.

Ummmm....the very subject of this thread? "My preference as a professional programmer is not to rely on the compiler's precedence rules."===and I asked, why not?

That is not the same question you asked previously. Are you now asking me why I do not rely on compiler precedence rules? Please clarify?

bobbo_the_Pragmatist wrote:I still enjoy that space probe that crashed into the planet instead of going into orbit because one team used the metric instead of the (English/American--?) system of notation. Given this one big example of confusion, you'd think some LOCK DOWN expression of what 100 was referring to would be right in the equation???

I assume you are referring to NASA's Mars Climate Orbiter ?

You might also enjoy the example of the Gimli Glider. The pilot of the Boeing 767 asked the refueling operator for X kilograms of jet fuel and the operator gave him X pounds instead. Not surprisingly, they ran out of fuel half way to their destination.

bobbo_the_Pragmatist wrote:Confusion always arises like this. A universal definition is being used and some little group wants to do something different......and things go downhill from there. A poor use of expertise?

In general, writing reliable (non trivial) software is hard. As you sometimes say, silly hoomans.

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Re: A Simple Calculation

Postby bobbo_the_Pragmatist » Wed Nov 11, 2015 7:51 pm

Gimli Glider was interesting. Pilot Error. Those planes have fuel gauges. GROSS ERROR to take off with half the fuel you ordered....and not to monitor while inroute.
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Re: A Simple Calculation

Postby xouper » Wed Nov 11, 2015 7:57 pm

bobbo_the_Pragmatist wrote:Gimli Glider was interesting. Pilot Error. Those planes have fuel gauges. GROSS ERROR to take off with half the fuel you ordered....and not to monitor while inroute.

Wrong. Apparently you did not read the whole thing.

I also have the official report from the Canadian government agency that investigated. The pilot was not at fault.

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Re: A Simple Calculation

Postby bobbo_the_Pragmatist » Wed Nov 11, 2015 8:09 pm

I didn't, but now have. The pilot was at fault and got demoted for his failures and assumptions. The MEL is there for a reason. You don't violate written procedure for some "custom" during a transition.

Not that I might have fallen for the same trap....but I never did violate the MEL even with a One Star spitting on me. I told him to get written authorization from 21st airforce and then I "might" consider taking off without radar. I think he knew I was Bs's him.....just for the fun of it.

Ha, ha. Every qualified pilot knows what the best glide range air speed for his aircraft is. Being a glider pilot hardly matters at all.

I almost hurt myself laughing at the popped circuit breaker maintenance fix. That has crashed more airplanes than anything else I can think of. A sequence of unlikely events........ yea, verily!!
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Re: A Simple Calculation

Postby xouper » Wed Nov 11, 2015 8:23 pm

bobbo_the_Pragmatist wrote:I didn't, but now have. The pilot was at fault and got demoted for his failures and assumptions. The MEL is there for a reason. You don't violate written procedure for some "custom" during a transition.

The pilots did not violate the MEL. And they were eventually exonerated. In fact, part of the problem was with the MEL.

bobbo_the_Pragmatist wrote:Ha, ha. Every qualified pilot knows what the best glide range air speed for his aircraft is. Being a glider pilot hardly matters at all.

Wrong. It takes more than merely knowing the best glide IAS to successfully land a glider.

wikipedia wrote:In 1985 the pilots were awarded the first ever Fédération Aéronautique Internationale Diploma for Outstanding Airmanship.[12] Several attempts by other crews who were given the same circumstances in a simulator at Vancouver resulted in crashes.[13]

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Re: A Simple Calculation

Postby bobbo_the_Pragmatist » Wed Nov 11, 2015 8:32 pm

xoup: I apologize for reading too fast even on the second attempt. They were exonerated AND the MEL was wrong? I didn't catch either of those........but.....whatever.

But please do tell me as I have flown heavy A/C as well as gliders: 'Wrong. It takes more than merely knowing the best glide IAS to successfully land a glider.//// Like what? ..... and this crew weren't flying a glider to begin (and end) with.

Edit: granted, knowing the best glide IAS doesn't tell you anything about how to land anything at all. best glide IAS is for most ground covered for altitude loss. Generally, you come in high and increase drag to land. Non-gliders have air and tire brakes that usually work even with one or more "other" complete systems failures. Two different issues, but on the issue stated: no difference as best glide IAS is well understood by commerical non-glider pilots.

I would not absolved the Pilot. THE PILOT IS ALWAYS RESPONSIBLE for aircraft safety. Taking other peoples word for fuel levels is just too basic. How was the MEL "wrong" as I did read just the opposite, that it was simply not followed or adhered to.
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Re: A Simple Calculation

Postby xouper » Wed Nov 11, 2015 8:55 pm

bobbo_the_Pragmatist wrote:xoup: I apologize for reading too fast even on the second attempt. They were exonerated AND the MEL was wrong? I didn't catch either of those........but.....whatever.

Perhaps those details are in the CTSB* report and not the wiki page. I didn't read the whole wiki article either. I am going from memory having read the official report. If you're interested, perhaps I can quote directly from that report.

bobbo_the_Pragmatist wrote:I would not absolved the Pilot. THE PILOT IS ALWAYS RESPONSIBLE for aircraft safety.

In general, I agree. My CFIs constantly pounded that into my brain, especially regarding never to run out of fuel.

However, in this particular set of highly complicated circumstances, the CTSB did not hold the pilots responsible and I agree with their opinion.


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* Footnote: The CTSB was called something else back in 1983, but I'd have to look it up.

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Re: A Simple Calculation

Postby TJrandom » Wed Nov 11, 2015 11:25 pm

Parentheses first, then left to right... So nine.

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Re: A Simple Calculation

Postby Poodle » Thu Nov 12, 2015 12:11 am

TJrandom wrote:Parentheses first, then left to right... So nine.


Ah - but what if it should read ...

6
______
2 (2+1) = x

ie 6 divided by 2(2+1)

in which case the answer would be 1.

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Re: A Simple Calculation

Postby Gord » Thu Nov 12, 2015 2:38 am

Poodle wrote:
TJrandom wrote:Parentheses first, then left to right... So nine.


Ah - but what if it should read ...

6
______
2 (2+1) = x

ie 6 divided by 2(2+1)

in which case the answer would be 1.

No no, it would be 6 divided by 2, then multiplied by (2+1). 6 divided by 2 = 3, then multiplied by 3 = 9.

You're thinking everything to the right of the "/" must be in the denominator, but the "/" is only in reference to the next single term in the equation, the 2.

Converting the equation into an English sentence, you might get: "Six, divided by two, multiplied by the sum of two plus one."
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Re: A Simple Calculation

Postby xouper » Thu Nov 12, 2015 3:51 am

Gord wrote:
Poodle wrote:
TJrandom wrote:Parentheses first, then left to right... So nine.


Ah - but what if it should read ...

6
______
2 (2+1) = x

ie 6 divided by 2(2+1)

in which case the answer would be 1.

No no, it would be 6 divided by 2, then multiplied by (2+1). 6 divided by 2 = 3, then multiplied by 3 = 9.

You're thinking everything to the right of the "/" must be in the denominator, but the "/" is only in reference to the next single term in the equation, the 2.

Converting the equation into an English sentence, you might get: "Six, divided by two, multiplied by the sum of two plus one."

I already explained all this (and more) in my first post. Did no one click the spoiler button?

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Re: A Simple Calculation

Postby TJrandom » Thu Nov 12, 2015 3:54 am

xouper wrote:
Gord wrote:
Poodle wrote:
TJrandom wrote:Parentheses first, then left to right... So nine.


Ah - but what if it should read ...

6
______
2 (2+1) = x

ie 6 divided by 2(2+1)

in which case the answer would be 1.

No no, it would be 6 divided by 2, then multiplied by (2+1). 6 divided by 2 = 3, then multiplied by 3 = 9.

You're thinking everything to the right of the "/" must be in the denominator, but the "/" is only in reference to the next single term in the equation, the 2.

Converting the equation into an English sentence, you might get: "Six, divided by two, multiplied by the sum of two plus one."

I already explained all this (and more) in my first post. Did no one click the spoiler button?


Your spoiler just said `show`... nothing about `click`....

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Re: A Simple Calculation

Postby Matthew Ellard » Thu Nov 12, 2015 4:43 am

Off topic
An economics tutorial on identifying and setting out abstract client scenarios, once had a silly nonsense wordy problem, that accidentally required us to deal with "11cents x $1".

11cents X 100cents = "1,100 cents squared", which makes no sense at all, but we all wrote it down, anyway.
:D

Yep. We were getting ready for the real world!

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Re: A Simple Calculation

Postby supervitor » Thu Nov 12, 2015 4:53 am

The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.

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Re: A Simple Calculation

Postby TJrandom » Thu Nov 12, 2015 7:55 am

supervitor wrote:The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.


I thought that the sequence was pre-determined by generally accepted rules.

http://www.mathgoodies.com/lessons/vol7 ... tions.html

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Re: A Simple Calculation

Postby supervitor » Thu Nov 12, 2015 8:21 am

TJrandom wrote:
supervitor wrote:The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.


I thought that the sequence was pre-determined by generally accepted rules.

http://www.mathgoodies.com/lessons/vol7 ... tions.html


No, it depends on the notation used, and the one it was used is ambiguous

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Re: A Simple Calculation

Postby TJrandom » Thu Nov 12, 2015 8:26 am

supervitor wrote:
TJrandom wrote:
supervitor wrote:The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.


I thought that the sequence was pre-determined by generally accepted rules.

http://www.mathgoodies.com/lessons/vol7 ... tions.html


No, it depends on the notation used, and the one it was used is ambiguous


Maybe so (ambiguous), but using the rules in that link provide the answer that I believe is correct. However - had I written it, I would have used clarifying parentheses...

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Re: A Simple Calculation

Postby supervitor » Thu Nov 12, 2015 8:50 am

TJrandom wrote:
supervitor wrote:
TJrandom wrote:
supervitor wrote:The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.


I thought that the sequence was pre-determined by generally accepted rules.

http://www.mathgoodies.com/lessons/vol7 ... tions.html


No, it depends on the notation used, and the one it was used is ambiguous


Maybe so (ambiguous), but using the rules in that link provide the answer that I believe is correct. However - had I written it, I would have used clarifying parentheses.

That link provides rules, which is what we need to remove the ambiguity. But you would care to notice the notation is not the same: they use ÷ instead of / and the multiplication is not implied.

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Re: A Simple Calculation

Postby xouper » Thu Nov 12, 2015 9:12 am

supervitor wrote:
TJrandom wrote:
supervitor wrote:
TJrandom wrote:
supervitor wrote:The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.

I thought that the sequence was pre-determined by generally accepted rules.

http://www.mathgoodies.com/lessons/vol7 ... tions.html

No, it depends on the notation used, and the one it was used is ambiguous

Maybe so (ambiguous), but using the rules in that link provide the answer that I believe is correct. However - had I written it, I would have used clarifying parentheses.

That link provides rules, which is what we need to remove the ambiguity. But you would care to notice the notation is not the same: they use ÷ instead of / and the multiplication is not implied.

I already addressed all these points in my first post. The question as written is not ambiguous to google calculator.

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Re: A Simple Calculation

Postby supervitor » Thu Nov 12, 2015 9:21 am

xouper wrote:
supervitor wrote:
TJrandom wrote:
supervitor wrote:
TJrandom wrote:
supervitor wrote:The question is ambiguous, because with the denotation you're using, you're not specifying which operation has precedence, the division or the multiplication. You need to use parentheses to make that clear or at least specify one of the operations with a higher precedence. Or if the same (as expected), if it associates left to right or right to left. Usually it would be same precedence, left to right, therefore 9. But one can't answer without those rules being defined.

I thought that the sequence was pre-determined by generally accepted rules.

http://www.mathgoodies.com/lessons/vol7 ... tions.html

No, it depends on the notation used, and the one it was used is ambiguous

Maybe so (ambiguous), but using the rules in that link provide the answer that I believe is correct. However - had I written it, I would have used clarifying parentheses.

That link provides rules, which is what we need to remove the ambiguity. But you would care to notice the notation is not the same: they use ÷ instead of / and the multiplication is not implied.

I already addressed all these points in my first post. The question as written is not ambiguous to google calculator.

Because google calculator uses google calculator's rules to remove the ambiguity. The question is ambiguous as is, and the assumption is that it was written like that on purpose.

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Re: A Simple Calculation

Postby supervitor » Thu Nov 12, 2015 9:25 am

supervitor wrote:the assumption is that it was written like that on purpose.


Hmmmm...
Apparently, dj was following this discussion (about gc's rules itself) and wanted our input:

https://productforums.google.com/forum/ ... ZkTv_WTSxA

Hmmmmm

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Re: A Simple Calculation

Postby Poodle » Thu Nov 12, 2015 9:59 am

I begin to understand why I'm a wordsmith rather than a mathematician.

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Re: A Simple Calculation

Postby xouper » Thu Nov 12, 2015 10:01 am

supervitor wrote:The question is ambiguous as is, and the assumption is that it was written like that on purpose.

I agree that it seems to be written that way for a purpose. But you passed the test, so all is good.

supervitor wrote:Because google calculator uses google calculator's rules to remove the ambiguity.

Google calculator uses the same rules as most mathematicians and most computer programming languages.

The reason for those rules is to have a consensus agreement on how to resolve those kinds of "ambiguities", thus rendering them unambiguous.

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Re: A Simple Calculation

Postby djembeweaver » Thu Nov 12, 2015 10:47 am

It's a problem that has been kicking about for years concerning how to apply the order of operations (or, more correctly, the ambiguity in how to apply them).

You can't talk about numerators and denominators since '/' doesn't imply a fraction but is a symbol for 'divide'. The order of operations that I know (BODMAS) puts division before multiplication but assumes they are the same 'level' and so one must work left to right in the calculation. Other acronyms put them the other way round. Since the problem arises out of an ambiguity in the terms, I would say both answers are right...

https://www.youtube.com/watch?v=cBWqYPwMivU

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Re: A Simple Calculation

Postby xouper » Thu Nov 12, 2015 12:13 pm


That's an excellent explanation. I have only one minor quibble with what he said.

djembeweaver wrote:You can't talk about numerators and denominators since '/' doesn't imply a fraction but is a symbol for 'divide'.

I have a quibble with that. A fraction is indeed semantically equivalent to division. They are merely two different notations for the same operation. In the video he shows how to translate between the two notations (inline division versus fraction bar). The issue he illustrates with the fractions is the ambiguity where to put the X.

Here's a simpler example of the two notations:

1/3

means the exact same thing as

1

3

The first is inline notation and the second notation uses a fraction bar. They represent the same exact mathematical operation and it is indeed valid to refer to the inline notation as a fraction. When I say 1/3, it is not unreasonable to say the denominator is 3, even though I used the inline notation. A denominator is just another name for a divisor.

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Re: A Simple Calculation

Postby Poodle » Thu Nov 12, 2015 1:49 pm

Whoa!!

So there is a quibble?? about notation? In which case my original point about interpretation stands. Make up your minds, guys.

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Re: A Simple Calculation

Postby xouper » Thu Nov 12, 2015 2:14 pm

Poodle wrote:So there is a quibble?? about notation? In which case my original point about interpretation stands.

In my first post, I addressed your original point. Was my explanation not satisfactory? Is there something in the video posted by djembeweave that needs clarification?

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Re: A Simple Calculation

Postby Poodle » Thu Nov 12, 2015 2:30 pm

Point taken, xouper. But I think my point about typographic interpretation still stands. I do not know what convention was used, if any, in the expression 6 / 2 (2+1) = x.

There is a disagreement between mathematical and typographical interpretation which I think is valid. I cannot know - and nor can you - if a strict interpretation has been followed. There is, surely, a better way to express it than the one given - the whole point of the post, I should imagine.

As a professional editor, I would have asked the originator to express himself more clearly. But then I'm a pedant - that's what editors do.

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Re: A Simple Calculation

Postby djembeweaver » Thu Nov 12, 2015 4:15 pm

xouper wrote:

That's an excellent explanation. I have only one minor quibble with what he said.

djembeweaver wrote:You can't talk about numerators and denominators since '/' doesn't imply a fraction but is a symbol for 'divide'.

I have a quibble with that. A fraction is indeed semantically equivalent to division. They are merely two different notations for the same operation. In the video he shows how to translate between the two notations (inline division versus fraction bar). The issue he illustrates with the fractions is the ambiguity where to put the X.

Here's a simpler example of the two notations:

1/3

means the exact same thing as

1

3

The first is inline notation and the second notation uses a fraction bar. They represent the same exact mathematical operation and it is indeed valid to refer to the inline notation as a fraction. When I say 1/3, it is not unreasonable to say the denominator is 3, even though I used the inline notation. A denominator is just another name for a divisor.


Hmmm...really? I'm not convinced. Just because they are functionally the same, that doesn't mean they are identical (does it?).

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Re: A Simple Calculation

Postby xouper » Thu Nov 12, 2015 5:04 pm

djembeweaver wrote:
xouper wrote:Here's a simpler example of the two notations:

1/3

means the exact same thing as

1

3

The first is inline notation and the second notation uses a fraction bar. They represent the same exact mathematical operation and it is indeed valid to refer to the inline notation as a fraction. When I say 1/3, it is not unreasonable to say the denominator is 3, even though I used the inline notation. A denominator is just another name for a divisor.

Hmmm...really? I'm not convinced. Just because they are functionally the same, that doesn't mean they are identical (does it?).

That's an interesting question. In mathspeak, that seems to be saying that in determining identity, an equivalent functionality is necessary but not sufficient. You may be right, which could be demonstrated with a counter example, if you have one.

However, I claim they are identical because they are merely two names for the same thing, not just because they function the same.

http://www.mathgoodies.com/glossary/term.asp?term=fraction%20bar
http://mathforum.org/library/drmath/view/58321.html
https://en.wikipedia.org/wiki/Algebraic_fraction
https://en.wikipedia.org/wiki/Fraction_%28mathematics%29

wikipedia wrote:Other uses for fractions are to represent ratios and to represent division.[1] Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).


As you saw in that video he shows three inline names (symbols) for division, the slash, the obelus, and the colon. Not only do they function the same, they are  the same because they are merely different names for the same function. In addition to those three "inline" names (notations), the fraction bar (sometimes also called a vinculum) is yet another name for division, although it is not an "inline" notation.

Example:

(x2 - 1)
———
(x -1)

means to divide (x2 - 1) by (x -1).

How is that not exactly identical to (x2 - 1) / (x -1)?

Another example of multiple names for the same thing:

x ^ (1/2)

x ** (1/2)

x1/2

√x

sqrt(x)

They are all different names for the same thing, the "square root of x". Not only do they function the same , they are  the same. They are indeed identical. I argue similarly that the fraction bar is just another name for division.

Does this answer your question, or have I misunderstood your point?


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