48/2(9+3) = ?
14 years ago
General
There's been a whole bunch of forum posts over the question of does 48/2(9+3) = 2 or 288
Some people have argued that multiplication and division have the same order of precedence resulting in the equation being resolved as...
48/2(9+3)
=48/2(12)
=24(12)
=288
However others, me included, maintain that multiplication by juxtaposition has a higher order of precedence, giving the result as
48/2(9+3)
=48/2(12)
=48/24
=2
As far as I've been able to tell there is no set convention on whether multiplication by juxtaposition has a higher priority or not, making both answers equally valid. However it does appear that giving multiplication by juxtaposition a higher priority is the most accepted practice.
This is what is stated on the Purplemath site. "The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations." -http://www.purplemath.com/modules/orderops2.htm
This was also stated on the American Mathematical Society guidelines in 2001 where they carried the instructions, "We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division." -http://replay.waybackmachine.org/20011201061315/http://www.ams.org/authors/guide-reviewers.html
This only applies to multiplication by juxtaposition however. Although at times multiplication has been considered to have a higher priority than division they are now considered equal and done in order from left to right. -http://jeff560.tripod.com/operation.html
The question has become famous now because of the different methods that various calculators and computers use when calculating their answers. This problem was recognised before it's current fame and in 2001 Mark Farris wrote a paper on 'Coping with Multiple Calculator Models in College Algebra' In it he makes two points that are particularly important in the current situation.
* "avoid using this construction of a division followed by an implied multiplication."
*"emphasis appropriate use of parentheses."
-http://archives.math.utk.edu/ICTCM/VOL13/C026/paper.pdf
Use of brackets can actually remove any need for an order of operations, although then losing simplicity. So what is being promoted now is the standard order of operations but it's your own responsibility to use brackets to indicate the correct grouping to make an unambiguous equation. 48/2(9+3) is ambiguous because depending on your treatment of multiplication by juxtaposition it can be read as either (48/2)(9+3) or 48(2/(9+3)). Of course this is only a problem when the entire equation is written out linearly. The dropping of the AMA guideline on linearising equations probably stems from the ease with which equations can now be written in the correct format, eliminating such misunderstandings.
Some people have argued that multiplication and division have the same order of precedence resulting in the equation being resolved as...
48/2(9+3)
=48/2(12)
=24(12)
=288
However others, me included, maintain that multiplication by juxtaposition has a higher order of precedence, giving the result as
48/2(9+3)
=48/2(12)
=48/24
=2
As far as I've been able to tell there is no set convention on whether multiplication by juxtaposition has a higher priority or not, making both answers equally valid. However it does appear that giving multiplication by juxtaposition a higher priority is the most accepted practice.
This is what is stated on the Purplemath site. "The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations." -http://www.purplemath.com/modules/orderops2.htm
This was also stated on the American Mathematical Society guidelines in 2001 where they carried the instructions, "We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division." -http://replay.waybackmachine.org/20011201061315/http://www.ams.org/authors/guide-reviewers.html
This only applies to multiplication by juxtaposition however. Although at times multiplication has been considered to have a higher priority than division they are now considered equal and done in order from left to right. -http://jeff560.tripod.com/operation.html
The question has become famous now because of the different methods that various calculators and computers use when calculating their answers. This problem was recognised before it's current fame and in 2001 Mark Farris wrote a paper on 'Coping with Multiple Calculator Models in College Algebra' In it he makes two points that are particularly important in the current situation.
* "avoid using this construction of a division followed by an implied multiplication."
*"emphasis appropriate use of parentheses."
-http://archives.math.utk.edu/ICTCM/VOL13/C026/paper.pdf
Use of brackets can actually remove any need for an order of operations, although then losing simplicity. So what is being promoted now is the standard order of operations but it's your own responsibility to use brackets to indicate the correct grouping to make an unambiguous equation. 48/2(9+3) is ambiguous because depending on your treatment of multiplication by juxtaposition it can be read as either (48/2)(9+3) or 48(2/(9+3)). Of course this is only a problem when the entire equation is written out linearly. The dropping of the AMA guideline on linearising equations probably stems from the ease with which equations can now be written in the correct format, eliminating such misunderstandings.
FA+

And I got through metric maths that way.
Do no... But that was what I was thought at high school.
then again, most people are stupid, so maybe it's not just me... no offense guys