48÷2(9+3) = ????

[Bill Verburg]

Originally Posted by winders

Bill, Bill, Bill......

Mathematical convention - AoPSWiki

Math 1010 on-line

A "convention" is not mathematics. It's not a basic property of mathematics. Solving from left to right is a convention. Tomorrow, the notation could be changed to right to left and the mathematics would still work. We use infix notation today. That is a convention. Sometime in the future, it could be decided that we will be using postfix (Reverse Polish) notation. The mathematics would still be the same.

If the mathematics community decided that implicit multiplication (juxtaposition) has precedence over division or explicit multiplication, only the notation would change. The mathematics would be unaffected.

Don't go confusing conventions with properties of mathematics or the foundations of mathematics.

Scott

When someone says 'In case of doubt, use parentheses!', they are talking to the author not the solver, the solver is restricted in his actions by the fundamental properties of the real # system under the operations defined for that system.

Those fundamental properties are well defined and not subject to ad hoc changes that someone dreams up as a convenience sure there are other notation system and other sets that they act upon. But here and now the discussion is not related to those other systems. You don't understand the basics of this one so I doubt that you will ever grasp the nuances of the others either.

You seem ready to believe any one that has posted on the subject, no matter how little fundamental basis they have for their assertions.

As I said I give up on trying to educate you, this is a religion w/ you and not a science based on fundamentals that can be written clearly and form a consistent unambiguous method of evaluating any expression w/o ad hoc rules and exceptions.

[RWebb]

the conventions exist b/c they work for real world problems

if you like other systems, winders, you might start here:

http://www.karlin.mff.cuni.cz/~stanovsk/math/nonassoc.pdf

[winders]

Originally Posted by Bill Verburg

Those fundamental properties are well defined and not subject to ad hoc changes that someone dreams up as a convenience...

Bill,

Precedence is not a fundamental property of mathematics. It is a convention.

Scott

[DARISC]

Originally Posted by winders

Bill, Bill, Bill......

Yeah, he gave up on you once. Beats me why he came back. After all his efforts you still think the equation is ambiguous/can be solved to give two different answers, both valid.

[winders]

Originally Posted by RWebb

the conventions exist b/c they work for real world problems[/url]

You seem to think that conventions are consistent and universal, Neither is the case.

Look here:

http://www.ets.org/s/gre/pdf/gre_math_conventions.pdf

Specifically:

Overview

The mathematical symbols and terminology used in the Quantitative Reasoning measure of the test are conventional at the high school level, and most of these appear in the Math Review. Whenever nonstandard or special notation or terminology is used in a test question, it is explicitly introduced in the question. However, there are some particular assumptions about numbers and geometric figures that are made throughout the test. These assumptions appear in the test at the beginning of the Quantitative Reasoning sections, and they are elaborated below.

Also, some notation and terminology, while standard at the high school level in many countries, may be different from those used in other countries or from those used at higher or lower levels of mathematics. Such notation and terminology are clarified below. Because it is impossible to ascertain which notation and terminology should be clarified for an individual test taker, more material than necessary may be included.

This kind of "conventions" primer is not uncommon in mathematical texts. Why? Because conventions in mathematics are not consistent nor universal.

Scott

[RWebb]

please don't try to change what I said

[DARISC]

This is post #5. Peppy nailed it.

Originally Posted by peppy

If you work the problem from left to right using basic algebra the answer is 288, if you over think it the answer is 2.

[dad911]

Originally Posted by DARISC

This is post #5. Peppy nailed it.

Yup.

Poor students (and teachers) of tomorrow:

Q. 48÷2(9+3) = X, Solve for x

A. The conventional answer is 288, however, respecting students using unconventional rules, and from various societies around the world, unconventional answers include '2' and '.5'

[DARISC]

LOL

Q. 2/(2*2)*2*(2/1)=2/(4)*2*(2)=1(.25*4)*.5*2*(-0+2)=X, Solve for X

[dad911]

Originally Posted by DARISC

LOL

Q. 2/(2*2)*2*(2/1)=2/(4)*2*(2)=1(.25*4)*.5*2*(-0+2)=X, Solve for X

Conventional answer is 2. My unconventional answers are 8 and 10.

[dad911]

Originally Posted by winders

.....
http://www.ets.org/s/gre/pdf/gre_math_conventions.pdf
Mathematical expressions are to be interpreted with respect to order of operations, which establishes which operations are performed before others in an expression. The order is as follows: parentheses; exponentiation;negation; multiplication and division (from left to right); addition and subtraction (from left to right).
......

Originally Posted by winders

.....
Mathematical convention - AoPSWiki

Mathematicians abide by conventions in order to allow other mathematicians to understand what they write without constantly having to redefine basic terms. (Imagine if every mathematical paper began with an explanation of PEMDAS!)

Math 1010 on-line
By convention, formulas are evaluated in the following sequence:

Exponentiation.
Multiplication and Division.
Addition and Subtraction
In the case of operations of the same level of precedence, evaluation proceeds from left to right......

If these conventions were absolute we would be severely stifled, To prevent this calamity, the conventions can be modified by the use of parentheses: Expressions in parentheses are evaluated first.

......

Scott,
The three sites you refer to all state (red print above) that absent of other instruction, as the original problem was presented, equations should be evaluated CONVENTIONALLY. Using these sites as a reference, thank you for clarifying that the correct solution of 48÷2(9+3) = 288

[diverdan]

Equation writer was ambiguous. (48/2)x(9+3) is clear. 48/2(9+3) is also clear. Hmmm.....

[DARISC]

Originally Posted by diverdan

...(48/2)X(9+3) is clear...

( )X...Ahh, hahaha! Totally superfluous.

But if ya really like parentheses (and wanna get rid of any pesky 'x''s), take THIS! ((6)(8))/((2)/(1))((9)+(3))=288

[dad911]

Originally Posted by diverdan

...
48/2(9+3) is also clear. Hmmm.....

Why do you find 48/2(9+3) clear as opposed to 48÷2(9+3)??

[winders]

Originally Posted by dad911

Scott,
The three sites you refer to all state (red print above) that absent of other instruction, as the original problem was presented, equations should be evaluated CONVENTIONALLY. Using these sites as a reference, thank you for clarifying that the correct solution of 48÷2(9+3) = 288

dad911,

You missed the point. What is "conventional" to you may not be "conventional" to others. In other words, conventions are not absolute nor are they universal. If they were, there equation would not be generating the debate that it has. This is exactly why the equation in question is termed "ambiguous". Conventions are not founded in mathematics fundamentals. They are the rules chosen to allow equations and expressions to be more universal and less likely to be misinterpreted. That doesn't mean they are universal and there can be no misinterpretation.

The author of this equation chose it because he knew that some people adhere to the convention that multiplication by juxtaposition takes precedence over division and explicit multiplication. You and Bill can argue all day that is a bogus convention. But the fact is that many people adhere to that convention.

Bill and others say those people that say the equation is "ambiguous" are wrong too. Well, that is an interesting opinion (that's all it is) but factually impossible to support since the mathematics community (i.e. not just the mathematics duffers) argue over the equation vehemently as well.

So, mathematicians are not surprised by the fact that people come up with two different answers. It is the job of the equation writer to make sure the equation is clear if their intent is not. They didn't do that (on purpose).

Scott

[DARISC]

Originally Posted by winders

The author of this equation chose it because he knew that some people adhere to the convention that multiplication by juxtaposition takes precedence over division and explicit multiplication. You and Bill can argue all day that is a bogus convention. But the fact is that many people adhere to that convention.

1. "The author of this equation chose it because he knew that some people adhere to the convention that "the Earth is flat? UH, ohhh! We have a problem!

Not really. Proceed to 2., below (in contrast to the 'left to right rule' in math, this post adheres to the 'top to bottom' rule in reading (I know; in some cultures that convention doesnt apply (however, those cultures still recognize the 'left to right' math rule (well, there are some who don't (and also probably believe there is the bogus convention you believe is in any way valid. HAHAHA ))).

2. The "convention that multiplication by juxtaposition takes precedence over division and explicit multiplication", as Bill Verburg pointed out to you, exists only in the minds of those who don't know math.

3. You cannot cite even one math text wherein that 'convention' is defined (nor can you cite one geography text that says the Earth is flat...

Hey! Besides adhering to bogus conventions when attempting to understand 8th grade algebra problems, are you also a member of "The Flat Earth Society"?!!

Oh yes, I need a citation for that reference:

Flat Earth Society - Wikipedia, the free encyclopedia

"The Flat Earth Society (also known as the International Flat Earth Society or the International Flat Earth Research Society) is an organization that seeks to further the idea that the Earth is flat instead of an oblate spheroid. The modern organization was founded by Englishman Samuel Shenton in 1956[1] and was later led by Charles K. Johnson, who based the organization in his home in Lancaster, California. The formal society was inactive after Johnson’s death in 2001 but was resurrected in 2004 by its new president Daniel Shenton."

Originally Posted by winder

and Bill can argue all day that is a bogus convention. But the fact is that many people adhere to that convention.

NOTE: The International Round Earth Research Society (also known as The Round Earth Society) research shows that members of The Flat Earth Society not only stubbornly adhere to their belief that the Earth is flat, they also stubbornly adhere to their belief that there is a valid math convention that states that "multiplication by juxtaposition takes precedence over division and explicit multiplication".

Originally Posted by winder

the mathematics community (i.e. not just the mathematics duffers) argue over the equation vehemently as well.

No, no, no! The mathematics community argues WITH the mathematics duffers/winders.

[DARISC]

Duffers/winders,

Here's some homework for y'all:

Order of Operations - PEMDAS
Operations
"Operations" means things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like ...

7 + (6 × 52 + 3)

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?

Warning: Calculate them in the wrong order, and you will get a wrong answer !

So, long ago people agreed to follow rules when doing calculations, and they are:

Order of Operations
Do things in Parentheses First. Example:

6 × (5 + 3) = 6 × 8 =
48

6 × (5 + 3) = 30 + 3 =
33
(wrong)
Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract. Example:

5 × 22 = 5 × 4 =
20

5 × 22 = 102 =
100
(wrong)
Multiply OR Divide before you Add or Subtract. Example:

2 + 5 × 3 = 2 + 15 =
17

2 + 5 × 3 = 7 × 3 =
21
(wrong)
Otherwise just go left to right. Example:

30 ÷ 5 × 3 = 6 × 3 =
18

30 ÷ 5 × 3 = 30 ÷ 15 =
2
(wrong)
How Do I Remember It All ... ? PEMDAS !

P
Parentheses first
E
Exponents (ie Powers and Square Roots, etc.)
MD
Multiplication and Division (left-to-right)
AS
Addition and Subtraction (left-to-right)

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)




After you have done "P" and "E", just go from left to right doing any "M" or "D" as you find them.

Then go from left to right doing any "A" or "S" as you find them.

You can remember by saying "Please Excuse My Dear Aunt Sally".
Or ...
Pudgy Elves May Demand A Snack
Popcorn Every Monday Donuts Always Sunday
Please Eat Mom`s Delicious Apple Strudels
People Everywhere Made Decisions About Sums
Note: in the UK they say BODMAS (Brackets,Orders,Divide,Multiply,Add,Subtract), and in Canada they say BEDMAS (Brackets,Exponents,Divide,Multiply,Add,Subtract). It all means the same thing! It doesn't really matter how you remember it, just so long as you get it right.

Examples
Example: How do you work out 3 + 6 × 2 ?
Multiplication before Addition:
First 6 × 2 = 12, then 3 + 12 = 15

Example: How do you work out (3 + 6) × 2 ?
Parentheses first:
First (3 + 6) = 9, then 9 × 2 = 18

Example: How do you work out 12 / 6 × 3 / 2 ?
Multiplication and Division rank equally, so just go left to right:
First 12 / 6 = 2, then 2 × 3 = 6, then 6 / 2 = 3


Oh, yes, and what about 7 + (6 × 52 + 3) ?

7 + (6 × 52 + 3) *
7 + (6 × 25 + 3) Start inside Parentheses, and then use Exponents First
7 + (150 + 3) Then Multiply
7 + (153) Then Add
7 + 153 Parentheses completed, last operation is an Add
160 DONE !

[dad911]

Originally Posted by winders

dad911,

You missed the point. ........

Nope, I believe you missed the point. Even in the articles you cited, they say to evaluate 'conventionally' (PEMDAS) by default, (see my post #629) unless other instructions are provided.

48÷2(9+3) - Wolfram|Alpha

If multiplication by juxtaposition took precedence, then you believe:

48÷2(9+3) = 48÷(9+3)2 ?????

[winders]

You guys are funny.....

I've never said that I think multiplication by juxtaposition takes precedence over division and explicit multiplication. I also said the answer, as I would calculate it, would be 288. But, and this where we differ, I understand that there are other conventions used by some people. This is why the equation is considered "ambiguous" by the mathematics community.

Scott

[winders]

Originally Posted by DARISC

Oh, yes, and what about 7 + (6 × 52 + 3) ?

7 + (6 × 52 + 3) *
7 + (6 × 25 + 3) Start inside Parentheses, and then use Exponents First
7 + (150 + 3) Then Multiply
7 + (153) Then Add
7 + 153 Parentheses completed, last operation is an Add
160 DONE !

And you call me a duffer? 160?? You transposed the 5 and the 2. The answer is 322. The parenthesis are superfluous:

7 + (6 × 52 + 3) = 7 + 6 × 52 + 3

Scott