Why you can't trust your calculator, or What is 48/2(9+3)?

Given the aforementioned equation: 48/2(9+3)....

I was taught that implicit multiplication, i.e. 2x, takes precedence over explicit multiplication, i.e. 2 * x.
In other words, the omission of the explicit function implies the implicit function occurs first.

The argument in favor of this would be to use something like this:
y = 48 / 2x where x = 9 + 3; solve for y. (literal mutation of the equation in the OP)
You cannot solve for y without the value of x, therefore 2x is evaluated first.
The result being 48 / (2 * 12) = 48 / 24 = 2.

Were the equation written such as:
y = 48 / 2 * x where x = 9 + 3...
... the explicit precedence rules of left to right would result in 48 / 2 * 12 = 24 * 12 = 288

Masterjaden
You're just straight up wrong. This is NOT a physics equation, it is an algebraic equation and in algebra, order of operations does NOT give precedence to multiplication over division and they are always done from left to right unless there is a delineation with parentheses or brackets. The answer is unambiguously 288.

Saying "they are always done left to right" is citing a rule, but strictly adhering to this rule is no more likely to give you a right answer with an ambiguous expression than typing the same expression into a calculator. Here is some more background on why this is so, which was only partially excerpted in the OP:

“Order of operations” and other oddities in school mathematics

An example is the convention known as the Rules for the Order of Operations, introduced into the school curriculum in the fifth or sixth grade:

(1) Evaluate all expressions with exponents.
(2) Multiply and divide in order from left to right.
(3) Add and subtract in order from left to right.

In short, these rules dictate that, to carry out the computations of an arithmetic expression, evaluate the exponents first, then multiplications and divisions, then additions and subtractions, and always from left to right. To sixth graders, these rules must appear random and therefore meaningless, and the meaninglessness of it all gives rise to the infamous Please Excuse My Dear Aunt Sally mnemonic device.

In this short article, I will first briefly discuss the mathematical background of these Rules in order to make sense of them, and then explain why it is unprofitable to pursue these rules vigorously in the arithmetic context the way it is done in most school classrooms. Finally, I will present an argument against using assessment items in standardized tests that assess nothing more than students’ ability to commit certain definitions and conventions (like Rules for the Order
of Operations) to memory.

The mathematical origin of Order of Operations

To understand how the Rules for the Order of Operations came about, one has to consider polynomial expressions in algebra. Look at the following polynomial function of degree 8 with coefficients 17, 2, 1, 0, 0, 6, 0, 3, 4:

17x^8 + 2x^7 + x^6 + 6x^3 + 3x + 4, (1)

where x is any number. The notational simplicity of this expression must be obvious to one and all, and this simplicity is the result of a common agreement that what this expression really says is:

(17(x^8)) + (2(x^7)) + (x^6 ) + (6(x^3 )) + (3x) + 4. (2)

To be precise, the symbolic notation in (1) is free of the annoying parentheses in (2) because the convention of performing the operations in the order indicated by the parentheses in (2) is universally accepted, namely,

(A) exponents first, then multiplications, then additions.

To drive home the point, the sum 17x^8 + 2x^7 does not mean
[[(17x)^8 + 2]x]^7 ,

which would be the case if the operations were uniformly performed from left to right, but rather

(17(x^8 )) + (2(x^7))

as indicated in (2).

The rule of “multiplications before additions” may sound simple, but these three words contain more than meets the eye. Because we are in the realm of algebra, “division by a (nonzero) number c” is the same as “multiplication by
1/c”. Moreover, “minus c” is the same as “plus (−c). Therefore if one rewrites what is in statement (A) above in the language of arithmetic, then one would
have to expand it to:

(B) exponents first, then multiplications and divisions, then additions and
subtractions.

Except for the stipulation about performing the operations “from left to right”, (B) is seen to be exactly the same as the Rules for the Order of Operations. It is important to note that this stipulation about “from left to right” is entirely extraneous, because the associative laws of addition and multiplication ensure that it makes no difference whatsoever in what order the additions or multiplications are carried out.

Hopefully you can understand "why it is unprofitable to pursue these rules vigorously in the arithmetic context the way it is done in most school classrooms" referring to the order of operations. If the expression is ambiguous, we really need to resolve the ambiguity, which as I said some physics journals do by saying multiplication before division. I didn't say it was a physics problem, but if you remove something like the "multiplication before division" rule in physics journals, and have no other way to resolve the ambiguity, the "left to right" really isn't a satisfactory way to resolve it, which you can hopefully understand from the explanation above, but if not that's ok too, you're entitled to your opinion.

abecedarian
Given the aforementioned equation: 48/2(9+3)....

I was taught that implicit multiplication, i.e. 2x, takes precedence over explicit multiplication, i.e. 2 * x.
In other words, the omission of the explicit function implies the implicit function occurs first.

The argument in favor of this would be to use something like this:
y = 48 / 2x where x = 9 + 3; solve for y. (literal mutation of the equation in the OP)
You cannot solve for y without the value of x, therefore 2x is evaluated first.
The result being 48 / (2 * 12) = 48 / 24 = 2.

Yes a lot of people were taught this about implicit multiplication, and if you look at the picture of the Casio calculator in the OP, it looks like that rule is programmed into the calculator. So, that's one way to resolve the ambiguity, but it's not universally taught or accepted. Where it is taught and accepted, it's certainly valid and results in the answer "2" as you say.

reply to post by abecedarian

AthlonSavage
reply to post by Arbitrageur

 

The general accepted rules of mathematics Multiplication, Division, addition and subtraction is that it occurs in that order. Therefore /2 operand takes place before 9+3.

Absolutely incorrect.

9+3 is in parenthesis, which means you absolutely must do the addition first.

Or you can distribute out the 2 over the parenthesis and get

48 divided by (18+6) which still gives you 24 on the denominator, and two as the final answer.

There is no debate here. 2 is the correct answer. The only debate is by people who forgot how to properly do math, or by people who don't realize calculators aren't perfect and you must be liberal with parenthesis to get the correct answer.

mojo2012
what is 48/2(x+y)? i typically assume that 48 is being divided by 2(x+y). I guess it should be written like this 48/[2(x+y)].

Great way of looking at it.

That would be 48/2x+2y

x=9
y=3

Then you get 48/18+6 or 48/24 or TWO.

I can't believe some people still think it's anything other than two. Sad state of education for sure.

reply to post by James1982


a lot of folks, i would say the majority, don't really have occasion to continue their practice with algebraic math once they leave school and enter the workforce.

And since school isn't about understanding, but rather memorization instead, it is easy to see how things can be forgotten or become more opaque in peoples memory.

Olivine


I guess I'm just a "everything below/behind the slash is in the denominator" kind of gal.

And that's the proper way to approach it, simply looking at the calculations in a linear fashion is not the correct way to do math, that's not how it works, anything other than 2 is incorrect.

bigfatfurrytexan
reply to post by James1982

 

a lot of folks, i would say the majority, don't really have occasion to continue their practice with algebraic math once they leave school and enter the workforce.

And since school isn't about understanding, but rather memorization instead, it is easy to see how things can be forgotten or become more opaque in peoples memory.

Math classes are what you make them. Every single one I've ever been in had teachers that wanted students to actually learn and comprehend the material. If students are lazy and simply try to pass tests instead of learning the material, that's their fault.

I'd agree with you on every other subject, but math is not political and it's not subjective. It just IS.

Given the current standard precedence, as used in programming languages, the result would be 288.

We best be distrustful of those machines.

I suppose not being clear on order of operations (including implied in some cases) can be as bad as not using correct units for scientific calculations. Although I haven't done much programming other than occasional scripts as part of some hobby stuff, I'm pretty sure I've heard the phrase "If in doubt, bracket it out." At least that way you can be very specific about which order the operations take place without any ambiguity.

And the way I was taught, brackets implied a distributive property for whatever was adjacent. I would have gotten 2. So when did they decide to change it? (I also have a 1990's vintage TI-85 somewhere in near mint condition too.)

reply to post by James1982


memorizing formulas is not knowing math. it didn't work for me. I now do math as a career, but have had to teach myself everything.

Math principles are taught (the basics) but we don't really learn until we have an application for it and then it tends to be only as much as is needed.

Just for fun, evaluate 48/2/(9+3)
I'm just interested in whether that can be considered ambiguous as well (to me it isn't)

I tested on some calculators:

Casio Classpad 330: 288
Casio FX-82MS: 2

Windows 7 Calculator: 288
Wolfram Alpha: 288
Google: 288
C/C++ (had to change it to 48/2*(9+3)): 288

I'm assuming computers tend to give 288 because it's probably more efficient to simply go from left-to-right when evaluating equations - rather than check BODMAS/PEMDAS correctly then deduce the correct order of operations. At least, if I were doing something like that in assembly that's how I would do it.

Funny how I work with C/C++ a lot and got the same answer even though it's possibly actually incorrect if PEMDAS are followed correctly. I could probably also try viewing it in assembly to see exactly how it is doing it.

EDIT: C does indeed do it left-to-right.

James1982

AthlonSavage
reply to post by Arbitrageur

 

The general accepted rules of mathematics Multiplication, Division, addition and subtraction is that it occurs in that order. Therefore /2 operand takes place before 9+3.

Absolutely incorrect.

9+3 is in parenthesis, which means you absolutely must do the addition first.

Or you can distribute out the 2 over the parenthesis and get

48 divided by (18+6) which still gives you 24 on the denominator, and two as the final answer.

There is no debate here. 2 is the correct answer. The only debate is by people who forgot how to properly do math, or by people who don't realize calculators aren't perfect and you must be liberal with parenthesis to get the correct answer.

James1982

mojo2012
what is 48/2(x+y)? i typically assume that 48 is being divided by 2(x+y). I guess it should be written like this 48/[2(x+y)].

Great way of looking at it.

That would be 48/2x+2y

x=9
y=3

Then you get 48/18+6 or 48/24 or TWO.

I can't believe some people still think it's anything other than two. Sad state of education for sure.

That is exactly what I said. The two is a common factor of the elements of the sum within the brackets and so expanding will give you (18+6) which as brackets are solved first gives a denominator of 24. I see no ambiguity here.
original post

reply to post by CthulhuMythos




reply to post by James1982


No,

48/2x+2y

x=9
y=3

equals 8 and two thirds.

By following PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction)

=> (48/(2*x))+(2*y)
= (48/(2*9))+(2*3)
= (48/18)+(6)
=8.666666666666666666

By following PEMDAS you do indeed get 2 for the 48/2(9+3) because:

=> 48/(2(12))
= 2.

Alternatively, how you should have done it:

=> 48/2(9+3)
= 48/(2(9)+2(3) )
.....

I did some searching and couldn't really find much more information beyond which Arbitrageur found.

However, BODMAS and PEDMAS are taught in plenty of countries. Some sources say that in BODMAS, division is before Multiplication, then:

=> 48/2(9+3)
= 48/2(12)
= (48/2)(12)
= (24)(12)
= 288.

Others say that Multiplication and Division have equal precedence, and the problem is solved left-to-right (with multiplication and division a higher priority than subtraction and addition). This is what most calculators and computer programs use and gives 288. And while this isn't a reason for it actually being correct, it does mean that anyone who is expecting 2 will get an unexpected result sometimes when they use a calculator or computer. I do most math using a calculator or computer, so assuming multiplication and division are equal and go left-to-right seems most practical in the real world.

Perhaps the reason why there is no authoritative source on this is because real professionals do not write things in ways which are ambiguous.

I'm still going with 288, as it's what is taught here (in multiple levels of education) and it's the result you're likely to get if you use any technology to get the answer.

reply to post by Arbitrageur


Without a doubt 288 is the answer. I did it in my head based on every math class ive ever had.

Parenthesis first (gives 12), you multiply the final outside number (after solving the problem outside the parenthesis) by the number that is left after the parenthesis is solved. Then you do division and multiplication in whatever order they appear so 48/2 is 24, then 24 x 12 = 288.

reply to post by C0bzz


This is incorrect. You are adding an extra parenthesis that is not in the problem which changes the solving order.

The answer is 288.

reply to post by GogoVicMorrow


All brackets in the post you replied to were added according to PEMDAS, therefore the order of operation was not changed.