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

reply to post by 3mperorConstantinE


The ambiguity arise when people use calculator and they found its 288 which is wrong. Instead of using () to encapsulate the numerator, they just key in the numbers. Calculators on the other hand, auto calculate the input without precedence of the 2nd (). The answer is correct if for these numbers, however when one of the numbers changed it will yield a different result.

A proper way to avoid ambiguity is to use algorithm, or algebra. Algebra consist of variables that can be replace but still will not affect the formula, this is the reason scientist talk in letters instead of number. And also the reason the 2 people resort to algebra to solve the ambiguity.

For now, stop believing the calculator, because they did not know the precedence before hand, 2*(9 + 3) was key in at later stage, while in actual calculation, it must be solved first. To show the flexibility and benefit of algebra, I have $48 dollars, I have 2 kids, each have 9 sons and 3 daughters, each of them will get $2.
By using algebra, you can calculate how many grandchildren allowable for them to receive $4.

This is why algebra learnt later than arithmetic and why algebra is specific yet still changeable.
You can have 288 as the answer, but you MUST add (), like this
(48/2)(9 + 3), of course the above will scenario cannot be use now. Ambiguity arise when people forgot to add () to the equation thus making them enter the incorrect input thus have "weird" result.

You can inspect my previous post about finding the number and it remain true for any number (Its not limited to multiply of 3).

reply to post by NullVoid


I don't use calculators (for basic stuff). But if you are coming at this from a Computer Science angle then it's still wrong because you misunderstand how parsers work.

I glanced at your gibberish for the brief amount of time needed to deduce that you need to go "back to the basics" in re: algebraic manipulations involving the parenthesis operator. i.e. meaning learn what you can/cannot do with them and when —then you can see how you doomed your examples above to failure based on how you set them up.

Do that and then actually read the last couple posts on page 8 carefully.

reply to post by 3mperorConstantinE


Show me please.

Ahh I just clicked on Google, it show the needed () in the numerator. I think I'm correct afterall.
Google sided with me
Google answer

Still, do show me where I'm wrong. I left algebra looong time ago, allergic to it.


Bahh I'm quitting this thread. cause I already found the correct answer.

C0bzz
reply to post by Arbitrageur

 

You keep stating that solving the problem left-to-right is incorrect, you reach this conclusion because you did not read your own source properly.

I said it doesn't solve the ambiguity, and people robotocally regurgitate a left to right rule without understanding the origins of it and without doing enough thinking, which as the author explains is not such a great idea. I'm agreeing with the author that it's not such a great idea.

Division is not the same as multiplication in that regard, changing which way it is read does tend to change the answer unless you change division by c with a multiplication by (1/c).

My point is that when you try to convert divide by c to multiply by 1/c, you still don't know whether c is 2, or whether c is 2(9+3), and if you're trying to pick one or the other, there are a lot better reasons to pick the 2(9+3), but at best it's still ambiguous. As the Berkeley author stated the left-to-right rule is of questionable value to more advanced students, and several things are better as already mentioned when they are understood, though they are not universally accepted (like multiplication before division in physics journals, or implied before explicit multiplication, which would lead you to define "c" as 2(9+3) when you choose to multiply by 1/c as an equivalent expression).

As I said if I wanted the answer to be 288 I would write the expression as 48(9+3)/2 which doesn't require any extra parentheses and if it's ambiguous someone will have to explain how. You can still change the divide by c to multiply by 1/c and since there is no ambiguity about the value of c (the denominator which is 2), the order of operations is irrelevant.

3mperorConstantinE
I can't believe that this question even made it to the internet.

288

We really do live in that universe where the movie “Idiocracy” becomes a true story.

Still, S&F for the thread~

—Mathematical Physicist

It's not apparent to me you read the opening post, where I explained why in some physics journals the answer is unambiguously 2.

Multiplication by juxtaposition, as used in algebra, suggests 2(9+3) is one term. This is supported by many mathematical societies and professionals.

If you were to come across ab+c or a+bc, is there any doubt ab or bc must be solved for before anything else? And what about ab/c or a/bc? Same thing- ab or bc must be solved before the division occurs. Same here, though it would be different if it were written as 2(12), as then the implied multiplication would require explicit symbols attached as in 2 * 12, and this changes the interpretation of the equation, and is what leads to the confusion, ambiguity and arguments.

At best, the equation presented in the OP is vague and ambiguous. And depending on your 'training' either 288 or 2 is the correct answer. Higher math's consensus is 2 is the correct answer while basic / menial arithmetic suggests 288 is correct.

Issues like this are best avoided by being concise and clear when presenting things, as in "mean what you say and say what you mean".

So, it should be written as either (48/2)(9+3) or 48/(2(9+3)) so as to remove all ambiguity.

In the mean time, millions of hours have been spent trying to solve a problem whose solution is only based on opinion, as there is no single, globally accepted rule establishing a standard regarding how this should be interpreted. And personally, I will side with the physicists and such who arrive at 2 being the proper answer, and I was taught that back in 1983 in my pre-algebra class in junior high school, by a math-major with a PhD.

A mathematician's opinion on this problem:
math.berkeley.edu...

I have to go with what the majority of computer software does. If it were obvious wrong then it would get changed.

I keep finding refs to multiplication and division being of equal precedence.

roadgravel
I have to go with what the majority of computer software does. If it were obvious wrong then it would get changed.

I keep finding refs to multiplication and division being of equal precedence.

And I keep finding references to implicit multiplication (multiplication by juxtaposition) as encountered in algebraic equations taking higher precedence over explicit multiplication and division.

And hence this conundrum we're arguing over.

But I do find it odd I was taught the implicit takes precedence back in the '80s, in 7th / 8th grade, yet here we are 30 years later being told 5th grade math ranks above established college level "technique".

reply to post by abecedarian


This whole thread is basically about this: 42/2(3) != 42/2*3.
Or: 42/2(1+2) != 42/2*(1+2).

The distinction between an implied and explicit multiplication creates misinterpretations. IMHO, it 'd be a lot easier to just say that it's one or the other.

And it's easier to write this: 42/2(1+2)...
Than this: 42/(2(1+2)).

It's also easier to write this: 2(1+2)...
Than this: 2*(1+2)...

Convenience usually wins.

I'd also like to say in Algebra books, in the example sections, there's a clear divider between the numerator and denominator and in most if not all cases there's no parenthesis because it's implied. You just know which is which. A computer doesn't know unless you use parenthesis or give implied multiplication higher precedence than * or /. And apparently many people don't either!!

One more thing I noticed is this produces an error: (4)4/2.
This doesn't: 4(4)/2.

Alright listen—to everyone following thus far in this interesting thread—I wanted to provide a simplified yet exhaustive analysis of what's going on here. This topic extends even deeper than what some of the math pros I've seen commenting online are saying.

this "ambiguity" is contrived because it is all based around the modern "inline" way of writing expressions
i.e. 1/2x = 1÷2x ~being either~> (½)x ~or~> 1÷(2x).
In the past there was no possible ambiguity because there were only FRACTIONS (ratios), not inline operators.

Since I see a few people who seem to have some understanding or at least exposure to more advanced mathematics, I wish to note my conjecture on why this debate has proceeded for so long and appears so confusing:

Because of the fact that this debate can be had on a multitude of different levels, with those on the below list apparently talking past each other.
We have the following "groups":

• those with a pre-algebraic, arithmetical, school-level understanding (e.g. rule-based order of operations codified by the various mnemonic devices in use, such as P.E/MD/AS).
  
• the technical minded who have a solid understanding of elementary algebra (engineers, etc).
  
• the pragmatic, historical minded people who remind us that this would be a non-issue if we weren't writing this equation on a single line.
  
• computer programmers proficient in one or more computer languages.
  
• the doctoral-level computer scientists (e.g. those with an extensive understanding of compiler theory, including all postfix, infix, and circumfix operators).
  
• those who at least understand that there is more than one "algebra".
  
• the exceedingly few who can discuss this topic in the rarefied realm of abstract algebra (e.g. algebra over a commutative ring, algebra over a field, group-theory, Lie algebra, etc.). Even the non-algebraist Professional Mathematicians do not fully understand these topics. Well, at least no more than one being a Professional Physicist implies that they ipso facto understand what the hell (k>n-2)-fold subprojective connexions of an Einstein Space signify!
  

Anyhow, the bottom line (from my humble POV, anyway) is that the ‘correct’ answer
(making stupid finger-quotes in the air) is:

288

And, instead of talking around in circles, I will give you the clues you need to kick the discussion up a notch and maybe even come close to determing an answer. Because I WILL TELL YOU THIS:
If you wish to argue it merely using PEMDAS, and left-to-right rules, THEN YOU WILL BE HAVING THIS DISCUSSION UNTIL THE END OF TIME

Now, for the technically proficient, I invite you to reduce the following expression:

1/(a+b)(a-b), (where a=1, b=ⅈ where ⅈ²= −1)

—Seriously, try this out!

Bringing complex numbers into it, is a possible path to the solution by seeing how the brackets must be evaluated in the above equation.
Remember, all ℝ (the field of real numbers) are, by definition, a subset of ℂ.
If φ is regarded as a complex number, then φ = (a+(b)√-1). In Cartesian form, this gives:
—> φ = Re(φ) + Im(φ) • ⅈ
Now if φ ⊂ ℝ (φ is a real number), then b=0
…and so
—> φ=(a+(0)ⅈ)

For those who already “get” what I just asked (and why), the following paragraph is for you:

When one distills the act of solving an equation, and thus computing a solution to a provided equation, ambiguously expressed though it may be, like say, 48/2(9+3)), then one must wind up in the field of logic. And by logic, of course I do not mean to imply "logic" in a vague, psychological sense. No, I mean in a precise, mathematical sense.

So what branch of mathematics is the ultimate domain for discussing this question?

– welcome to the Heyting algebra.

Do you know what a topological space is, but the above link is veritable gibberish to you?
Then try this instead.

Finally, here is a (simple) synopsis for everyone else.

Question ~~> using the Switch Algebra, why would the answer be “288”?

~E.

/thread

I also got 288, calculated in my head. However, I see why others got 2.

Looking at it, the way I was taught, gave me 288. But, looking at it again, I can see how you could get 2.

Six of one, half a dozen of another. This isn't a "new" debate, BTW...

reply to post by 3mperorConstantinE

i.e. 1/2x = 1÷2x ~being either~> (½)x ~or~> 1÷(2x).

From my rustic algebra..
i.e. 1/2x = 1÷2x = 1/(2x)
if I want this (½)x I would write it as (1/2)/x
Less beautiful than yours but look, you even have the () thrown in, just like mine.

Again, we have to accept this:
So, it should be written as either (48/2)(9+3) or 48/(2(9+3)) so as to remove all ambiguity.

Ok, I think we got the solution, so both proponents are wrong, the correct one should be as above.
The equation is wrong and ambiguous, to correctly arrive at 288 or 2, we have to put (), no matter which side you are fighting, you have to put (). The position then will determine either 288 or 2.

Now, who the goddamn person who so lazy to put () in the first place ? I got my pitchfork, burn all witches and their false teaching!

NullVoid
reply to post by 3mperorConstantinE

 

i.e. 1/2x = 1÷2x ~being either~> (½)x ~or~> 1÷(2x).

From my rustic algebra..
i.e. 1/2x = 1÷2x = 1/(2x)
if I want this (½)x I would write it as (1/2)/x
Less beautiful than yours but look, you even have the () thrown in, just like mine.

Well, the point was that as written:
1/2x = 1÷2*x but is not automatically interpreted as 1/(2x) rather than ½x (= x/2)
unlike what some people have suggested:
2x = (2)x = 2(x) =2*x
So removing the division symbol by using reciprocals:
1*(2)ˉ¹*x or 1*(2*x)ˉ¹

Again, we have to accept this:
So, it should be written as either (48/2)(9+3) or 48/(2(9+3)) so as to remove all ambiguity.

Ok, I think we got the solution, so both proponents are wrong, the correct one should be as above.
The equation is wrong and ambiguous, to correctly arrive at 288 or 2, we have to put (), no matter which side you are fighting, you have to put (). The position then will determine either 288 or 2.

Now, who the goddamn person who so lazy to put () in the first place ? I got my pitchfork, burn all witches and their false teaching!

You are absolutely correct. In terms of real-world pragmatism, the equation is very poorly formatted, hence terminally ambiguous.
There are many similar examples of apparently indeterminate expressions.

The thing is though, as I made mention to above, many of these same indeterminate expressions end up having unambiguous solutions when seen through the lens of higher branches of mathematics (number-theory, group theory, Lie Algebra, etc)

So, in the case of the ill-formatted expression: 48/2(9+3), I was just pointing out that you can reach the ultimate conclusion of "288" being the "correct" answer using either:

• operator-theory, which is the reason why all modern calculators and programming languages give 288 as the solution.
  
• complex number theory
  
• Heyting Algebra
  

But all of the above is essentially just math gibberish, and for all practical purposes the formatting of the expression = FAIL, lol

Hmm, in Spotlight on Mac OS X 6.8

48/2(9+3)=2

but 48/2*(9+3)=288

In Unix terminal using bc, the top expression yields a parse error, which makes some sense as Unix generally doesn't accept implied stuff -- and well so.

I guess some convention needs to be made regarding implied multiplication, to do away with this ambiguity/inconsistency.

When I looked at the problem, I assumed that implied multiplication is like explicit multiplication, i.e. I obtained 288 as the answer.

Obviously parentheses can be used to remove ambiguity, but they're superfluous if one has a complete convention.

MrInquisitive
Hmm, in Spotlight on Mac OS X 6.8

48/2(9+3)=2

but 48/2*(9+3)=288

In Unix terminal using bc, the top expression yields a parse error, which makes some sense as Unix generally doesn't accept implied stuff

The issue with Spotlight is a parser bug in the implementation of the Cocoa Framework's NSValueTransformer class. It's because Spotlight was designed to accept Logical predicates like:
(x == "A") || ( x == "B")
in the text box. It's complicated.

And in "bc":
echo "4(2)" | bc -l

Gives a parse error as well. In bc you must use explicit multiplication symbols.

reply to post by 3mperorConstantinE


It is NOT ambiguous nor is it poorly formatted.

It can be confusing, especially to those who have studied higher maths and should be written out more implicitly to avoid that confusion but that does NOT make it ambiguous.

The reason that it is unambiguous is that the way it is written does not REQUIRE higher maths or algebraic expression to solve.

Higher maths have a purpose and a necessity that requires they be used. This is a simple expression that does not require higher algebraic interpretation or physics based formulaic interpretation to solve.

Because it is simple (KISS) there is no need to use higher maths to solve it, so you would use basic expressions and precedence, i.e. multiplication and division are treated equally and solved from left to right.

You are not solving for any variables, so only in the case that there WERE variables is it in the least ambiguous.

You aren't inserting imaginary numbers, because it is NOT a formula giving license to insert ANYTHING. If it were then it would have a different answer.

AS it is expressed, simple arithmetic is ALL that is needed to solve it, so it is unambiguously, 288.

So, again as it is listed, it is unambiguously 288. If, however, it was an expression with a variable it would be ambiguous without additional qualifiers and would be 2 as an answer if you were inserting the same numbers in the expression as the variables.

So if the expression were 48/n(9+3) and n=2, then the final answer would be 2 or 288 and it could be considered sort of ambiguous
While the expression 48/2(n+3) where n=9, the answer would again either be 2 or 288 and it would be ambiguous.

In both of the above cases if instructions were given to simplify the expression prior to inserting the variable, the answer would only be 2 and it would be unambiguous.

As it stands where simple arithmetic is all that is necessary to solve it, it is 288.

Quit nuking it for christ's sake.

Jaden

reply to post by Masterjaden

I'm the one arguing that question is not ambiguous when calculated using the standard arithmetic, Left-2-Right rule.
==288.
That's why all serious mathematical programming languages come up with:
=+288.

I wasn't advocating turning it into an algebraic problem, but others were using that to prove a point of it being ambiguous. If you know anything about mathematics (which is unclear) then you'd realize that there is no such demarcation between algebra and arithmetic.

48/2(9+3) =DOES EQUAL= x/y(z+a)
To say anything but that is pure ignorance.

x/y(z+a) simplified =MAY EQUAL= (2) ~OR~ (288)

Wrong, ^ still reduce to: x(a/y + z/y)
== 288
The algebraically-inclined posters are NOT necessarily wrong to look at it as unambiguous.
They ARE WRONG if they conclude that the answer is unambiguously "2", however.
I say it IS AMBIGUOUS and POORLY FORMATTED because it IS.
People are obviously in disagreement right??? Lol.

BTW: YOU CAN SOLVE ALL REAL equations using imaginary numbers, you don't "need permission". Lol
Seriously J$—you should learn of why they even came about, historically speaking.

Everything else I said was intended to those equipped for analysis of the problem under discussion.

Arbitrageur
Here is the same formula 48/2(9+3) typed in two calculators from Texas Instruments, the TI-85 and the TI-86, and the answers are different:

This is a silly argument.

because it all comes down to how the device interprets the input.

For example.

In Excel / is the operator for division. if you bracket something () then that is a separate calculation that if used with =sum would be added to the calculation.

So the answer to your equation 48/2(9+3) according to Excel would be 36.

48 divided by 2 = 24
+
9 + 3 = 12

= 36

If you are using a calculator and don't understand how you are building the equation you will likely make a mistake and the device will give you an answer to a different question entirely.

Peace,

Korg.

reply to post by Korg Trinity


How Excel interpretations forumulas is completely non-standard and application-specific.

GetHyped
reply to post by Korg Trinity

 

How Excel interpretations forumulas is completely non-standard and application-specific.

Exactly my point....

If you are using a device or application then you have to understand how it interprets the input... or the answer you get would be for a completely different question...

This is also true of any language translation but especially true of Math.

Peace,

Korg.

reply to post by Korg Trinity

Thank you for understanding the point of the thread that so many missed. This was the point, and the reason I titled the thread the way I did:

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

However it was entertaining to see so many provide what they felt was the "right answer", in some cases by doing no more than entering the formula in Google Calculator.

Some calculators are running internal computer code so this concept also extends to code and various computer programs but Excel may be one of the worst offenders in using non-standard conventions in what it does, like the nested exponent issue I mentioned previously for example, which does have a fairly well established convention that Excel ignores.