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

reply to post by 3mperorConstantinE


Yeah I guess I started nuking it too. There is no way to get 2 from that without additional parentheses or without assuming it is

48
-------
2(9+3)

It is just straight up non ambiguous without considering it relative to the task at hand.

If you are looking at it relative to a calculator, it is ambiguous. If you are looking at it relative to computer programming, it would depend on the software as to whether it is even usable or not.

That's why it is always best to use formatting that is explicit.

As to whether or not you can insert imaginary numbers into an equation, if there are no variables in an equation, you can't insert ANYTHING.

You solve it as is.

Jaden

Masterjaden
reply to post by 3mperorConstantinE

 

Yeah I guess I started nuking it too. There is no way to get 2 from that without additional parentheses or without assuming it is

48
-------
2(9+3)

It is just straight up non ambiguous without considering it relative to the task at hand.

It's not ambiguous regardless of the task at hand if it's published in an American Physical Society journal, and I haven't seen you acknowledge that the answer is unambiguously 2 in those journals, where multiplication is done before division.

Arbitrageur
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)?"

You're Welcome

Korg.

Arbitrageur
t's not ambiguous regardless of the task at hand if it's published in an American Physical Society journal, and I haven't seen you acknowledge that the answer is unambiguously 2 in those journals, where multiplication is done before division.

It's unfortunate that programming languages didn't follow this, they all treat multiplication and division at the same precedence level in which case they go left to right.

I think Fortran was responsible for the original sin, and nobody corrected it later.

Masterjaden
reply to post by 3mperorConstantinE

 

As to whether or not you can insert imaginary numbers into an equation, if there are no variables in an equation, you can't insert ANYTHING.

You solve it as is.

Jaden

You don't have to "insert them" into the equation. The set of REAL numbers (-5,1,100, etc) are by definition a subset of COMPLEX numbers (just with the “imaginary” part being set equal to 0)

ƒ(w,x,y,z) —> (w+√-1•0) / (x+√-1•0) ((y+√-1•0) + (z+√-1•0))

ƒ(48,2,9,3) —> 288

It would be a stupid waste of time to do that for this problem, but it's still perfectly valid.
I'd only mentioned complex numbers to point out the fact that in formal mathematics (such as e.g. the study of complex numbers), there is a convention for handling the precedence concerns for expressions containing [÷,/] and [*,•,()].

For example, in my original expression on the other page, it has to be evaluated like:
((1 + i) / (1 - i))ˉ¹ —> reducing to minus the square root of negative one
= -i

Which is, of course, awesome. But for the problem at hand STILL gives the answer as 288

unless a physics or math journal specifies differently at the beginning of an article that implicit multiplication has stronger precedence than division. Which may happen. But not often.

This problem, to me, isn't dissimilar to:
1-3+4 =

1. 2✓
   
2. -6
   
   
   edit on 6-3-2014 by 3mperorConstantinE because: (no reason given)
   

The answer is 42.

Kidding. It's 288. No doubt.

reply to post by Arbitrageur


With regards to physics journals the issue referenced is inline expression involving implicit multiplication such as
1/2x

Now, the interpretation will usually always be clear from context;
due to the fact that the articles in Physics Journals are formatted using LaTeX and by convention utilize reduced-form expressions.
So if by:
1/2x
they really meant (½)x —> they would write it as x/2

You'd never see this (written inline)
48/2(9+3) = 2

in a physics journal.

~E

mbkennel
It's unfortunate that programming languages didn't follow this, they all treat multiplication and division at the same precedence level in which case they go left to right.

I think Fortran was responsible for the original sin, and nobody corrected it later.

The programs would be hard to change, but some students already misinterpret the PEMDAS mnemonic and think the fact that M comes before D implies that multiplication comes before division, though that's not the intent of educators. But if they changed the intent to give multiplication a higher priority, they wouldn't even need to change the mnemonic. Well there are variants like BEDMAS and BOMDAS where the M and D are sometimes interchanged so they aren't all consistent.

With the programming languages, maybe it's easier to make them do left to right operations, and could that be part of the reason they do it? I'm not really sure how or why the APS decided to prioritize multiplication over division, but if I had to guess, I suspect it's using the inline slash symbol in this context, where it represents the horizontal line:

48
-------
2(9+3)

reply to post by 3mperorConstantinE

I've certainly never seen that expression in a physics journal, but if a similar expression was made using say, variables instead of numbers, you would perform the multiplication, then the division, in the APS journals according to the submission guidelines, which would make the answer 2. But if presented with a similar expression you'd perform multiplication before division. It's not universally accepted but I don't see how it would be a bad thing if it was.

As I've said several times already if I wanted the answer to be 288 I'd write the expression 48(9+3)/2. I think it's fairly unambiguous even without adding additional parentheses.

288 all the way.

reply to post by Arbitrageur


Unfortunately, you are misunderstanding the style guides…

From your OP:

So in conclusion, we can't say that either 2 or 288 is the wrong answer. However if you're following prominent physics textbooks or reading journals from the American Physical Society, if you said the answer is 2, you would be right, because of the conventions followed in those sources.

This is incorrect. You may have missed this word, also from your OP:

…changed in 2013 to treat implied multiplication the same as explicit multiplication (formerly, implied multiplication without parentheses was assumed to bind stronger than explicit multiplication).

48/2(9+3) is using implied multiplication with parentheses.

There is another interpretation which is pretty common that states the answer is 2, based on the implied multiplication having priority over explicit multiplication issue mentioned in the quote above regarding a change in 2013 in Wolfram Alpha. I couldn't find much documentation on this, but one reason it's not even an issue at the American Physical Society is, they don't even use multiplication signs for multiplication, rather they are used only for vector products so all multiplication is the "implied" type:

48/2(9+3) - implied
48/2x(9+3) - explicit

This isn't correct.
First, the use of an explicit multiplication symbol DOES occur at times (usually for clarity), both in journals and elsewhere. But for this we use the dot ("•") operator (technical term for it is the "interpunct") for all scalar products.

In both mathematics and physics we do not EVER use the cross symbol, "×", for scalar multiplication.
The cross symbol is ONLY for Cartesian products, and vector analysis’ related Cross (Vector) product (to differentiate from the scalar (dot) product "•")

Second, the above two ways of writing the expression:
48/2(9+3)
48/2•(9+3)
are exactly equivalent.

What the sources such as Wolfram et al. are talking about is:

2x/2x = 2*x/2*x = 2(x)/2(x)

all being interpreted as 2•(x/2)•x
= x²
rather than as (2x)/(2x)
= 1

—that's the kind of "multiplication before division" they are referring to.
Basically, that whole issue is more akin to "binding rules" in programming languages, not how people interpret basic arithmetic.

~E

3mperorConstantinE
What the sources such as Wolfram et al. are talking about is:

2x/2x = 2*x/2*x = 2(x)/2(x)

You are repeating stuff I already said, but you appear to be mixing up Wolfram Alpha and APS. APS is not "Wolfram Alpha, etc". I wasn't citing the guidelines from Wolfram Alpha and APS as the same, because they aren't, but here you seem to be lumping them together with "etc" which I did not do and is either a misunderstanding, misdirection, or something like that.

Maybe you can clarify what you mean by referring to this without discussing Wolfram Aplha:
Physical Review Style and Notation Guide (pdf) p21


Are you saying following that would give you 288?

reply to post by Arbitrageur


This was amusing! Whoever decided that using the forward slash as the division expression totally fubared mathematics as we know it...too funny.

BTW: Being an old fart I got 2 as the answer simply because the problem was written as a fractal to my tired old eyes.

Wow...some of you are going to end up in the psych ward if you keep on. Lol

Here's the deal...unfortunately I'm old enough to remember when the 1st handheld calculators came out and we were taught to solve equations using them(just standard, not the scientific calcs). In order to solve an equation with a regular calculator you had to use the star key(*) and make it explicit because they had no way of reading an equation or implied multiplication. You had to be able to read the equation using the orders of operation and enter it the best way you could on a calculator.

Anyway, pre calculators 48/2(9+3) was always 288. The TI 85 or any other calculator that gives you 2 is simply wrong. In order to get 2 as a answer your abandoning the orders of operation. You're saying it's 48 divided by 2 times 9 plus 3. You're using division as your 1st order. There's no math on the planet to justify that. The correct way is 9 plus 3, then 48 divided by 2, then 24 times 12.

I agree the expression could have been written differently to make it easier to read but that doesn't mean you can read it anyway you want just so the calculator can understand it.

The bottom line is like the title suggest; don't trust your calculator.

Arbitrageur

3mperorConstantinE
What the sources such as Wolfram et al. are talking about is:

2x/2x = 2*x/2*x = 2(x)/2(x)

You are repeating stuff I already said, but you appear to be mixing up Wolfram Alpha and APS. APS is not "Wolfram Alpha, etc". I wasn't citing the guidelines from Wolfram Alpha and APS as the same, because they aren't, but here you seem to be lumping them together with "etc" which I did not do and is either a misunderstanding, misdirection, or something like that.

Maybe you can clarify what you mean by referring to this without discussing Wolfram Aplha:
Physical Review Style and Notation Guide (pdf) p21

Are you saying following that would give you 288?

Yes, that's 100% what I'm saying.

I'll spell it out for you below...
I pointed out what "the sources such as Wolfram et alia" (translation: "Stephen and friends" not "etc") were really saying, because that was a reference you gave. I was making clear what THAT specific issue was about, which you seem to have misunderstood. Or else what was the point of your Wikipedia quotation?

Now. As written, you have:
48/2(9+3)
which is the same as:
48/2•(9+3)
You left the • out of the expression, but I don't see how you could possibly think that they're different, because they are not.

Now, while "proceed from left to right" is, as you correctly pointed out, merely a convention, what is not merely a convention is that multiplication AND division BOTH have what is called left-associativity (a/k/a "fixity").

Which means: 48/2•(9+3) =
(48/2)•12 = 288
NOT 48/(2•(12)) = 2

Left-associative operators work like so:
a/b/c = (a/b)/c
NOT a/b/c = a/(b/c)

Now, because multiplication and division are left-associative, when they are combined in the same expression they are interpreted like:
a/b•c/d•e = ((a/b)•(c/d))•e

So essentially your particular confusion stems from the fact that, by leaving the multiplication parenthetically implied, the left-associativity of the multiplication and division operators is obscured.

Once you make explicit what was implicit, and parenthesize the expression using the precedence and associativity properties of the operators, then it's pretty easy to see how to go from there.

Hope this helps,
—E

reply to post by 3mperorConstantinE

As both of us already pointed out (I think) APS journals only use implied multiplication expressions because the dot and "x" symbols are reserved for vector products, etc.

It seems to me like you're trying to perform the 48/2 division, before the multiplication.

If the rule is multiplication before division as in APS journals, you would do the division after the multiplication, wouldn't you? Or maybe you wouldn't, but that's how I interpreted their guidelines.

If you do the multiplication first, you multiply 2 times 12, so the denominator becomes 24. Then you do the division, so 48/24 = 2.

reply to post by Arbitrageur


While yes, by convention the multiplication is performed first, here in the specific case, there are some overriding conditions:
From the link I gave above, after noting that both division and multiplication are left-associative:

To prevent cases where operands would be associated with two operators, or no operator at all, operators with the same precedence must have the same associativity.

This means that you cannot do the multiplication first in this equation, because the /'s left-associativity has already "captured" the 2 (i.e. Which you can make explicit by adding parentheses; which is what associativity actually does)
(48/2)•(9+3) =
(48/2)•12 =
24•12 = 288
The fixity of the operators must always be respected.

If, due to this, an operation cannot be performed, then you move on to the next operation on the list, and then cycle back around, if that makes sense.
Now should you have wanted the equation to equal 2, remember it rests upon the denominator to clarify itself.

The APS refers to ambiguous denominators, here:

Use the solidus (/) or negative exponents for fractions in running text, and in displayed equations when this does not reduce clarity. When the extent of a denominator is ambiguous, use appropriate bracketing to ensure clarity.

Cheers,
~E.

3mperorConstantinE
reply to post by Arbitrageur

 

While yes, by convention the multiplication is performed first, here in the specific case, there are some overriding conditions:
From the link I gave above, after noting that both division and multiplication are left-associative:

To prevent cases where operands would be associated with two operators, or no operator at all, operators with the same precedence must have the same associativity.

This means that you cannot do the multiplication first in this equation

That refers to "operators with the same precedence", and if APS specifies multiplication before division, then those two operators (multiplication and division) don't have the same precedence, right?

The APS refers to ambiguous denominators, here:

Use the solidus (/) or negative exponents for fractions in running text, and in displayed equations when this does not reduce clarity. When the extent of a denominator is ambiguous, use appropriate bracketing to ensure clarity.

Right. I can't argue with that, since that was sort of the point of this thread.

Hmm... let me have a go at it...

48/2(9+3)

BEDMAS (Brackets, Exponents, Multiplication/Division(L to R), Addition/Subtraction(L to R).


48/2(12) or 48/2*12

This where things go wrong, remember those brackets are not true brackets... Multiplication/Division takes precedence according to whichever comes first.

This case Division comes first.

24(12) or 24*12.

Answer is 288.



I would say this question is asked in a bad way. If i was a teacher i would give check mark to both.

i remember my first calculator fondly. red LEDs magnified by little circular plastic lenses. oh such fun (....) to 'write' words such as 'hello' and 'leeds' by turning it upside down. alas it had zero scientific functions. did i trust it? to a degree. i certainly wouldn't have lent it money.

luciddream
I would say this question is asked in a bad way. If i was a teacher i would give check mark to both.

This Berkeley mathematician agrees with you on that point, as that's pretty much what he said:

Berkeley mathematician (Ring theorist) is asked if he would accept either 2 or 288 as the correct answer due to the ambiguity, and he says he would, as parentheses which would resolve the ambiguity are missing (He says that near the end):