Math Problem: 1 or 16?

[bostonman]

I don't understand, the article states it depends on which method a person is taught (BODMAS or PEMDAS) will depend on the answer.

How can the order of operations have changed from division first to multiplication first (or vice versa)? The Internet isn't "divided" because the answer is 16; and confirmed by Excel.

So how can people be taught two different methods thus changing the order of operations? This would mean math problems would have different answers.

https://www.inc.com/dave-kerpen/this-basic-math-problem-is-breaking-internet.html

[awallin]

that notation is just ambiguous - there's no real math question here...

[Rerouter]

8 / 2 * (2 + 2) =?

Both methods describe division / multiplication to be interchangeable, and to solve them in left to right order when not otherwise marked out, it would only be if you forgot the rul for the PEDMAS acronym, that MD and AS are local groupings treated left to right, not a fixed order.

so

8 / 2 * (4)
4 * (4)
16

[ciccio]

I really disagree.

I don't know what BODMAS or PEMDAS mean (I'm too old for it) , but my anser is 1 or maybe 8, but please ask a correct question.

The original question is badly written. There should be more brackets... Missing them, it can be read in many ways:
 -  8/(2*(2+2)) = 8/(2*4) = 8/8 =1  (this was my first answer) or
 -  (8/2) * (2+2) = 4 * 4 = 16

I'm open to any explanation about the second  being correct one and the first wrong.
I could not have an answer from Excel, my copy (2003) cannot solve the question by writing the equation in a single cell, but maybe it's my fault...

edit: I've read the article quoted by the OP, and I find interesting that the majority of readers gave the wrong (according to the author ) answer.
This coud open  a Pandora box of discussions, so I will not go further...

Best regards

[Nominal Animal]

confirmed by Excel

 

[mathsquid]

 

8 / 2 * (2 + 2) =?
...

8 / 2 * (4)
4 * (4)
16

Exactly. This is unambiguous, but horribly written. It reminds me of the Obfuscated C contest. https://www.ioccc.org

[nixxon]

The top answer seems reasonable:
https://www.quora.com/Should-we-follow-BODMAS-or-PEMDAS-in-mathematical-calculations-One-says-to-do-multiplication-first-and-then-division-and-the-other-says-the-reverse-To-evaluate-an-expression-like-6-2*3-which-set-of-rules-are-to-be-followed

Only division/addition is the problem. Then left to right solving seems sensible.

[Nominal Animal]

It is ambiguous.  It is exactly the same thing as whether 1/2x is equal to (1/2)x or 1/(2x).  There is no universal rule which one is correct (even if you happen to have been taught one, others have been taught a different one, and the rest of us that the order depends on the context).

The correct answer is that the stated math problem is underspecified: no answer exists unless the exact precedence of multiplication and division is specified. The easiest, and most common, way to do that is to add one set of parentheses to remove the ambiquity.

And before you ask, no, (1/2)x is not always equal to 1/(2x).  For example, when x = i,  i.e. x² = -1, (1/2)x = -1/(2x).

[Mr. Scram]

What many people don't realize is that BODMAS and PEMDAS can't be mathematically proven or constructed. They're taught their local version as gospel in school and too much weight gets attached to. In reality it's just agreed upon shorthand to imply absent information. People get confused and subsequently annoyed when they realize what they've learnt as truth is arbitrary and not an intrinsic part of mathematics and that's why you see these discussions. As is so often the case it's not a matter of one being right or the other. It's a matter of having a badly defined and constrained problem and therefore more than one valid output. A few brackets would have eliminated any ambiguity and nipped any discussion in the bud.

[ajb]

How can the order of operations have changed from division first to multiplication first (or vice versa)? The Internet isn't "divided" because the answer is 16; and confirmed by Excel.

Order of operations isn't a mathematical law, it's a convention for interpreting expressions.  A much simpler convention would have been to just evaluate left to right, but for whatever reason the dominant convention became that you always multiply or divide before adding or subtracting. That's why 2+3*4 is conventionally 12 instead of 20. The fact that you can use brackets to override the standard order of operations is only possible because the standard way is arbitrary to begin with.

I didn't realize that there were two popular conventions that placed multiplication and division in different orders. I vaguely recall being taught that those were supposed to be done left-to-right, but maybe my elementary school education and programming experience are just voting together at this point.

[SparkyFX]

8 / 2 * (2 + 2) =?

it is either
      8     
2 * (2+2)

or

8
--- * (2+2)
2

No idea what BODMAS or PEMDAS might mean, but a slash is an improper division line here, and thats the whole "problem". Which means whoever wrote it needs to clarify what he or she means.

[SiliconWizard]

Yeah it's absolutely not a math problem and is not breaking anything except asses who think that pressing keys on a calculator is doing math.

Good lord... 

[Mr. Scram]

Yeah it's absolutely not a math problem and is not breaking anything except asses who think that pressing keys on a calculator is doing math.

Good lord... 

Ask an ambiguous question everyone has an ill informed opinion about and profit. I've seen multiple versions of this go around and nothing is breaking the internet. It's just simple clickbait to incentivize traffic. I guess we fell for it too although I'm reassured by the amount of sense in this thread.

[NivagSwerdna]

I just feel sorry for people who don't represent everything in RPN and for cases where RPN isn't available I fallback on what I was taught in my BCPL lecture!

[Nominal Animal]

I wouldn't say everyone has an "ill informed opinion"; I think it is more about human tendency to seek authority or authoritative answers rather than accept ambiguity.

For example, if you check the history of bugs in Excel, you know it cannot be trusted as an authority for, well, anything. Or, whether colour, or color, is the correct spelling.  Or whether one should use ß or ss.  Or how many letters are there in the alphabet.

Even mathematical notation is just a matter of agreement.  We have several, including reverse Polish (RPN) or postfix notation; no single one is more correct than any other.  (Some are less ambiguous than others, however; postfix notation being among the most unambiguous ones.)

In fact, what most people call math, is itself a system based on root axioms which we must accept as "truths" for it to make sense; other mathematics are equally possible.

[tggzzz]

I just feel sorry for people who don't represent everything in RPN and for cases where RPN isn't available I fallback on what I was taught in my BCPL lecture!

BCPL? Pah.

Importantly neither BCPL nor ALP is maths, but use APL to evaluate

      7
7

      7-7
0

      7-7-7
7

      7-7-7-7
0

https://tryapl.org/

[schmitt trigger]

The New York Times has an article written by a mathematician about this debate:

https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html?nl=todaysheadlines&emc=edit_th_190803?campaign_id=2&instance_id=11317&segment_id=15843&user_id=10ca517e806ee8b2b6c2d219d309c7d0&regi_id=371077490803

[SiliconWizard]

Again that's a problem of people thinking that writing a succession of digits and operators like they would press them on a typical calculator is writing correct algebraic expressions. It's not.

Learning how to use a calculator, be it in algebraic or RPN mode, is not learning math, and it doesn't define correct math notation either.

Now obviously every time people don't have easy access to a proper math notation tool for publishing an algebraic expression, they can fall into the trap. The way to go is pretty simple though, every time the computer-like (pure "ASCII" we could say) notation could be ambiguous compared to the proper math notation, use parentheses. And don't try to be overly clever about the use of operator precedence, because in many cases it won't end up clever at all.

But sure this kind of clickbait "problems" are ubiquitous on the internet, it goes from simple math stuff like here, to erroneous "proof" that 1=-1, to ambiguously perceived colors on some ugly pictures, etc.
The list is endless. It does create huge traffic because it all works on the same principle: it drives people into a competitive mode in which everyone will be dead sure they are right and, the other side, wrong, regardless of the relevance of the question.

[soldar]

Didn't we do this topic quite recently?

There is no right or wrong. The order of operations is a convention. Whatever is the convention being used that is the right answer.

[IanB]

Some considerable discussion here:

https://youtu.be/vaitsBUyiNQ

Apparently the conventional answer would have been different 100 years ago.

What is wrong with this question is it asks people to apply rules defined by other people. If you follow the rules you are "right", and if you don't follow the rules you are "wrong". This divides people into two camps: rule followers or free thinkers. So this question is actually not a math question, it is a social question.

[tggzzz]

Some considerable discussion here:

https://youtu.be/vaitsBUyiNQ

Apparently the conventional answer would have been different 100 years ago.

What is wrong with this question is it asks people to apply rules defined by other people. If you follow the rules you are "right", and if you don't follow the rules you are "wrong". This divides people into two camps: rule followers or free thinkers. So this question is actually not a math question, it is a social question.

Thank heavens for that. I knew I was right to think the answer is 42.

[tggzzz]

But sure this kind of clickbait "problems" are ubiquitous on the internet, it goes from simple math stuff like here, to erroneous "proof" that 1=-1,

Here's nice proof that 1=-1. It doesn't involve "hidden" zeros, most people can correctly state where the error occurs, but too few can say what the error actually is

now 1: sqrt(-1) = sqrt(-1)
therefore 2: sqrt(-1/1) = sqrt(1/-1)
therefore 3: sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)
therefore 4: sqrt(-1)*sqrt(-1) = sqrt(1)*sqrt(1)
therefore 5: -1 = 1
Q.E.D.

EDIT: correct that in the light of IanB's correct observations. I am annoyed (at myself)

[Nusa]

Multiple issues here. Note that the original posters linked example is "8 / 2(2+2)"

First is that many people are mis-using the mnemonic they may have learned, assuming they ever learned it correctly. The PEMDAS, BODMAS, BIDMAS and similar mnemonics actually all agree with each other. It's just that some have forgotten that multiplication and division are grouped as equals, as are addition and subtraction.
PEMDAS = Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction.
BODMAS = Brackets, Order, Division and Multiplication, Addition and Subtraction
BIDMAS = Brackets, Indicies, Division and Multiplication, Addition and Subtraction
Interestingly, most people that have trouble treating division and multiplication as equals don't have that same problem with addition and subtraction. For instance, does 10-5+2 equal 7 or 3?

Second is that there is another issue in the original example, which is whether implied multiplication has a higher precedence than normal multiplication. Namely, is 8/2x the same as 8/2*x? Even some professionals have disagreed on this one. It's even better to use brackets for clarity.

[HighVoltage]

I am more confused about the fact that this little math question confused people !

[Mr. Scram]

I am more confused about the fact that this little math question confused people !

That's why it's fertile clickbait. The other camp can't imagine the confusion either but has another solution and thus a holy war is born. Note that quoting Excel, calculators or Wolfram Alpha doesn't prove a point either way for reasons already discussed.