I have been reading arguments about a simple math problem recently, and it is seriously frustrating me.
20/5(2*2)=
There is, for some reason, a large backing of people who state the solution is 1. BEDMAS would dictate that:
22=4 -----20/54
20/5=4 -----44*
4*4=16
How does the argument of assumed parenthesis after the division hold so much support? Why is BEDMAS so contested in this situation? What is the TRUE answer? (as much as I know the answer, I want affirmation damnit)
Because for many places multiplication is performed before division. Are you outside the US?
This isn’t a math question, it’s a language question.
It all depends on how you learned to interpret written math into Order of Operations
For me, the answer is 1. Implied multiplication, the type of which appears in “xy+4” or “5(x^2)” acts as parenthesis. If the author intended the latter half to be in the numerator, he’da written 20/5*****(2*2).
You must be outside the US, or you’d be writing PEMDAS, not BEDMAS. Regardless, notice that radicals and fraction bars aren’t included in the acronym. Neither are implied functions. It’s just the way it is.
Where would this be? That goes against PEMDAS/BEDMAS.
The OOO page on Wikipedia says that the only time where things would be interpreted outside of BEDMAS is in the physical sciences, which would support my interpretation (since this is not a physical science issue).
RE: the above post, 20/5(44) = 20/5(2*2). It is the exact same thing.
I’d like to know why the physical sciences use a different approach though, if anyone knows.
I’m assuming you meant “20/5(22) = 20/5(2*2)”
They aren’t the same thing. The former is 20 divided by 5*4, which is 1. The latter is 20 divided by 5, times 4, which is 16. The difference is that the 4 is in the numerator in one and denominator in the other.
I don’t think that your argument of the use of a * defining numerator or denominator is useful. No matter how you understand that interaction, you are multiplying 5 by 4.
How can you justify computing 5(4) first? If you do that first you are explicitly not using BEDMAS.
I larnt me my basic arithmetic in grammar school, circa 1958-1963, in the United States. The rule was:
[list=a][li] Evaluate any parenthesized group before combining that result with anything outside those parentheses.[/li][li] Perform all multiplications and divisions in order from left to right. Note that multiplications and division have equal precedence, neither of them having priority over the other![/li][li] Finally, perform all additions and subtractions in order from left to right. Again, note that addition and subtraction have equal precedence, neither having priority over the other.[/list][/li](Exponents were introduced later, having greater precedence than multiplication and division, and associating from right-to-left, unlike + – × ÷ which associate from left-to-right.)
I have an Associate of Science degree majoring in Math. Nowhere in all that education did I ever learn otherwise. And in particular, nowhere have I ever heard that “Implied multiplication, the type of which appears in “xy+4” or “5(x^2)” acts as parenthesis”; these two particular examples given by Chessic Sense don’t rely on any such rule anyway.
The original problem given by OP, to-wit:
20/5(22)=
is a weird way to write it. As given, I agree with OP’s analysis, that the answer is [del] 47 [/del] 16. But, being weirdly written, it would be most clearly written as either:
(20/5)(22)
or
20/[5(2*2)]
according to the author’s intention.
ETA: I’m getting some Spidey-tingly-like sense of deja vu here. Didn’t we discuss exactly this same arithmetic problem in a thread once before?
This is another facebook-style notation question, in which the main confusion comes from using ASCII text to approximate real math notation. On a chalkboard, the parenthetical would unambiguously be either in the denominator or next to the fraction.
In plaintext, I would assume this equates to 1, and would have written it (20/5)(22) had I meant for it to simplify to 16. (I would use parentheses in either case to make it clear. The "5(22)" to me necessarily implies 5*4, NOT necessarily (20/5)*4, so I would definitely clarify.)
You should use parentheses when writing expressions in such a way that you cannot be misunderstood. That’s what they are for. The fact that it is ambiguous in this case means the writer failed.
Dr[sup]3[/sup] and I may disagree about how the given expression should be evaluated, but we do seem to agree that it isn’t perfectly clear, and we agree on how it could be more clearly written.
Okay, I found some prior threads on the same topic (although the specific given problem is different in each):
6÷2(1+2)=? Math problem - Factual Questions - Straight Dope Message Board
6-1X0+2/2=? Math question (order of operations) - Factual Questions - Straight Dope Message Board
One can at best guess what the original writer of those symbols meant. Even though one can invoke a set of rules to come up with a definite answer, the elephant in the room is that the expression isn’t written in a very natural way. There are lots of ways to write any expression, but fluency with the language of math (like with any language) carries with it some expectations. This expression is out of the norm enough that the only right answer is to take pause and ask for clarification from whoever is trying to communicate with that jumble of symbols.
An English analogy: If someone had a son and a daughter and no other children, they could say to you “Can you pick my son’s sister up from soccer practice?” There’s nothing wrong with that sentence, and grammar rules would suggest that he means his daughter, but it’s such a non-standard way to say it that you would naturally ask for clarification.
There are multiple non-canonical aspects to the expression in the OP, and it would be odd to assume that the writer was strictly obeying a particular set of order-of-operations rules when he or she clearly ignored expectations for how that quantity should be written. (If it’s to equal 16, then it should be something like “(20/5)(22)” or maybe “(20/5)(22)". If it’s to equal 1, then it should be something like "20/(522)" or "20/(5(22))”. Just because you can make an interpretation of “20/5(2*2)” doesn’t mean you should. It’s just too far from a normal way to write it.)
I agree that in a perfect world this is a bad example of a problem. Thanks for finding older examples!
This seems odd to me. Why would you use parenthesis (20/5)(22) to denote the value of 16, when the parenthesis do not change the value if you were to follow BEDMAS? Of course 5(4) implies 54! BEDMAS still dictates that this results in 16 because 20/5 must be done before the 5*4. I would think that if you wanted to imply the answer was 1, you would add parenthesis to achieve that, rather than add parenthesis which do not change the equation as it is written.
Is the purpose of BEDMAS not to make it so that equations can not be misunderstood? BEDMAS applied properly ensures you always get the proper answer.
On a related topic, there is sometimes some discussion about how to best write that expression that, in modern usage, is commonly written
sin[sup]2[/sup] x
In some textbook or other, I saw a quote from one of the major math sages (I forget who; maybe Leibniz?) that
sin x[sup]2[/sup]
was a reasonable notation, on the grounds that the interpretation
sin (x[sup]2[/sup])
was unlikely ever to appear in any real-life application.
Seems like the mathematical purists won that argument, saying that sin x[sup]2[/sup] must be interpreted literally as sin (x[sup]2[/sup]) whether that case ever really arises or not. Thus, the modern notation of sin[sup]2[/sup] x evolved to be the accepted notation, winning out over the equally correct but more cumbersome and thus less common (sin x)[sup]2[/sup]
I think we should all begin by thoroughly purging our minds, and the world’s textbooks, of that abomination of a mnemonic PEDMAS, or BEDMAS, or BODMAS, or however you spell that. This acronym seems to say that we should do division before multiplication, and addition before subtraction.
That’s just plain flat-out wrong. No wonder our children am confuzzed.
In Spain it’s “whatever is in the parenthesis before anything else”, but with the formula as originally given we’d still run into that issue of “where is that parenthesis supposed to be”. We also learned that it was better to pile up the parenthesis (using different symbols, preferably) than to leave a formula unclear. There was no precedence between multiplication and division, nor between addition and substraction. Since multiplication-division and addition-substraction are inverse commutative pairs (division is multiplication by the inverse, substraction is addition of a negative value), you should get the same result no matter which of the two you begin with - if you don’t, you’ve made a mistake.
This same week I had to ask a Swedish customer to clarify one of his units, because as written it wasn’t clear whether its dimentions included T[sup]-2[/sup] or T[sup]2[/sup]. The solution came in the form of a parenthesis showing that the “*s[sup]2[/sup]” was part of the denominator.
You keep going on about BEDMAS. Why do you assume the author is using it, and why do you assume the reader is? Further, why do you assume that either of them knew the other was using it?
BEDMAS is not universal. In fact, I, for one, didn’t even know it existed until a few years ago and I still think it’s as strange as saying “Why do you think this?” or “I’m off to phone me mum.”
So let me get this straight: In the expression “F/ma=1,” you think the a is in the numerator? That is, you think it has a positive exponent?
Meant to add this to my latest post above but missed edit window:
ETA: BTW, nowhere in all my years of math education (1st grade through lower-division college math with DiffEq) did I evah hear or see any variant of that mnemonic. And now that I have seen it, I’m glad I never did before now.
“Dolly Madison Says Compare” (or “Daughter, Mother, Sister, Brother; Divide, Multiply, Subtract, Bring Down”) are okay with me (although each one seems to leave out a step that the other includes).
I’m also glad I never got taught SOH-CAH-TOA which I would have had as much trouble remembering as I would have had simply, you know, memorizing the actual definitions.
Yes. Just like the purpose of the USSC is to resolve unambiguously and for all time the meaning, intent and application of the US constitution.
And just like the analogy, it performs decently a lot of the time, but there’s always people setting out to bend it until it breaks. 
Point is, you’ll probably never get utterly perfect notation rules because the better they are, the more determined people will be to find a way to screw them up. For a more mathematical analogy, I suppose it’s a bit like the 18-19th century attempts to rigorously define the idea of a ‘function’. Every time someone thinks they’ve nailed it, along comes some smartass like Weierstrass. In fact now that I think of it, mathematics must be one of the few professions in which trolling is not only tolerated but actually insisted upon.
Getting back to your problem, the best way imo to resolve the ambiguity is to use a horizontal fraction line (apparently a ‘vinculum’, as opposed to a ‘solidus’… huh). Those are really good at making really clear what’s on the top and what’s on the bottom. Of course you can’t use plain ASCII, have to go to LaTeX or something, but the takeaway lesson is that ASCII is not for maths.
I don’t mean BEDMAS as in the actual acronym, I mean the proper OOO. The OOO is universal, is it not? I was only saying BEDMAS as a simple way to refer to order of operation. I would like to assume that all parties involved have an understanding of the order of operations. If they didn’t, the whole conversation becomes meaningless.
As I noted above, physical sciences are the one field which specifically and intentionally go against the standard OOO in this way. If this were a physical science question rather than a simple math one, you may have a point in your f/ma comment. I still want to know why this is so… Anyone?
No it’s not, see my previous post. My math teachers would have a collective apoplexy if they encountered a student who thought you need to perform division before multiplication or addition before substraction.