6÷2(1+2)
6÷2(1+2)
I wrote a (very long) blog post about those viral math problems and am looking for feedback, especially from people who are not convinced that the problem is ambiguous.
It's about a 30min read so thank you in advance if you really take the time to read it, but I think it's worth it if you joined such discussions in the past, but I'm probably biased because I wrote it :)
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You lost me on the section when you started going into different calculators, but I read the rest of the post. Well written even if I ultimately disagree!
The reason imo there is ambiguity with these math problems is bad/outdated teaching. The way I was taught pemdas, you always do the left-most operations first, while otherwise still following the ordering.
Doing this for 6÷2(1+2), there is no ambiguity that the answer is 9. You do your parentheses first as always, 6÷2(3), and then since division and multiplication are equal in ordering weight, you do the division first because it's the left most operation, leaving us 3(3), which is of course 9.
If someone wrote this equation with the intention that the answer is 1, they wrote the equation wrong, simple as that.
The calculator section is actually pretty important, because it shows how there is no consensus. Sharp is especially interesting with respect to your comment because all scientific Sharp calculators say it's 1. For all the other brands for hardware calculators there are roughly 50:50 with saying 1 and 9.
So I'm not sure if you are suggesting that thousands of experts and hundreds of engineers at Casio, Texas Instruments, HP and Sharp got it wrong and you got it right?
There really is no agreed upon standard even amongst experts.
Hi, expert here, calculators have nothing to do with it. There's an agreed upon "Order of Operations" that we teach to kids, and there's a mutual agreement that it's only approximately correct. Calculators have to pick an explicit parsing algorithm, humans don't have to and so they don't. I don't look to a dictionary to tell me what I mean when I speak to another human.
Thanks for putting my thoughts into words, that's exactly why I hate math. It was supposed to be the logical one, but since it only needs to be parsed by humans it failed at even that. It's just conventions upon conventions to the point where it's notably different from one teacher/professor to the next.
I guess you can tell why I went into comp-sci (and also why I'm struggling there too)
there’s a mutual agreement that it’s only approximately correct.
No there isn't. I've never seen a single Year 7-8 Maths textbook that is in the slightest bit ambiguous about it. The Distributive Law has to literally always be applied (hence why it's a law). dotnet.social/@SmartmanApps/110819283738912144
The order of operations is not the same as the distributive law.
The first step in order of operations is solve brackets. The first step in solving unexpanded brackets is to expand them. i.e. The Distributive Law. i.e. the ONLY time The Distributive Law ISN'T part of order of operations is when there's no unexpanded brackets in the expression.
The distributive law has nothing to do with brackets.
The distributive law can be written in PEMDAS as a(b+c) = ab + ac, or PEASMD as ab+c = (ab)+(ac). It has no relation to the notation in which it is expressed, and brackets are purely notational.
The distributive law has nothing to do with brackets
BWAHAHAHA! Ok then, what EXACTLY does it relate to, if not brackets? Note that I'm talking about The Distributive LAW - which is about expanding brackets - not the Distributive PROPERTY.
a(b+c) = ab + ac
a(b+c)=(ab+ac) actually - that's one of the common mistakes that people are making. You can't remove brackets unless there's only 1 term left inside, and ab+ac is 2 terms.
ab+c = (ab)+(ac)
No, never. ab+c is 2 terms with no further simplification possible. From there all that's left is addition (once you know what ab and c are equal to).
brackets are purely notational
Yep, they're a grouping symbol. Terms are separated by operators and joined by grouping symbols.
You are off your meds
Noted that you were unable to tell me what The Distributive Law relates to (given your claim it's not brackets).
You are unhinged
it shows how there is no consensus
Used to not be. Except for Texas Instruments all the others reverted to doing it correctly now - I have no idea why Texas Instruments persists with doing it wrong. As you noted, Sharp has always done it correctly.
There really is no agreed upon standard even amongst experts
Yes there is. It's taught in literally every Year 7-8 Maths textbook (but apparently Texas Instruments don't care about that).
No, those companies aren't wrong, but they're not entirely right either. The answer to "6 ÷ 2(1+2)" is 1 on those calculators because that is a badly written equation and you(not literally you, to be clear) should feel bad for writing it, and the calculators can't handle it with their rigid hardcoded logic. The ones that do give the correct answer of 9 on that equation will get other equations wrong that it shouldn't be, again because the logic is hardcoded.
That doesn't change the fact that that equation worked out on paper is absolutely 9 based on modern rules of math. Calculate the parentheses first, you then have 6 ÷ 2(3). We could solve from here, but to make the point extra clear I'm going to actually expand this out to explicit multiplication. "2(3)" is the same as "2 x 3", so we can rewrite the equation as "6 ÷ 2 x 3". All operators now inarguably have equal precedence, which means the only factor left in which order to do the operations is left to right, and thus division first. The answer can only be 9.
If you'd ever taken any advanced math, you'd see that the answer is 1 all day. The implicit multiplication is done before the division because anyone taking advanced math would see 2(1+2) as a term that must be resolved first. The answer still lies in the ambiguity of the way the problem is written though. If the author used fractions instead of that stupid division symbol, there would be no ambiguity. It's either 6/2 x 3 = 9 or [6/(2x3)] = 1. Comment formatting aside, if someone put 6 in the numerator, and then did or did NOT put all the rest in the denominator underneath a horizontal bar, it would be obvious.
TL;DR It's still a formatting issue, but 9 is definitely not the clear and only answer.
If you’d ever taken any advanced math, you’d see that the answer is 1 all day
Don't need to do advanced Maths - every rule you need to know for this problem is taught in Year 7.
"Always remember to solve using PEMDAS once you've used the distributive property!" Link%20and%20subtraction%20(S).)
(emphasis mine)
And...? Not sure what your point is, but the link is VERY badly worded...
- The Distributive Law and The Distributive Property aren't the same thing - he's applying The Distributive Law, but mistakenly calling it The Distributive Property (a lot of people make that mistake). The latter is merely a property in Maths (like the commutative property, the associative property, etc.), the former an actual rule of Maths The Distributive Law
- Applying the Distributive Law - i.e. expanding brackets/parentheses - is part of solving brackets. i.e. the first step in BEDMAS/PEMDAS. There's no "once you've used", you've already started!
- As I already said, this is taught in Year 7, so I'm not sure what your point is?
That you're still wrong? As I said, the true answer is that the problem is written poorly due to the obelus and thus is open to interpretation. You're entitled to your own interpretation since it's written poorly, I just find it pretty obviously less logical than multiplying using the distributive property first to resolve the term with the parentheses fully as you would in any advanced math.
Also, distributive law and distributive property are the same thing per Khan academy "The distributive property is sometimes called the distributive law of multiplication and division."
Wait till you hear that "i before e except after c" wasn't true either. It's wild that you think 7th grade math overrules grad school math though lol.
That you’re still wrong?
About? You haven't pointed out anything that's wrong.
the problem is written poorly due to the obelus and thus is open to interpretation
Oh, you're one of those people. Good, maybe we can finally get an answer then (this was also talked about in the blog). What other interpretation of an obelus is possible other than division? People keep saying it's ambiguous, but no-one has ever said why (other than some stuff that makes no sense in the context, as explained in the blog)
The distributive property is sometimes called the distributive law of multiplication and division
Yes, and sometimes people call Koalas "Koala bears", but that doesn't mean they're bears. Now bearing that in mind, read again what Khan said - the page which is called "Distributive property explained", not "Distributive Law explained".
Wait till you hear that “i before e except after c” wasn’t true either
Wait till you hear that's not a rule of Maths.
It’s wild that you think 7th grade math overrules grad school math though
Umm, never said anything of the kind...
The answer still lies in the ambiguity of the way the problem is written though
But it's not ambiguous, as per the reason you already gave.
If the author used fractions instead of that stupid division symbol
If you use fractions then the whole thing is a single term, if you use division it's 2 terms.
9 is definitely not the clear and only answer
1 is definitely the only answer.
"The obelus is treated differently,” Church said. "It could mean ratios, division or numerator and denominator, and these all tweak the meaning of the symbol.”
This is the only symbols I've ever seen used (but feel free to provide a reference if you know of any where it isn't - the article hasn't provided any references)...
Ratio is only ever colon.
Division is obelus (textbooks/computers) or slash (computers, though if it's text you can use a Unicode obelus).
Fraction is fraction bar (textbooks) or obelus/slash inside brackets (computers). i.e. (a/b).
those calculators because that is a badly written equation
It's not badly written, and the reason Texas Instruments gets it wrong is right there in their manual (disobeys The Distributive Law).
modern rules of math
The order of operations rules haven't changed in at least 100 years, and more likely at least 400 years. Don't listen to Youtubers who can't cite a single Maths textbook.
“2(3)” is the same as “2 x 3”
No, it's the same as (2x3), as per The Distributive Law and Terms.
There has apparently been historical disagreement over whether 6÷2(3) is equivalent to 6÷2x3
As a logician instead of a mathemetician, the answer is "they're both wrong because they have proven themselves ambiguous". Of course, my answer would be RPN to be a jerk or just have more parens to be a programmer
There has apparently been historical disagreement over whether 6÷2(3) is equivalent to 6÷2x3
No, there hasn't - that's a false claim by a Youtuber (and others who repeated it) - it is equal to 6÷(2x3) as per The Distributive Law and Terms, and even as per the letter he quoted! Here is where I debunked that claim.
Are you referring to Presh Talwalkar or someone else? How about his reference for historical use, Elizabeth Brown Davis? He also references a Slate article by Tara Haelle. I've heard Presh respond to people in the past over questions like this, and I'd love to hear his take on such a debunking. I have a lot of respect for him.
Your "debunk" link seems to debunk a clear rule-change in 1917. I wouldn't disagree with that. I've never heard the variant where there was a clear change in 1917. Instead, it seems there was historical vagueness until the rules we now accept were slowly consolidated. Which actually makes sense.
The Distributive Law obviously applies, but I'm seeing references that would still assert that (6÷2) could at one time have been the portion multiplied with the (3).
And again, from logic I come from a place of avoiding ambiguity. When there is a controversiallly ambiguous form and an undeniablely unambiguous form, the undeniably unambiguous form is preferable.
Yes, the guy who should mind his own business.
How about his reference for historical use
Are you talking about his reference to Lennes' letter? Lennes' letter actually completely contradicts his claim that it ever meant anything different.
Elizabeth Brown Davis
Haven't seen that one. Do you have a link?
He also references a Slate article by
...a journalist. The article ALSO ignores The Distributive Law and Terms.
I wouldn’t disagree with that.
Thank you. And also thank you for being the first person to engage in a proper conversation about it here.
I’ve heard Presh respond to people in the past over questions like this
I've seen him respond to people who agree with him. People who tell him he's wrong he usually ignores. When he DOES respond to them he simply says "The Distributive Property doesn't apply". We're talking about The Distributive LAW, NOT the Distributive Property. It's called "law" for a reason. i.e. ALWAYS applies. I've only ever seen him completely unwilling to engage in any conversation with anyone who points out he's wrong (contradicting his claim that he "welcomes debate").
I have a lot of respect for him
Really?? Why's that? I'm genuinely curious.
I’ve never heard the variant where there was a clear change in 1917
Me either. As far as I can tell it's just people parroting his misinterpretation of Lennes' letter.
Instead, it seems there was historical vagueness until the rules we now accept were slowly consolidated
I can't agree with that. Lennes' letter shows the same rules in 1917 as we use now. Cajori says the order of operations rules are at least 400 years old, and I have no reason to suspect they changed at all during that time period either. They're all a natural consequence of the way we have defined the symbols in the first place.
The Distributive Law obviously applies
Again, thank you.
I’m seeing references that would still assert that (6÷2) could at one time have been the portion multiplied with the (3)
If it was written (6÷2)(1+2), absolutely that is the correct thing to do (expanding brackets), but not if it's written 6÷2(1+2). If you mean the latter then I've never seen that - links?
leaving us 3(3)
You just did division before brackets, which violates order of operations rules. 6÷2(3)=6÷(2x3)=6÷6=1