# Elementary Mathematical Equation

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• #2179447
ujm
Participant

Since one poster challenged another on elementary math, perhaps we can all participate in the following simple mathematical equation by answering it.

Please don’t show how you reached your equation for at least 24 hours after this thread was posted, in order to allow as many participants to answer without seeing another’s work.

No using calculators, computers or other aids (unless you refrain from answering the mathematical equation). This is a really simple question.

And here it is:

8÷2(2+2) = ?

#2179463

why not start with an easier challenge: simply find the smallest positive integer solution of x^n + y^n + z^n for n>2? I think I saw an answer on the margins of some old sefer, but can’t locate it right now.

#2179467
Reb Eliezer
Participant

Mods: In order to follow the above request which I did not see, don’t post my previous reply.

#2179478

Come on please post some harder questions this is fourth grade level

#2179479
Participant
Participant

16

#2179509

BIDMAS

#2179510
Zetruth
Participant

#2179571
ubiquitin
Participant

We have come full circle.

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 – 1 + 1 + 1 + 1 + 1 + 1 + 1 x 0 = ?

I found it fascinating that even after the correct answer (with detailed explanation) ws pointed out to people they still insisted that there was more than one correct approach.

I love this place

Of course in this case the answer is 16

#2179602
midwesterner
Participant

23.7

#2179616
Ploni Almoni18
Participant

Original equation:
8÷2(2+2) = ?
Parenthesis first
8÷2(4)
Which is another way of saying:
8÷2X4= ?
And then you just go in order from left to right
4X4
16

#2179622
Yserbius123
Participant

These questions are all kind of silly, because the way it was written is ambiguous. It’s like if I asked you “I met to friends, Chaim, and Shmiel, and he gave an apple to him. Who gave who an apple?” In academic and science writing, the author would make sure to either surround everything with multiple sets of parenthesis, or split it into smaller equations so that there’s only one way to read it.

There is a way of writing math problems called “Polish Notation” where you put the operand at the beginning of the problem. It’s how computer code works behind the scenes actually. It has the advantage of not being ambiguous when it comes to writing it out. So “2 + 2” becomes “+ 2 2” and “(6 – 4) x 3” becomes “x – 6 4 3”.

Using Polish Notation, this problem can be either “÷ x + 8 2 2 2” or “÷ 8 x 2 + 2 2”
8÷2(2+2)

#2179654
Quayboardwarrior
Participant

Something something the DemonRats

#2179607
zen3344
Participant

The answer is 16. It has always been 16 and will continue to be 16.

#2179671
Dr. Pepper
Participant

Do you remember the one about FLT from August of 2010?

It’s hard to follow since the conversation was taking place in two different threads so it’s not continuous.

#2179681
Reb Eliezer
Participant

I say it is 1. 8÷2(2+2) = 8 / [2 x (2 + 2)] = 8 / (2 x 4) = 8 / 8 = 1 According to rules of math of precedence multiplication and division come before addition and subtraction. It is not 8/2 x (2+2) = 4 x 4 = 16

#2179724

you can also take a practice SAT (math or English) on khanacademy and post your result here if you want the kavod when you correct others.

#2179725

if you _really_ not sure of the answer – copy it into google

#2179735
FlamingBaalTeshuvah
Participant

@ujm thanks for the post, I was pleasantly surprised to enjoy it. You are either aware of the pitfalls in this and testing us, or were genuinely intrigued by the debate it can potentially engender.

I’d like to post (what I believe is) *the* definitive answer. So no other answers need be given following this. Only your thanks, and praises are necessary.

8÷2(2+2) = ?
This question seems simple-enough. And anyone who proudly quotes their “PEMDAS” or “BODMAS” will quickly lament those whom they think don’t even know that much.
But, to such a potentially over-confident person, there are two potential pitfalls that one needs to be aware of:

Pitfall #1: PEMDAS doesn’t mean PEMDAS
Just because the “M” is written before the “D” (or vice versa for BODMAS (which proves this point)), doesn’t necessarily mean that M[ultiplication] is performed before D[ivision]. They, in fact, hold equal priority, and so those two operations must be executed from left to right.
So in 8/2*4, you do the division first.

Pitfall #2: 10y doesn’t mean 10*y
In 8÷2(2+2), there is no symbol given for the multiplication, just the juxtaposition of the “2” to the parentheses. By convention that is, and usually can, be regarded as a shorthand for multiplication. So when we see 10y in a formula, we can rightly treat it as 10*y. But, since these two are co-dependent they should more correctly be written with parentheses around them, so 10y is really (10*y) within parenthesis. This would make a difference to something like this example: 20/10y = 20/(10*y).

So too in the problem given, 8÷2(2+2), it is natural to want to treat this as 8÷(2*4).
BUT, strictly speaking, there is only this juxtaposition between a constant coefficient and a variable, never a constant and another constant.
So, the 2(2+2) of the question is just 2×4 – without parentheses – and not (2×4) – as a single unit.

So, in summary:
8÷2(2+2) is more rightly:
8÷2*(2+2), which =
8÷2*(4)
And since PEMDAS doesn’t mean “M” before “D”, but rather apply them left-to-right, it =
4*4
=16

That said, the faulty is really in the question; it was technically incorrect to write 2(2+2) without a symbol between them. So if someone went along with the questioner’s mistake at treated it as 8÷(2*4) and said the answer is 1, I would fault the questioner, not the one giving the answer. (Unless they made a mistake in Pitfall #1, in which case I would blame their school teacher.)

Now that I have given the definitive answer to this question, I would like to also state, for the record, that Trump isn’t as bad as some people think. Even his middos aren’t as despicable as some believe. Those that do think so are the kind of idiots who are swayed by headlines and can’t read the body of an article, or certainly can’t divorce themselves from their pre-conceived perceptions formed upon reading the misleading headlines. He absolutely didn’t lead and insurrection, not that J6 was anything remotely close to an insurrection. And those on this forum who are quick to criticize the frum yidden, are low-lives, descendants of Lavan haRosha.

Wishing you all a chag kasher v’sameach.

#2179754
amiricanyeshivish
Participant

FlamingBT

About the critisizim of frum yidden I 100% agree.
It is dispicable and nausiating to see how some frum yidden are willing to publicly smear another frum yid in public. And of course it’s only leshaim shomayim.
And about my spelling see above regarding math.

#2179757
ubiquitin
Participant

Flaming (and Yserbius)

you said “BUT, strictly speaking, there is only this juxtaposition between a constant coefficient and a variable, never a constant and another constant.”
(Yserbius didn’t say exactly that but implication is the same “These questions are all kind of silly, because the way it was written is ambiguous. “)

I was under the impression that 2×2; 2*2; (2)(2); 2(2) are all equally valid ways of expressing the same thing, namely: two multiplied by two.
In which case there is absolutely no ambiguity and nothing remotely incorrect about the OP’s problem. (assuming you recall how to perform order of operations)

can you provide a source for your assertion that the latter two equations above are ” technically incorrect”
(FWIW google had no problem solving them which seems odd for an equation that is not written correctly)

#2179784
@fakenews
Participant

I only discussed second or third grade level math/maths (once we’re getting into thorny debates nobody can really win) you brought this to at least third grade honors…

#2179870
ubiquitin
Participant

Fake

no thorny debate.

You made an assertion. Is it correct? I like learning new things .

If you can provide a source please supply it.
If you vaguely remember hearing it but aren’t exactly sure where or when, say that
If you just made it up say that

#2179894
Reb Eliezer
Participant

#2179900

Maybe, due to the lack of the multiplication sign, the answer is – incorrect syntax?

#2179917
frumnotyeshivish
Participant

#2179919
Yserbius123
Participant

@ubiquitin I don’t have a source. PEMDAS is a convention, not a hard and fast rule. The ÷ combined with the “2(” is really what’s throwing the equation since in general when you’re using a more shorthand way of writing it without multiplication signs, you would also put the numerator above the denominator instead of using a division sign. So there’s no real right answer since the notation is odd and there are different ways of reading it depending on who you ask.

#2179921
Dr. Pepper
Participant

@ujm

I’m currently auditing the calculation of a financial product and part of a formula in the documentation is given as a/b/c. Is it (a/b)/c or a/(b/c)?

20/10/5 = ?
(20/10)/5 = .4
20/(10/5) = 10

#2179960
ubiquitin
Participant

“PEMDAS is a convention, not a hard and fast rule.”

In order to have meaningful symbolic math we all have to use the same conventions, thus making them hard and fast rules.

Otherwise suppose I say + means divide or that 5 means take the square root , or when unsure just take the last number convert it to a letter and reply with a bird starting with that letter

In order for math to work we all have to agree what to do first when faced with say: 1 + 2 × 3

If there is no “rule” then 9, 7 or for that matter 11, 5.4, 6.3333456 and chicken all become equally correct depending on the individual’s “convention”

#2180053
Dr. Pepper
Participant

@ubiquitin

I agree that we all need to be on the same convention but who has the authority to make this convention and what happens if someone doesn’t follow it? (I.e. what are the consequences going to be? Can an architect be sued if the builder uses a different convention and the building collapses?)

(When documenting a process, I personally put in parentheses very generously to avoid any ambiguity.)

#2180051
emes nisht sheker
Participant

Reading the comments here amaze me how convoluted people are in their thinking.

Very simply… mathematical formulas have to follow “grammar” rules that the person writing and the person reading both agree on or understand.

So 8/2(2+2) depending on which system of “grammar” you are using will get you one of two results.

1. Following the system where multiplication is deemed a higher order operation than division:
(8)/(2(2+2)) = 1

2. Following the system where multiplication and division are deemed equal in order of operations you just compute left to right with this result:
(8/2)(2+2) = 16

Neither answer is more correct or better, they just reflect a different set of rules for parsing the equation.

#2180057
ubiquitin
Participant

Dr P

“but who has the authority to make this convention”

I’m not sure. but the fact that every math textbook says same is a good indication.

” and what happens if someone doesn’t follow it? (I.e. what are the consequences going to be? Can an architect be sued if the builder uses a different convention and the building collapses?)”

I would think for sure. If Architect said using my convention 3 + 2 = 6, or 1 + 2 × 3 = 9 . Of course he could be sued. neither of those are correct.

(When documenting a process, I personally put in parentheses very generously to avoid any ambiguity.)

emes

“1. Following the system where multiplication is deemed a higher order operation than division:… Neither answer is more correct or better, they just reflect a different set of rules for parsing the equation.”

Can you provide a source for this system? I have never heard of it (except from people who misremembered/ never learnt PEMDAS)

“Reading the comments here amaze me how convoluted people are in their thinking.”

Same!

This very topic amazed me when I first encountered it. I still find it fascinating

#2180103
Yserbius123
Participant

@ubiquitin @emes-nisht-sheker You would be surprised how much of math is just conventional rather than a hard and fast grammar. Anyway, like I’ve demonstrated before, just like a sentence can be perfectly grammatically correct and still be interpreted in two completely different ways, so too can math. Which is why when mathematicians write out their equations, they do it in a way that makes it unambiguous.

#2180108
Curiosity
Participant

The smallest positive integer solution of the equation x^n + y^n + z^n for n>2, is 1.
We can assign x=y=0 which gives us x^n+y^n=0^n+0^n=0.
Then, we want to assign z^n=1 (since 1 is the smallest positive integer, and the ideal solution to the problem), which can be achieved by setting z=1, or 1^n=1 for n>2
Thus, 0+0+1=1

#2180184
Dr. Pepper
Participant

@ubiquitin

If there’s one standard used by all textbooks then I’d agree with you but from the different answers I see listed above it seems as if different textbooks teach the order of operations differently. (I can not recall seeing a textbook that gave an order of operation that would yield a number other than 1 but from Google searches it seems like there are.)

Well, I think everyone would agree that if it says 2+3=6 that it shouldn’t be followed but there are other situations that are more complicated. A college professor of mine told our class of a story where a builder followed the engineer’s directions while building an oil rig and it collapsed. After the investigation was completed it turned out that one (or more) of the formulas required an integral that couldn’t be solved in closed rectangular format and the software used a numerical approach which the builders used without changing the internal default tolerance. Anyway- there was a massive lawsuit with everyone blaming everyone else and the lawyers and expert witnesses making tons of money.

#2180189
emes nisht sheker
Participant

https://cdn .journals.aps.org/files/styleguide-pr.pdf

Page 21, little e says multiplication is higher order than division.

“The Feynman Lectures on Physics” also use this convention.

Bottom line is it is simply a convention and as long as both writer and reader know which system is being used it is fundamentally correct. Of course, if you work in an industry where a certain convention is expected and varying from it can have consequences, then you may have no choice but to follow those rules to avoid violating professional standards.

#2180232

Ok, so main conclusion is that unless people unite in their reading of the formulas the world will come to an end. This is because a theorem that says if you can prove that 1 = 2, then you can prove any false statement using valid operations (i.e. 1+1 = 2 + 1, 1*4 = 2*4, etc).

What is the source of math? While people figured out counting and addition on their own (Noach – animals, Moshe – accounting for Mishkan), subtraction is already clearly min hashamayim – Avraham asks whether there will be 5 tzadikim less than 50, and Hashem teaches him that this is 45.

#2180236
ubiquitin
Participant

Dr P
” it seems as if different textbooks teach the order of operations differently.”

This hasnt been my experience. And today although people calim there are nobody has ever been able to identify these alleged textbooks (although IVe asked)
I think people (myself included) forget the order of operations (most of us havent done these kind of problems in decades) So we get the question wrong. Then when reminded we feel silly that we got an elemntray school problem wrong.
Instead of saying “” wow cant beleive I forgot the rule” The responses are “There must be other rules” “the rules are made up its up to each person if they should follow them ” etc (these arent verbatim quotes from this thread)

“(I can not recall seeing a textbook that gave an order of operation that would yield a number other than 1 but from Google searches it seems like there are.)”

I could not find any such rule for order of operations. (a few posters said “PEMDAS is a convention, not a hard and fast rule.” and “Following the system where multiplication is deemed a higher order operation than division:” These are verbatim quotes. and not based on any thing real

BUT

however I did find some debate over Flaming’s Pitfal #2 with some fellow (who seemed well reaosned) arguing that 8÷2(4) was not the same as 8÷2×4 (which would absolutly be 16) rather that 2(4) was one unit equaling 8 (thus leaving 8÷8 = 1) not two seperate units 2×4 (which would require first dividng the 8 by 2)

For interesting sources see Slate’s “What Is the Answer to That Stupid Math Problem on Facebook?
And why are people so riled up about it?” (3/12/13) where he argues (wrongly in my opinion) “none is incorrect.” the thrust of the article is that histoically there are different ocnventions, and sure in 1919 a different answer may have been correct but that has no bearing on today.

#2182445
ujm
Participant

Dr. Pepper: …”a/b/c. Is it (a/b)/c or a/(b/c)?”

What logic could argue for a/(b/c)? (a/b)/c is following your original equation left-to-right.

In the OP’s example you could invoke left to right. Or you could invoke the basis of prioritizing the touching rule. The left to right only became prevalent with the advent of calculators. Prior to the widespread use of calculators (i.e. within the last less than hundred years) the generally accepted convention was to prioritize touching — such as 2(2+2). Of course that’s only applicable where they otherwise hold equal priority, such as multiplication and division.

#2182446
ujm
Participant

What is 2x/3y-1 if x=9 and y=2 ?

#2182546
Yserbius123
Participant

It’s not so much a question of different conventions of order of operations, but that there are certain contrived cases where the order of operations is different. Like PEMDAS doesn’t tell you what to do for 6÷3/10. Also there are different ways of writing out equations (like ignoring the multiplication sign, or using ÷ instead of a superscript/subscript) that sometimes have different rules of orders of operations. So when you mix and match those signs, you give your equation a level of ambiguity that PEMDAS cannot answer.

#2182553
Reb Eliezer
Participant

(2 x 9) / (3 x 2) – 1 = 18/6 – 1 = 3 – 1 = 2

#2182614
ubiquitin
Participant

Well put Yserbius

PEMDAS isn’t optional it is a rule like + means addition

using / for division is ambiguous . It would be like saying solve this equation

40 + 2O – 0 x O = ?
where O = 2

while solving it here is doable, imagine it handwritten

#2182634
ytvalumni
Participant

R’ Eliezer

(2 x 9 / 3)2 -1 = (18/3)2-1= 6(2)-1 = 12-1 = 11

#2182735
Reb Eliezer
Participant

ytvalumni, according to algebra, 3y is one entity which cannot be broken up.

#2182736
Reb Eliezer
Participant

ubi, 40 + 20 – 2 x 2 = 60 – 2 x 2 = 60 – 4 = 56 not (60 – 2) x 2 = 58 x 2 = 116

#2182753
ubiquitin
Participant

RE

Nope you got it wrong. You didn’t look carefully and mixed up zeros and O’s (which was my point I’m just surprised the point came across so well )

Question was
40 + 2O – 0 x O = ?
where O = 2

Translates to
40 + 2(2) – 0 x 2 = ?
40 + 4 – 0 =44

But the truth is its not your fault it is hard to tell zero’s and O’s apart, and (near?) impossible to do so in writing. So the fault is not in you but in the unclear question.

Similarly 2x/3y-1 if x=9 and y=2 ?

Is unclear:
is it 2x/(3y-1) ? ( = 3.6)
is it (2x/3y) -1 ? (= 2)
(2x/3)y-1 (=11)
2(x/3)(y-1) (= 6)
among others

Granted some of these make morse sense than others but more than one could have been meant. when typed out it is impossible to convey in a way that is clear to all what the numerator/dividend is and what the denominator/divisor .

So none of these answers (3.6, 2, 11, 6) are incorrect per se*. The question is unclear much like 40 + 2O – 0 x O = ? (especially when handwritten) is unclear

*Ok some are silly, but definitely more than one especially the first 2 can be equally valid

#2182764
ujm
Participant

Apparently you can have a legitimate machlokes even in mathematics.

#2182802
ChatGPT
Participant

This is a math problem that requires the application of the order of operations or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Using PEMDAS, we should first solve the expression inside the parentheses:

2 + 2 = 4

Then, we multiply 2 and 4, since multiplication and division have the same level of precedence in the order of operations:

8 ÷ 2 × 4 = 4 × 4 = 16

#2182804
Reb Eliezer
Participant

Looking carefully I see the point. In writing the zeroes are overwritten with a slash.

#2182817
Reb Eliezer
Participant

Just as parentheses precede exponents shouldn’t multiplication precede division? So the answer should be 1.

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