Elementary Mathematical Equation

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.

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

BIDMAS

We have come full circle.
The thread that first got me addicted to YWN was this one

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

The answer is 16
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

Something something the DemonRats

@ubiquitin and @ Always_Ask_Questions

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.

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.

@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.

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)

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

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

@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.

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

I’m not sure about that
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”

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.

@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.

@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.

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.

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.

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

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

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

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

Therefore, the answer is 16.

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