Agree completely. Old school calculator is wrong, but why? Pemdas wasn’t really big in school curriculums until around the turn of the century, but the order of operations existed at the previous turn of the century, and should operate correctly on every digital calculator ever made…
You don’t do multiplication before division, they’re equal operations, so you go left to right. 8 x 0.5 (2 + 2) is the same from a mathematical point of view.
Multiplication and division are same level just as addition and subtraction are same level. So it would be worked multiplication and division in order from left to right.
My public school education on pemdas is that for multiplication/division and addition/subtraction, you do them on order from left to right. Doing it that way gets me 16, which I believe to be right, but I’m also very bad at math. The way you had explained is also technically correct, if you do the multiplication out of order. Now that I think about it, you could solve for the parentheses by multiplying 2+2 by two, giving you 8/8 quicker and still yielding 1. I’m now having more doubts about my math capabilities, both are right, but I know that’s wrong, I just don’t know why
In PEMDAS M does not get priority over D so the equation has to be executed in order: 8/2=4, 4*4=16. You would be correct if all PEMDAS were a priority list., but it is not.
Parenthesis comes first, do everything in each of them as though they were a whole equation to themselves.
8/2(2+2) = 8/2(4)
Then you do your exponents. The equation doesn’t have any, so we can go ahead and skip those.
Multiplication and division are the same operation, just flipped around, so you go left to right and do those as you come across them. A number next to a parenthesis means multiplication, so to simplify:
8/2(4) = 8/2x4
8/2x4 = 4x4
4x4 = 16
Addition and subtraction don’t have any weird effects on the outcomes of each other, so you go left to right and do them as they come up. This equation has no more addition or subtraction to do, so we can consider what we have left our answer.
Therefore: 8/2(2+2)=16
This is straight from the textbook. You are wrong, and so are your purpose-built calculators.
You can do this without needing to replace by using a backslash. 1*2 comes from 1*2.
Anyway, the problem with your logic is that it’s using rules designed for primary school by one random primary school teacher many decades ago. Not a rigorous mathematical convention.
In real maths, mathematicians frequently use juxtaposition to indicate multiplication at a higher priority than division. Rather than BIDMAS, something like BIJMDAS might work. But that isn’t as catchy, and more to the point: it requires understanding of an operation that doesn’t get used in primary school, so would be silly to put in to a mnemonic designed to aid probably school children.
There’s quite a few calculators that get this wrong. In college, I found out that Casio calculators do things the right way, are affordable, and readily available. I stuck with it through the rest of my classes.
Casio does a wonderful job, and it’s a shame they aren’t more standard in American schooling. Texas Instruments costs more of the same jobs, and is mandatory for certain systems or tests. You need to pay like $40 for a calculator that hasn’t changed much if at all from the 1990’s.
Meanwhile I have a Casio fx-115ES Plus and it does everything that one did, plus some nice quality of life features, for less money.
TI did the same thing Quark and Adobe did later on – got dominance in their markets, killed off their competition, and then sat back and rested on their laurels thinking they were untouchable
EDIT: although in part, we should thank TI for one thing – if they hadn’t monopolized the calculator market, Commodore would’ve gone into calculators instead of computers
Huge failure my ass. Come at me on munch man, Alpiner, or Tombstone City. Or coding vaguely racist things like Mr. Bojangles, one of the first codes in the early books.
Had one at home and used the hell out of it, don't get me wrong. Was my first computer. Played the Zork series on that thing. But, it had issues and wasn't a financial success.
It had fewer issues than almost anything I’ve owned since. I bet it would still work if I got the right adaptors. Wasn’t a huge financial success though. They seemed content with early coding and games, and didn’t move into word processing etc.
If you’re lucky, you can find these TI calculators in thrift shops or other similar places. I’ve been lucky since I got both of my last 2 graphing calculators at a yard sale and thrift shop respectively, for maybe around $40-$50 for both.
That’s wrong. Multiplication and division have equal precedence, same as addition and subtraction. You do them left to right. PEMDAS could be rewritten like PE(MD)(AS). After parentheses and exponents, it"s Multiplication and division together, then addition and subtraction together. They also teach BODMAS some places, which is “brackets, order, division and multiplication, addition and subtraction” Despite reversing the division and multiplication, it doesn’t change the order of operations. They have the same priority, so they are just done left to right. PEMDAS and BODMAS are the different shorthand for the same order of operations.
They were right but for the wrong reason. Implied multiplication–that is, a(b) or ab–often comes before explicit multiplication and division. Apparently it’s up to the person writing the equation, so the meme is intentionally and explicitly ambiguous
They’re still wrong, in my humble opinion. I’m aware of this notion, and I’ve even had people share a snip from some book that states this as fact. However, this is not standardized and without the convention being widely understood and recognized as the standard in the world of mathematics (which generally doesn’t use the symbol (÷) at all at post-algebra levels), there is no reason to treat it as such just because a few people assert it is should be.
It doesn’t make sense at all to me that implied multiplication would be treated any differently, let alone at a higher priority, than explicit multiplication. They’re both the same operation, just with different notations, the former of which we use as shorthand.
There are obviously examples that show the use of the division symbol without parentheses sometimes leads to misunderstandings like this. It’s why that symbol is not used by real mathematicians at all. It is just abundantly more clear what you’re saying if you use the fraction bar notation (the line with numerator on top and denominator on bottom). But the rules as actually written, when followed, only reach one conclusion for this problem and others like it. x÷y(z) is the SAME as x÷y*z. There’s no mathematical or logical reason to treat it differently. If you meant for the implicit multiplication to have priority it should be in parentheses, x÷(y(z)), or written with the fraction bar notation.
Implicit multiplication being before regular multiplication/division is so we can write 2y/3x instead of (2y)/(3x). Without priority, 2y/3x becomes (2y÷3)•x.
Coefficients are widely used enough that mathematicians don’t want to write parentheses around every single one. So implicit multiplication gets priority.
I think one could argue a coefficient on an unknown variable, like 2y, should take higher priority simply because it cannot be any further resolved or simplified. That is not the case with, say, 2(3+1). Although that does still leave you with potential ambiguity with division/multiplication, such has 1/7y. Is the coefficient 7, or is it 1/7? i.e. Is that 1/(7y)? Or (1/7)y? Either way, if that’s not the the standard understood by everyone, then it is a non-standard, inconsistent rule. And as demonstrated, if you do use that rule, it needs to be more clearly defined. That is the source of this “ambiguity”. If you don’t include it, the order of operations rules, as written, are clear.
I agree it needs to be more clearly defined, but one of the reasons it wasn’t clearly defined was because mathematicians thought it was so universal it didn’t need defining, like how parentheses work to begin with.
Casio tried not doing umplicit multiplication after some american teachers complained, then went back to doing it after everyone else complained. Implicit multiplication is the standard.
There’s an argument to be made that implicit multiplication comes before division, resulting in the answer 1, but all multiplication? That’s wrong, full-stop. You calculate (explicit) multiplication and division in one step, left to right. Reason being that division is technically just multiplying by the reciprocal.
That’s exactly where the calculators in the op differ. For more examples, Casio calculators do implicit multiplication first, while ti’s treat it the same as explicit multiplication and division. I think that the latter is more predictable personally, but really you just need to know your calculator.
I don’t mean when they’re explaining “here’s how the order of operations works”. I mean in the basic way that they write more advanced problems and the answers they give for them.
This video, and the prequel to it linked in the description, go into some detail showing who uses what convention and why.
FWIW I went to school in Asia, using an internationally-focused curriculum, rather than going through the Australian curriculum here in Aus.
The video I linked includes some discussion with a calculator manufacturer who apparently is under the impression that teachers in North America are asking for strict BIDMAS, so the calculator manufacturer actually switched their calculators to doing that. Until they then got blowback from the rest of the world’s teachers, so they switched back to BIDMAS with juxtaposition being prioritised over division. The video also presents the case that outside of teachers—among actual maths and physics academics—prioritising juxtaposition is always preferred, even in North America.
prefix notation doesn’t need parentheses either though, at least in this case. lisp uses them for readability and to get multiple arity operators. infix doesn’t have any ambiguity either if you parenthesize all operations like that.
spoilerFor anyone like me who has math as their worst subject: PEMDAS. > PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations. PEMDAS stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction. So we gotta do it in the proper order. And remember, if the number is written like 2(3) then its multiplication, as if it was written 2 x 3 or 2 * 3. So we read 8/2(2+2) and need to do the following; - Read the Parentheses of (2 + 2) and follow the order of operations within them, which gets us 4. - Then we do 2(4) which is the same as 2 x 4 which is 8- 8 / 8 is 1. The answer is 1. The old calculator is correct, the phone app which has ads backed into it for a thing that all computers were invented to do is inaccurate.
EDIT: Turns out I’m wrong, but I haven’t been told how or why. I’m willing to learn if people actually tell me, instead of call me wrong and keep it at that. I was told PEMDAS and did PEMDAS.
Not quite, pemdas can go either from the left or right (as long as you are consistent) and division is the same priority as multiplication because dividing by something is equal to multiplying by the inverse of that thing… same as subtraction being just addition but you flip the sign.
8×1/2=8/2 1-1=1+(-1)
The result is 16 if you rewrite the problem with this in mind: 8÷2(2+2)=8×(1/2)×(2+2)
I’ve always heard it that way too but I think it is for consistency with students, imo Logically, if you are looking at division = multiplying by inverse and subtraction = adding the negative, you should be able to do it both ways. Addition and multiplication are both associative, so we can do 1+2+3 = (1+2)+3 = 1+(2+3) and get the same answer.
But subtraction and division are not associative. Any time you work on paper, 2 - 2 - 2 would equal -2. That is, (2-2)-2=0-2=-2. If you evaluate right to left, you get 2-2-2=2-(2-2)=2-0=2
Correct, subtraction and division are not associative. However, what is subtraction if not adding the opposite of a number? Or division if not multiplying the inverse? And addition and multiplication are associative.
2-2-2 can be written as 2 + (-2) + (-2) which would equal -2 no matter if you solve left to right, or right to left.
In your example with the formula from right to left, distributing the negative sign reveals that the base equation was changed, so it makes sense that you saw a different answer.
not to be That Guy, but the phone is actually correct… multiplication and division have the same precedence, so 8 / 2 * 4 should give the same result as 8 * 4 / 2, ie 16
The problem with this is that the division symbol is not an accurate representation of the intended meaning. Division is usually written in fractions which has an implied set of parenthesis, and is the same priority as multiplication. This is because dividing by a number is the same as multiplying by the inverse, same as subtracting is adding the negative of a number.
8/2(2+2) could be rewritten as 8×1/2×(2+2) or (8×(2+2))/2 which both resolve into 16.
The latter is correct, Multiplication/Division, and Addition/Subtraction each evaluate left to right (when not made unambiguous by Parentheses). I.e., 6÷2×3 = 9, not 1. That said, writing the expression in a way that leaves ambiguity is bad practice. Always use parentheses to group operations when ambiguity might arise.
Ignore the idiots telling you you’re wrong. Everyone with a degree in math, science or engineering makes a distinction between implicit and explicit multiplication and gives implicit multiplication priority.
The problem is that the way PEMDAS is usually taught multiplication and division are supposed to have equal precedence. The acronym makes it look like multiplication comes before division, but you’re supposed to read MD and as one step. (The same goes for addition and subtraction so AS is also supposed to be one step.) It this example the division is left of the multiplication so because they have equal precedence (according to PEMDAS) the division applies first.
IMO it’s bad acronym design. It would be easier if multiplication did come before division because that is how everyone intuitively reads the acronym.
Maybe it should be PE(M/D)(A/S). But that version is tricky to pronounce. Or maybe there shouldn’t be an acronym at all.
Inaccurate, this has nothing to do with the mnemonic PEMDAS, this has to do with the actual order of operations it tries to instill. That order of operations is not ambiguous, there is a correct way to solve simple equations like the one above, and there is one and only one correct answer to it. That answer is 16.
No, 2( does not bind more tightly than ÷. 2( is simply 2×(…, and ÷ and × occur at the same level of priority. After resolving the addition in the parentheses, the remaining operations are resolved left to right.
No, the fact that a good many people are incorrect about how math works does not entail that math is an open question. It’s not, math has actual rules to its equations and an unambiguous right answer. In this case, that answer is 16.
I’m well familiar with math and the rules by which it works. Those who persist in arguing the case here could save the rest of us the bother by admitting they were stumped by a simple gotcha equation and are embarrassed, rather than wasting everyone’s time by insisting that math is nothing but a lawless, rules-free wasteland where the answer to an equation depends on your feelings at the time.
Fortunately, the rules necessary to resolve the equation in this post are extremely elementary, so none of what you’re referencing has any bearing whatever.
There are exactly three things to consider in here to determine priority: parentheses, multiplication/division, and addition. The addition happens first due to the parentheses, and the remaining is evaluated left-to-right. The only correct answer here is 16.
All your deflection from your embarrassment at misreading a simple equation doesn’t detract from this.
Fortunately, the rules necessary to resolve the equation in this post are extremely elementary, so none of what you’re referencing has any bearing whatever.
this would be like trying to tell a chemical engineer they didn't know what they were doing based on your understanding of the atom as a ball of protons with electrons wooshing round it like they were moons
very cute
unfortunately, if you give the expression 1 / 2x to anybody who knows what they're doing they'll interpret it as 1 / (2x) because it would be absurd not to
for reference, that's why the calculator works like this. because it's a tool designed primarily for people who actually know what they're doing with numbers, so it works how they expect it to work
And there you’ve proven exactly what I’ve been saying all along. 2x works the way it does because there’s a variable involved, and natural reading of that treats it as a single entity. There are no variables in the equation in the post, there are only definite numbers, parentheses, and simple mathematical operations. 8/2(2+2) is nothing more than 8/2×(2+2). There is nothing special about 2(…, this is not the equivalent of 2x.
No, what I’m explaining to you is the facts behind what every calculator with any modicum of computing power will tell you, namely that 2(2+2) is identical to 2×(2+2).
Yeah, kind of. The crappier calculator is the one generating the incorrect answer. Any calculator with any real level of oomph behind it can parse this correctly to get the correct answer, 16.
The good calculator is the one using the processing power of the phone to handle the programming necessary to correctly interpret the order of operations and arrive at the correct answer, whereas the bad calculator - despite having no ads - is a cheap piece of trash unable to contain the necessary computational logic to arrive at the correct answer.
No need. The fact that you’re incapable of comprehending it at this point indicates that any further attempts to explain it to you are equally likely to fall on deaf ears.
And you’ve long since given up trying to defend your incorrect position, so I can only assume that you’re childishly continuing in an attempt to get the last word. That’s fine, go right ahead and have the last word, I won’t reply further since it’s that important to you.
You know that a couple has two children. You go to the couple’s house and one of their children, a young boy, opens the door. What is the probability that the couple’s other child is a girl?
Oops, I changed it to a more unintuitive one right after you replied! In my original comment, I said “you flip two coins, and you only know that at least one of them landed on heads. What is the probability that both landed on heads?”
And… No! Conditional probability strikes again! When you flipped those coins, the four possible outcomes were TT, TH, HT, HH
When you found out that at least one coin landed on heads, all you did was rule out TT. Now the possibilities are HT, TH, and HH. There’s actually only a 1/3 chance that both are heads! If I had specified that one particular coin landed on heads, then it would be 50%
But you do know which one it is, because you said “other child”. As soon as you ask the question, you assign a specific outcome to a specific child eliminating HH and HT (or in the new example, BB and BG). “What are the odds they have a female child” and “what are the odds the other child is female” are not the same question.
it depends on how you know this information. did you look at one of the coins? then it’s 50/50. did you ask a friend to look and yell you whether at least one of them is heads? then it’s 2/3.
No. It’s still 50-50. Observing doesn’t change probabilities (except maybe in quantum lol). This isn’t like the Monty Hall where you make a choice.
The problem is that you stopped your probably tree too early. There is the chance that the first kid is a boy, the chance the second kid is a boy, AND the chance that the first kid answered the door. Here is the full tree, the gender of the first kid, the gender of the second and which child opened the door, last we see if your observation (boy at the door) excludes that scenario.
1 2 D E
B B 1 N
B G 1 N
G B 1 Y
G G 1 Y
B B 2 N
B G 2 Y
G B 2 N
G G 2 Y
You can see that of the scenarios that are not excluded there are two where the other child is a boy and two there the other child is a girl. 50-50. Observing doesn’t affect probabilities of events because your have to include the odds that you observe what you observed.
I was about to reply to you with a comment that started with “oh shit you’re right!” But as I wrote it I started rethinking and I’m not sure now.
Because I actually don’t think it matters whether we’re BB1 or BB2. They’re both only one generation of the four possible initial states. Which child opens the door is determined after the determination of which child is which gender. Basically, given that they have two boys, we’re guaranteed to see a boy, so you shouldn’t count it twice.
Still, I’m now no where near as confident in my answer as I was a moment ago. I might actually go and write the code to perform the experiment as I outlined in an earlier comment (let me know if you think that methodology is flawed/biased, and how you think it should be changed) to see what happens.
That’s a great idea let me know how it turns out. If you randomly pick the genders and randomly pick who opens the door, I think it will be 50-50. With these kinds of things they can get pretty tricky. Just because an explanation seems to make sense doesn’t mean it’s right so I’m curious!
That’s cool. Always nice to see a practical example of theory. Thanks to you I got to brush up on my Python too! I think it all makes sense. Everything is random, you exclude examples where s girl comes to the door. The odds are 50-50. Looks like you were right in the end though, this DID stir quite a bit of conversation lol!
There’ve been a lot of times when I simply didn’t believe something in statistics until I simulated it. Like this problem:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
I wanted to simulate that because the answer seems absurd. 13/27? Where does that even come from? I’m scared of snakes, so I use Baby’s First Programming Language: Tasker.
Variable randomize %sex Min:1 Max:2
Variable randomize %day Min:1 Max:7
Variable set %child1 “%sex%day”
Variable randomize %sex Min:1 Max:2
Variable randomize %day Min:1 Max:7
Variable set %child2 “%sex%day”
Goto 1 IF %child1 != 11 AND %child2 != 11
Now I’ve generated two random children, at least one of which is 11–a specific sex born on a specific day.
Variable add %BoyGirl IF %child1 = 2* OR %child2 = 2*
Variable add %BoyBoy IF %child1 = 1* AND %child2 = 1*
If either child is a girl, it adds one to the BG bucket. If neither one is a girl, it adds one to the BB bucket.
Variable add %Counter
Goto 1 IF %Counter > 1,000
Hit play and yep, about 48% of families were BB. But remove the Tuesday part and just simulate the question “I have two children, and at least one of them is a boy,” and it drops down to 33% again. I don’t understand it, but apparently the math maths.
Ha. As someone from the “mathematics is shortened to maths” part of the world, this sounds weird to me. I’d probably say “the maths mathses”. I just thought you might enjoy that.
Anyway, I Googled the problem you presented, and came across this excellent answer:
There are even trickier aspects to this question. For example, what is the strategy of the guy telling you about his family? If he always mentions a boy first and not a daughter, we get one probability; if he talks about the sex of the first born child, we get a different probability. Your calculation makes a choice in this issue - you choose the version of “if the father has a boy and a girl, he’ll mention the boy”.
What I’m aiming to is this: the question is not well-defined mathematically. It has several possible interpretations, and as such the “problem” here is indeed of the language; or more correctly, the fact that a simple statement in English does not convey enough information to specify the precise model for the problem.
The whole answer is worth reading, but that part there is the crux of it. It goes back to one of my earliest comments on this topic in this thread. The problem isn’t that maths is weird, it’s that language’s ability to describe mathematical problems is lacking. There are so many different ways to translate the described problem into mathematical formulae and they necessarily carry assumptions. Even far more subtle assumptions than I at first thought.
Since using “maths” as a verb is very much nonstandard, I would argue against trying to apply any rigorous rules to it. It’s about the vibes of the thing.
And this interaction is why lemmy (and others) is superior. No “fuck you, you’re wrong” just “well I think it’s this” followed by “okay, let me try that” 10/10
Yes! They responded to my comment before the edit, where I gave the coin example: “I flipped two coins, at least one of them was heads. What is the probability that both of them are heads?”
Before I read their reply, I edited it to the more confusing and infuriating two kids example. It’s annoying because it seems like it should be the same as saying “I have two children, and at least one is a boy. What is the probability that I have two boys?” In both the coin case and this one, the answer is 1/3, but when one child answers the door, it’s like sliding one quarter out from behind my hand. Now you know a particular (child, coin)'s (sex, face), and the answer is 50% again.
The issue is that you weren’t told a particular one was heads, only that at least one was heads. If I flipped a nickel and a dime, then the four possibilities are NtDt, NtDh, NhDt, and NhDh. If I say that at least one of them is heads but don’t tell you which one, then there are three possibilities: I flipped NhDt, NtDh, or NhDh. It’s only when I tell you that the nickel landed on heads that it collapses to NhDt and NhDh.
Sorry if the acronyms are hard to read, they’re much faster than typing something like “Heads (nickel) tails (dime)”
Again, you are assuming that every occurrence has the same chance. When in fact, they have not. There are 3 random events happening here:
Flip of one coin (50% chance each)
Flip of the other coin (50% chance each)
The coin that you told me (let’s say it’s 50% nickel 50% dime for simplicity’s sake)
Also, I am assuming that these 3 events are completely unrelated. That is, the result of a coin flip won’t determine whether you tell me the nickel or dime. A complete list of events is as follows:
T T N
T T D
H T N
H T D
T H N
T H D
H H N
H H D
After telling me that one of them is heads, the list is as follows:
H T D
T H N
H H D
H H N
H H is 50% chance, and the sum of HT + TH is the other 50%
This is very easily simulated. I use Tasker, you can use Python or something. Assume 1 is heads and 2 is tails
Randomize variable %dime to 1 or 2
Randomize variable %nickel to 1 or 2
This flips two coins. At this point there’s a 25% chance of each TT, TH, HT, and HH occurring.
Goto 1 IF %dime != 1 AND %nickel != 1
This is the point where I tell you that at least one of them is heads, but not which. It flips the coins again only if they landed TT, which means there are three possible sets of coins now: TH, HT, or HH.
Add 1 to variable %HeadsTails IF %dime = 2 OR %nickel = 2
Add 1 to variable %HeadsHeads IF %dime = 1 AND %nickel = 1
This keeps track of what sets made it past line 3. If either is tails, it adds one to %HeadsTails. If neither is tails, then it adds one to %HeadsHeads
Add 1 to variable %Counter
Goto 1 IF %Counter <1000
You can set your counter as high as you like. Python would probably handle 10,000 flips faster than Tasker can handle 1,000, but I’m on my phone and also python illiterate.
Press play, and I get 357 sets with two heads, and 643 sets that have a tails. The longer it runs, the closer to a ratio of 1:2 it’ll get.
If you simulate it like that, it leads to a contradiction.
According to the problem, 2 coins are flipped, and we all agree that it leads to an event pool of {TT,TH,HT,HH}, where all 4 events are equally as likely.
In the simulation, however, you just ignore the “TT” situation, which leads to a total event pool of {HT,TH,HH}. Where all events are equally likely.
The way the problem was phrased was “2 coin flips happen, and I have a machine that tells me either if both are Tails or not, this time it turns out that there is at least 1 heads”. But the way you simulated it is “I will make coinflips until I have at least 1 heads”.
The way the problem was phrased was “2 coin flips happen, and I have a machine that tells me either if both are Tails or not, this time it turns out that there is at least 1 heads”.
That is exactly what my program simulates. We’re only interested in the times when this time, it turns out that there is at least one heads. If they both land tails, then we don’t record anything about the flip, because the question is “if they aren’t both tails, what is the probability that they are both heads?”
Think of it this way: generate n pairs of flipped coins, and put them in Bucket 0. Take every pair of coins that has at least one heads, and put it in Bucket 1. You’ll be leaving 25% of the pairs in Bucket 0–specifically, the TT pairs. The TH, HT, and HH pairs are all in Bucket1. Now, 33% of the pairs in Bucket 1 are HH.
When I tell you that I’ve flipped a pair of coins, there’s a 25% chance that they landed HH. But when I tell you that at least one is heads, it’s like pulling a random pair out of Bucket 1. We don’t know whether that pair was in Bucket 1 because of the nickel, dime, or both.
I tried to answer but idk why Lemmy failed to post it, so I’ll make a tldr instead.
TLDR:
Instead of reasoning I used actual statistics equations and you are correct: the chance in the coins case is 1/3.
However, I was misguided assuming that both the “girl and boy” problem and “coins” problem are the same, when in fact they are not.
In the “coins” case, the statement “at least one of them is heads” has a probability of 3/4. In the “girl and boy” case, the statement “the child that opened the door was a boy” has a probability of 1/2.
This is basically Monty Hall right? The other child is a girl with 2/3 probability, because the first one being a boy eliminates the case where both children are girls, leaving three total cases, in two of which the other child is a girl (BG, GB, BB).
It doesn’t, though. The Monty Hall problem utilizes the fact that there were more possibilities before one was eliminated AND that it cannot eliminate the “best” outcome. No such qualities are at play here.
The question being asked here is “what is the gender of the second child?” The gender of the first child is completely irrelevent. Observed or unobserved, door open or closed, it doesn’t impact the outcome of the second child.
I suspect it’s not the question OP intended to ask, but it’s the question they asked nonetheless.
The Monty Hall problem utilizes the fact that there were more possibilities before one was eliminated
So does this problem. There was the GG possibility.
AND that it cannot eliminate the “best” outcome.
True, this problem doesn’t have that element.
The question being asked here is “what is the gender of the second child?” The gender of the first child is completely irrelevent. Observed or unobserved, door open or closed, it doesn’t impact the outcome of the second child.
I don’t agree. First, I’d say your use of the term “second child” is ambiguous, because normally that would mean “the younger of two children”, which obviously isn’t what’s meant here. What you mean to say here is “the child that we have not already seen”. It’s in that rephrasing that it becomes obvious that having observed the first child matters, because there cannot be a second until there has been a first. And it’s in that observation that the outcome is altered.
If we haven’t seen the first child and are asked “what will be the gender of the second child to walk through the door?” we would have to answer 50/50. But having seen one child, we eliminate one of four possibilities of gender pairs (BB, BG, GB, GG). This we are left with 3 equally possible cases, 2 of which will be the opposite of the gender of the child we saw first.
Of course, we could easily simulate this experiment to arrive at an empirical answer. Randomly generate 2 genders, randomly select one of those. If they’re a girl, end the experiment and move to another iteration (because they didn’t fit the parameters). If a boy, record the gender of the other child. Repeat a few dozen times and see how many times the second child was a girl.
Well, I guess OPs point is demonstrated. People will in fact argue about it.
What you’re trying to present has multiple holes, but only one matters: you’re not paying attention to the question that’s being asked. You can say first, second, alpha, beta, Leslie, whatever you want to assign the child in question as, but the question only asks you the gender of a singular child. The door opening child doesn’t matter, because it isn’t part of the question. No one asked what gender that child is. No one asked what the odds they have a female child is. It just isn’t a part of the question.
Yes, I referred to it as the second child because the question that was asked happens to have a child in it and ask you about another. Because we’re communicating in a hilariously precise language, we have to say “the other child”. But that doesn’t make the door opening child a part of the equation. The question could be “there is a child in a box. What are the odds the child is female? Oh, it has a brother by the way.” Cool, who cares, the sibling wasn’t a part of the question.
The Monty Hall problem spreads multiple outcomes across multiple choices and then eliminates one. The outcomes and options have a relation. This question just asks you about a singular variable with two possible outcomes and throws around an unrelated red herring.
You can’t just end the experiment if the randomly chosen child doesn’t “fit the parameters”, by doing that you aren’t accounting for half the girls in the whole event pool. Half of the girls have siblings that are girls.
Being 2 girls was a possible event at the start, you can’t just remove it. This time it happened to be a boy who opened the door, but it could’ve been as likely for a girl to open it.
If it was phrased like “there are 2 siblings, only boys can open doors. Of all the houses that opened their doors, how many have a girl in them?”, then it will be 2/3. In this example, there is an initial pool of events, then I narrowed down to a smaller one (with less probability). If you “just” eliminate the GG scenario, then the set of events got smaller without reducing the set’s probability.
It depends on what you classify as the “start”. If all households with two children everywhere? Sure. But the story explicitly starts with us knowing a boy came to the door. By the parameters of the story, we know that’s what happened.
We don’t just eliminate GG, we also eliminate any BG or GB where it just so happened that the girl came to the door. Because that’s what we already know is true, and we’re asking for the conditional probability given that this has already happened.
You didn’t eliminate BG and GB where a girl opens the door though. If you do that, then the answer is 50%. Because you remove half the probability from BG and GB and you remove none from BB.
I know you didn’t eliminate those cases because you said “That leaves us with 3 possibilities with equal probabilities”. That would be false, BB is twice as likely.
I know you didn’t eliminate those cases because you said “That leaves us with 3 possibilities with equal probabilities”. That would be false, BB is twice as likely.
I’m guessing you haven’t read the rest of the thread? My first comment was incorrect and the correction has been made.
On the one hand, you might be right. This could be akin to flipping two coins and saying that at least one is heads. You’ve only eliminated GG, so BG, GB, and BB are all possible, so there’s only a 1/3 chance that both children are boys.
On the other hand, you could say this is akin to flipping two coins and saying that the one on the left (or the one who opened the door) is heads. In that case, you haven’t just ruled out GG, you’ve ruled out GB. Conditional probability is witchcraft
I don’t think the problem is conditional probability, it’s translating word problems to maths problems.
If you make the assumptions I made, the maths is unambiguous. Namely, I assumed that a child has a 50/50 probability of being born a boy or a girl. I assumed the child who opens the door is random. I don’t think I made any other assumptions that could have been made any other way. With those assumptions, I’m pretty confident my answer is the only correct one, though I’d love to see an argument otherwise.
If the probability of a child being a girl is different, say, 52%, that will affect the result.
More interestingly, if the probability of which child opens the door is different, that will affect the result. If there’s a 100% chance the elder child opens the door, it goes to 50/50 of the gender of the second child. This makes it like the “coin on the left” example you gave.
If we said the elder child is going to open the door 75% of the time…well, the maths becomes more complicated than I can be bothered with right now. But it’s an interesting scenario!
Assuming the chance of either sex is equal, this problem can be broken down into multiple cases. The first is that there are two unseen kids in the house. What’s the probability they are both boys? 1/4. Now the door opens and you see two boys. The probability both are boys is 1/1. But if you only see one boy, the problem simplifies into the probability of a child being a boy. One of the probabilistic events postulated in the original problem is fixed at 1. So the answer is 1/2.
Think of it as the two coin flip, except one coin has two heads. That simplifies to a one coin flip.
Well not really, right? BG and GB are the same scenario here, so it’s a 50/50 chance.
Even if, say, the eldest child always opened the door, it’d still be a 50/50 chance, as the eldest child being a boy eliminates the possibility of GB, leaving either BG or BB.
Ironically you’ve got the right answer, but (as you can see in some of the other conversation here) not necessarily for the right reason. It’s not necessarily that BG and GB are the same, but that BB and BB are two different scenarios worthy of being counted separately.
Why is it that BB and BB are being counted separately? I thought that order didn’t matter: you could have two girls, a boy and a girl (or vice versa, same thing), or a two boys. (And then by eliminating two girls you’d have a 50/50 chance).
No, because knowing the first child is a boy doesn’t tell you any information about the second child.
Three doors, Girl Girl Boy
You select a door and Monty opens a door to show a Girl. You had higher odds of picking a girl door to start (2/3). So switching gives you better odds at changing to the door with the Boy because you probably picked a Girl door.
Here the child being a boy doesn’t matter and the other child can be either.
It is 50-50, though. The remaining possible states are BG and BB. Both are equally likely. Any further inference is narrative… not statistics.
The classic example of this is flipping 100 coins. If you get heads 99 times in a row… the last coin is still 50-50. Yes, it is obscenely unlikely to get heads 100 times in a row. But it’s already obscenely unlikely to get heads 99 times in a row. And it is obscenely unlikely to alternate perfectly between heads and tails. And it is obscenely unlikely to get a binary pattern spelling out the alphabet. And it is obscenely unlikely to get… literally any pattern.
Every pattern is equally unlikely, with a fair coin. We see 99 heads in a row versus 1 tails at the end, and think it narrowly averted the least-probable outcome. But only because we lump together all sequences with exactly one tails. That’s one hundred different patterns. 1-99 is not the same as 99-1. We just treat them the same because we fixate on uniformity.
Compare a non-binary choice: a ten-sided die. Thirty 1s in a row is about as unlikely as 100 heads in a row. But 1 1 1… 2 is the same as 1 1 1… 3. Getting the first 29 is pretty damn unlikely. One chance in a hundred million trillion. But the final die can land on any number 1-10. Nine of them upset the pattern our ape brains want. Wanting it doesn’t make it any more likely. Or any less likely.
It would be identically unlikely for a 10-sided die to count from 1 to 10, three times in a row. All the faces appear equally. But swap any two events and suddenly it doesn’t count. No pun intended.
If this couple had eight children, for some god-forsaken reason, and you saw seven boys, the eighth kid being another boy is not less likely for it. The possibility space has already been reduced to two possibilities out of… well nine, I suppose, if order doesn’t matter. They could have 0-8 boys. They have at least 7. The only field that says the last kid’s not a coin toss is genetics, and they say this guy’s chromosome game is strong.
You’re right, but it’s not a subversion of the Gambler’s Fallacy, it’s a subversion of conditional probability. A classic example is that I have two kids, and at least one of them is a boy. What is the probability that I have two boys?
The intuitive answer is 50%, because one kid’s sex doesn’t affect the other. But when I told you that I have two kids, there were four possibilities: GG, GB, BG, or BB. When I told you that at least one of them is a boy, all I did was take away the GG option. That means there’s only a 1 in 3 chance that I have two boys.
But by having one child answer the door, I change it yet again–now we know the sex of a particular child. We know that the child who opened the door is a boy. This is now akin to saying “I have two children, and the eldest is a boy. What is the possibility that I have two boys?” It’s a sneaky nerd snipe, because it targets specifically people who know enough about statistics to know what conditional probability is. It’s also a dangerous nerd snipe, because it’s entirely possible that my reasoning is wrong!
Two more for funsies! I flipped two coins. At least one of them landed on heads. What is the probability that both landed on heads? (Note: this is what my comment originally said before I edited it)
I have two children. At least one of them is a boy born on a Tuesday. What is the probability that I have two boys?
[...] the question is ambiguous. There is no right or wrong if there are different conflicting rules. The only ones who claim that there is one rule are the ones which are wrong!
As youngsters, math students are drilled in a particular
convention for the "order of operations," which dictates the order thus:
parentheses, exponents, multiplication and division (to be treated
on equal footing, with ties broken by working from left to right), and
addition and subtraction (likewise of equal priority, with ties similarly
broken). Strict adherence to this elementary PEMDAS convention, I argued,
leads to only one answer: 16.
Nonetheless, many readers (including my editor), equally adherent to what
they regarded as the standard order of operations, strenuously insisted
the right answer was 1. What was going on? After reading through the
many comments on the article, I realized most of these respondents were
using a different (and more sophisticated) convention than the elementary
PEMDAS convention I had described in the article.
In this more sophisticated convention, which is often used in
algebra, implicit multiplication is given higher priority than explicit
multiplication or explicit division, in which those operations are written
explicitly with symbols like x * / or ÷. Under this more sophisticated
convention, the implicit multiplication in 2(2 + 2) is given higher
priority than the explicit division in 8÷2(2 + 2). In other words,
2(2+2) should be evaluated first. Doing so yields 8÷2(2 + 2) = 8÷8 = 1.
By the same rule, many commenters argued that the expression 8 ÷ 2(4)
was not synonymous with 8÷2x4, because the parentheses demanded immediate
resolution, thus giving 8÷8 = 1 again.
This convention is very reasonable, and I agree that the answer is 1
if we adhere to it. But it is not universally adopted.
Everyone in this threading referencing PEMDAS and still thinking the answer is 1 are completely ignoring the part of the convention is left to right. Only way to get 1 is to violate left to right on multiplication and division.
The problem is that BIDMAS and its variants are lies-to-children. Real mathematicians don’t use BIDMAS. Multiplication by juxtaposition is extremely common, and always takes priority over division.
Nobody in their right minds would saw 1/2x is the same as (1/2)x. It’s 1/(2x).
That’s how you get 1. By following conventions used by mathematicians at any level higher than primary school education.
There are plenty of contexts where writing in a single flat line is necessary, so it’s still useful to address the issue.
Just using more brackets is always a solution, but it can become messy and hard to read if you take it to the extreme (there’s a Minute Physics video where he does this and it unintentionally shows you just how bad it is), so it eventually becomes a matter of agreeing on convention and using brackets judicially where there’s actual ambiguity.
The problem is that you’re thinking of BIDMAS as a set of hard rules, rather than the set of rough guidelines created in the early 20th century by one random teacher for the purposes of teaching 10-year-olds how to do the level of maths that 10-year-olds do.
This video and this one point to some examples of style guides in academia as well as practical examples in the published works of mathematicians and physicists, which are pretty consistent.
If you want to come up with a hard rule, doing BIJMDAS, adding in “multiplication indicated by juxtaposition” with the J, is a much better way to do it than what you learnt when you were 10. But even that’s still best to think of as a handy guideline rather than a hard and fast rule.
No fr I had no idea that those acronyms weren’t the whole picture, I just assumed some mathematicians a long time ago decided how that stuff should be written out and that BEDMAS/PEMDAS/whatever contained all the rules in it. Thank you for the info, Idk why this isn’t more widely taught, ig because those acronyms are what all the questions are already written for? It doesn’t seem that hard to just teach it as BEADMAS, where the A refers to numbers adjacent to variables
The problem isn’t math, it’s the people that suck at at it who write ambigous terms like this, and all the people in the comments who weren’t educated properly on what conventions are.
Yeah, you could easily make this more straightforward by putting parentheses around 8÷2. It’s like saying literature sucks because Finnegans Wake is incomprehensible.
lol, math is literally the only subject that has rules set in stone. This example is specifically made to cause confusion. Division has the same priority as multiplication. You go from left to right. problem here is the fact that you see divison in fraction form way more commonly. A fraction could be writen up as (x)/(y) not x/y (assuming x and y are multiple steps). Plain and simple.
The fact that some calculator get it wrong means that the calculator is wrongly configured. The fact that some people argue that you do () first and then do what’s outside it means that said people are dumb.
They managed to get me once too, by everyone spreading missinformation so confidently. Don’t even trust me, look up the facts for yourself. And realise that your comment is just as incorrect as everyone who said the answer is 1. (uhm well they don’t agree on 0^0, but that’s kind of a paradox)
Off topic, but the rules of math are not set in stone. We didn't start with ZFC, some people reject the C entirely, then there is intuitionistic logic which I used to laugh at until I learned about proof assistants and type theory. And then there are people who claim we should treat the natural numbers as a finite set, because things we can't compute don't matter anyways.
On topic: Parsing notation is not a math problem and if your notation is ambiguous or unclear to your audience try to fix it.
If we had 1/2x, would you interpret that as 0.5x, or 1/(2x)?
Because I can guarantee you almost any mathematician or physicist would assume the latter. But the argument you’re making here is that it should be 0.5x.
It’s called implicit multiplication or “multiplication indicated by juxtaposition”, and it binds more tightly than explicit multiplication or division. The American Mathematical Society and American Physical Society both agree on this.
BIDMAS, or rather the idea that BIDMAS is the be-all end-all of order of operations, is what’s known as a “lie-to-children”. It’s an oversimplification that’s useful at a certain level of understanding, but becomes wrong as you get more advanced. It’s like how your year 5 teacher might have said “you can’t take the square root of a negative number”.
I will happily point out that Wolfram Alpha does this wrong. So do TI calculators, but not Casio or Sharp.
Go to any mathematics professor and give them a problem that includes 1/2x and ask them to solve it. Don’t make it clear that merely asking “how do you parse 1/2x?” is your intent, because in all likelihood they’ll just tell you it’s ambiguous and be done with it. But if it’s written as part of a problem and they don’t notice your true intent, you can guarantee they will take it as 1/(2x).
According to Casio, they do juxtaposition first because that’s what most teachers around the world want. There was a period where their calculators didn’t do juxtaposition first, something they changed to because North American teachers were telling them they should, but the outcry front the rest of the world was enough for them to change it back. And regardless of what teachers are doing, even in America, professors of mathematics are doing juxtaposition first.
I think this problem may ultimately stem from the very strict rote learning approach used by the American education system, where developing a deeper understanding of what’s going on seems to be discouraged in favour of memorising facts like “BIDMAS”.
To be clear, I’m not saying 1/2x being 1/(2x) rather than 0.5x is wrong. But it’s not right either. I’m just pretty firmly in the “inline formulae are ambiguous” camp. Whichever rule you pick, try to apply it consistently, but use some other notation or parenthesis when you want to be clearly understood.
The very fact that this conversation even happens is proof enough that the ambiguity exists. You can be prescriptive about which rules are the correct ones all you like, but that’s not going to stop people from misunderstanding. If your goal is to communicate clearly, then you use a more explicit notation.
Even Wolfram Alpha makes a point of restating your input to show how it’s being interpreted, and renders “1/2x” as something more like
<span style="color:#323232;">1
</span><span style="color:#323232;">- x
</span><span style="color:#323232;">2
</span>
Even Wolfram Alpha makes a point of restating your input to show how it’s being interpreted
This is definitely the best thing to do. It’s what Casio calculators do, according to those videos I linked.
My main point is that even though there is theoretically an ambiguity there, the way it would be interpreted in the real world, by mathematicians working by hand (when presented in a way that people aren’t specifically on the lookout for a “trick”) would be overwhelmingly in favour of juxtaposition being evaluated before division. Maybe I’m wrong, but the examples given in those videos certainly seem to point towards the idea that people performing maths at a high level don’t even think twice about it.
And while there is a theoretical ambiguity, I think any tool which is operating counter to how actual mathematicians would interpret a problem is doing the wrong thing. Sort of like a dictionary which decides to take an opinionated stance and say “people are using the word wrong, so we won’t include that definition”. Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.
Linguists would tell you the job of a dictionary should be to describe how the word is used, not rigidly stick to some theoretical ideal. I think calculators and tools like Wolfram Alpha should do the same with maths.
You’re literally arguing that what you consider the ideal should be rigidly adhered to, though.
“How mathematicians do it is correct” is a fine enough sentiment, but conveniently ignores that mathematicians do, in fact, work at WolframAlpha, and many other places that likely do it “wrong”.
The examples in the video showing inline formulae that use implicit priority have two things in common that make their usage unambiguous.
First, they all are either restating, or are derived from, formulae earlier in the page that are notated unambiguously, meaning that in context there is a single correct interpretation of any ambiguity.
Second, being a published paper it has to adhere to the style guide of whatever body its published under, and as pointed out in that video, the American Mathematical Society’s style guide specifies implicit priority, making it unambiguous in any of their published works. The author’s preference is irrelevant.
Also, if it’s universally correct and there was no ambiguity in its use among mathematicians, why specify it in the style guide at all?
Mathematicians know wolfram is wrong and it was warned in my maths degree that you should “over bracket” in WA to make yourself understood. They tried hard to make it look like handwritten notation because reading maths from a word processor is typically tough and that creates the odd edge case like this.
1/2x does not equal 0.5x or it’d be written x/2 and I challenge you to find a mathematician who would argue differently. There’s no ambiguity and claiming there is because anyone anywhere is having this debate is like claiming the world isn’t definitely round because some people argue its flat.
The answer is 1, but the logic you’ve used to get there is a little off. Different groups actually follow different logic, but they usually arrive at the same end-point.
The American Mathematical Society goes:
Brackets
Indices
Multiplication indicated by juxtaposition
Regular multiplication and division
Addition and subtraction
While the American Physical Society does
Brackets
Indices
Multiplication
Division
Addition and subtraction
In both cases, addition and subtraction are equal in priority (this solves the problem brought up by a different comment where following primary school BIDMAS would mean 8-4+2=2). In one case (and this is the way I prefer to do it) they solve the problem by declaring that implicit multiplication is done before division, but explicit multiplication with the × sign follows the same rules you would have learnt in primary school. The other says all multiplication is done before division, including explicit multiplication.
The comment from subignition explains that the phone’s answer, 16, is what you get by strictly following PEMDAS: the rule is that multiplication and division have the same precedence, and you evaluate them from left-to-right.
The calculator uses a different convention where either multiplication has higher priority than division, or where “implicit” multiplication has higher priority (where there is no multiply sign between adjacent expressions).
The parentheses step only covers expressions inside parentheses. That’s 2 + 2 in this case. The times-2 part is outside the parentheses so it’s evaluated in a different step.
This is more language/writing style than math. The math is consistent, what’s inconsistent is there are different ways to express math, some of which, quite frankly, are just worse at communicating the mathematical expression clearly than others.
Personally, since doing college math classes, I don’t think I’d ever willingly write an expression like that exactly because it causes confusion. Not the biggest issue for a simple problem, much bigger issue if you’re solving something bigger and need combine a lot of expressions. Just use parentheses and implicit multiplication and division. It’s a lot clearer and easier to work with.
I was taught that negative numbers should be written as (-2) with the parentheses when using exponents. So I assume that the calculators are doing it right, or maybe it's just a measure against calculators doing it wrong? I cannot be sure. Also-2 = 0-2 so -2^2 = 0-2^2.
if you’ve touched polynomials ever, you’d expect the exponent to be before the negation. If you write x³-x² you don’t mean x³ + (-x)² = x³+x², you mean x³-(x²)
so far as I know, [BIDMAS] is a creation of some educator, who has taken conventions in real use, and extended them to cover cases where there is no accepted convention. So it misleads students; and moreover, if students are taught PEMDAS by rote without the proviso mentioned above, they will not even get the standard interpretation of a−b+c.
Ultimately it’s ambiguous and bad. But most mathematicians (at a level higher than primary school education) use multiplication by juxtaposition—aka implicit multiplication—at a higher priority than division. BIDMAS, as you might have been taught in primary school, is an oversimplification that doesn’t even account for the possibility of juxtaposition, because you didn’t learn about that until secondary school.
Most mathematicians would use logic that lets you arrive at the answer of “1”, while also saying it should have been written better. Brackets are cheap. Use them!
In some countries we’re taught to treat implicit multiplications as a block, as if it was surrounded by parenthesis. Not sure what exactly this convention is called, but afaic this shit was never ambiguous here. It is a convention thing, there is no right or wrong as the convention needs to be given first. It is like arguing the spelling of color vs colour.
This is exactly right. It’s not a law of maths in the way that 1+1=2 is a law. It’s a convention of notation.
The vast majority of the time, mathematicians use implicit multiplication (aka multiplication indicated by juxtaposition) at a higher priority than division. This makes sense when you consider something like 1/2x. It’s an extremely common thing to want to write, and it would be a pain in the arse to have to write brackets there every single time. So 1/2x is universally interpreted as 1/(2x), and not (1/2)x, which would be x/2.
The same logic is what’s used here when people arrive at an answer of 1.
If you were to survey a bunch of mathematicians—and I mean people doing academic research in maths, not primary school teachers—you would find the vast majority of them would get to 1. However, you would first have to give a way to do that survey such that they don’t realise the reason they’re being surveyed, because if they realise it’s over a question like this they’ll probably end up saying “it’s deliberately ambiguous in an attempt to start arguments”.
So are you suggesting that Richard Feynman didn’t “deal with maths a lot”, then? Because there definitely exist examples where he worked within the limitations of the medium he was writing in (namely: writing in places where using bar fractions was not an option) and used juxtaposition for multiplication bound more tightly than division.
Yep with pen and paper you always write fractions as actual fractions to not confuse yourself, never a division in sight, while with papers you have a page limit to observe. Length of the bars disambiguates precedence which is important because division is not associative; a/(b/c) /= (a/b)/c. “calculate from left to right” type of rules are awkward because they prevent you from arranging stuff freely for readability. None of what he writes there has more than one division in it, chances are that if you see two divisions anywhere in his work he’s using fractional notation.
Multiplication by juxtaposition not binding tightest is something I have only ever heard from Americans citing strange abbreviations as if they were mathematical laws. We were never taught any such procedural stuff in school: If you understand the underlying laws you know what you can do with an expression and what not, it’s the difference between teaching calculation and teaching algebra.
So 1/2x is universally interpreted as 1/(2x), and not (1/2)x, which would be x/2.
Sorry but both my phone calculator and TI-84 calculate 1/2X to be the same thing as X/2. It’s simply evaluating the equation left to right since multiplication and division have equal priorities.
X = 5
Y = 1/2X => (1/2) * X => X/2
Y = 2.5
If you want to see Y = 0.1 you must explicitly add parentheses around the 2X.
Before this thread I have never heard of implicit operations having higher priority than explicit operations, which honestly sounds like 100% bogus anyway.
You are saying that an implied operation has higher priority than one which I am defining as part of the equation with an operator? Bogus. I don’t buy it. Seriously when was this decided?
I am no mathematics expert, but I have taken up to calc 2 and differential equations and never heard this “rule” before.
I can say that this is a common thing in engineering. Pretty much everyone I know would treat 1/2x as 1/(2x).
Which does make it a pain when punched into calculators to remember the way we write it is not necessarily the right way to enter it. So when put into matlab or calculators or what have you the number of brackets can become ridiculous.
I’m an engineer. Writing by hand I would always use a fraction. If I had to write this in an email or something (quickly and informally) either the context would have to be there for someone to know which one I meant or I would use brackets. I certainly wouldn’t just wrote 1/2x and expect you to know which one I meant with no additional context or brackets
I think when a number or variable is adjacent a bracket or parenthesis then it’s distribution to the terms within should always take place before any other multiplication or division outside of it. I think there is a clear right answer and it’s 1.
Firstly, don’t forget exponents come before multiply/divide. More importantly, neither defines wether implied multiplication is a multiply/divide operation or a bracketed operation.
It’s BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
But, since your rule has the D&M as well as the A&S in brackets does that mean your rule means you have to do D&M as well as the A&S in the formula before you do the exponents that are not in brackets?
But seriously. Only grade school arithmetic textbooks have formulas written in this ambiguous manner. Real mathematicians write their formulas clearly so that there isn’t any ambiguity.
Grade school is a US synonym for primary or elementary school; it doesn’t seem to be used as a term in England or Australia. Apparently, they’re often K-6 or K-8; my elementary school was K-4; some places have a middle school or junior high between grade school and high school.
I don’t know why you’re getting lost on the pedantry of defining “grade school”, when I was clearly discussing the fact that you only see this kind of sloppy formula construction in arithmetic textbooks where students are learning the basics of how to perform the calculations. Once you get into applied mathematics and specialized fields that use actual mathematics, like engineering, chemistry and physics, you stop seeing this style of formula construction because the ambiguity of the terms leads directly to errors of interpretation.
For example, if f(x,y)=x2+yx, then (∂f/∂x)y=2x+y, and (∂f/∂y)x=x. We can extend this idea to higher derivatives: ∂^2f/∂y^2 or ∂^2f/∂y∂x. The latter symbol indicates that we first differentiate f with respect to x, treating y as a constant, then differentiate the result with respect to y, treating x as a constant. The actual order of differentiation is immaterial: ∂2f/∂x∂y=∂2f/∂y∂x.
Notice: ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x).
What an interesting error to point out in support of pemdas.
Clearly the formatting of a paragraph of text in a textbook full of clearly and unambiguously written formulas discussing the very order of operations itself compared to the formatting of an actual formula diagram is going to be less clear. But here you’ve chosen to point to a discussion of why the order is irrelevant in the case under question.
Your example is the conclusion of a review of mathematics.
First we shall review some mathematics.
…
The actual order of differentiation is immaterial:
The fact that the example formula is written sloppy is irrelevant, because at no point is this going to be an actual formula meant to be solved, it’s merely an illustration of why, in this case, the order of a particular operation is “immaterial”.
Even if ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x), it doesn’t matter because “∂2f/∂x∂y=∂2f/∂y∂x”. So long as you’re consistently applying pemdas, you’re going to get the same answer whether you derive x first or y.
However, when it’s time to discuss the actual formulas and equations being taught in the example text, clearly and unambiguously written formulas are illustrated as though copied from Ann illustration on a whiteboard instead of inserted into paragraphs that might have simply been transcribed from a lecture. Which, somewhat coincidentally, is exactly what your citation is.
I was taught that division is just inverse multiplication, and to be treated as such when it came to the order of operations (i.e. they are treated as the same type of operation). Ditto with addition and subtraction.
And the other error present is the incorrect pluralisation. Mathematica means the entire area or domain of knowledge, while mathematics sounds like several lines of thinking, which is weird when we use it as a singular. Maths doesn’t refer to several kinds of math, and that’s confusing.
Ok, but the British also shortened television and made it tele. That makes sense because they took part of the word to do it. If you were going to shorten the word mathematics, why wouldn’t it be math, especially when that would follow what you did with television. Why shorten the word and then add the s from the end for no reason?
Television isn’t a plural word tho? And if they were talking about more than one television I feel like they would absolutely say teles (tellies?), altho not British so 🤷♂️
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