Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.
The “order of operations” tells us the sequence in which we should carry out mathematical operations. It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. You could come up with a different convention, but why bother? Let other people waste their time coming up with arbitrary conventions so that you can merely waste time reading posts about them.
The rules to follow are:
1. Things in parenthesis get computed before things not in parenthesis (this is because of the (little known fact) that parentheses kick ass).
2. Then, exponents get dealt with (including roots). Hence a^b*c = (a^b)*c.
3. Next, multiplication and division get carried out from left to right. So a*b/c*d is ((a*b)/c)*d. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another.
4. Finally, additions and subtractions get carried out from left to right. Subtraction can be thought of as being like addition since a – b = a + (-b) so it makes sense that subtraction is treated on an equal footing with addition.
People sometimes use the expressions PEMDAS, “Please Excuse My Dear Aunt Sally” or “Puke Exhaust Mud Dolls Autumn Sasquatch” to remember these rules. These all stand for “Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction”. It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right). That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch”
One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever. But that implies there is something interesting about them. This is a contradiction, and hence the order of operations form a paradox. They are also about as fun as getting to watch a fish crawl up the stairs as a reward for answering math questions.
But, that won’t stop me from giving an example. Consider:
a^b*c/d+e-f.
Without a convention, it has many possible interpretations, such as
a^(b*(c/(d+e)))-f
or
(a^b)*((c/(d+e))-f)
which could be horribly confusing for people trying to communicate with each other. But, the correct interpretation according to the rules mentioned above, is:
(((((a^b)*c)/d)+e)-f).
When in doubt about what the orders of operation are for an expression, there is a simple solution which works even better than tattooing PEMDAS to your face. Just paste the expression into the google search bar and hit “search”. For instance, if I paste in:
4^2*3/6+1-5
it gives back
Not only is this the correct answer, but it shows you explicitely the order of operations that were used, so it’s useful for learning.
Google even gets this one right:
4^2*3/6 + 1 – 5/14*3 + 6/10 – 4*2/14*6^3/18 + 14
which is the kind of question little sisters give you when they are old enough to finally sort of get what a mathematician is.
It’s important to note that not all calculators handle the order of operations in the standard way. Why? Because people who make calculators hate children. The preceding sentence was not intended to be a factual statement.
Also, a lot of programming languages use the standard order of operation rules, but then you have exceptions like Smalltalk.
Oh, and in case you were wondering:
a^b^c = 
This is not a complicated problem, nor is it ambiguous. 6 is a monomial, and so is 2(1+2). We could write the problem as a÷b, where a=6 and b = 2(1+2).
Don’t even need to use PEMDAS. Just reduce the monomial 2(1+2) to its simplest form (6) and you get 6 ÷ 6 =1.
Want proof? 2(1+2)=6. Divide both sides by 2(1+2) and you get 1=6 ÷ 2(1+2).
Some will tell you that 2(1+2)= 2*(1+2), and so it does, by itself. But when used with other numbers and operations, you need to add [] so as not to lose track of its identity as a monomial. Thus, 6 ÷ 2(1+2) = 6 ÷ [2*(1+2)], not 6 ÷ 2*(1+2).
The whole thing has come about by Excel verbose syntax becoming the new rules.
1/2π is 1 over 2π, but a boat load of argument that it’s one half pi.
6/2(1+2) =
the way this is written could be worked the same way as distributing the 2 to what is inside the ( ) … therefore: first step is to distribute the 2*1 + 2*2 = 2+4 = 6, then working the 6/6 = 1
Of course. Distribution at its most basic.
6/2(1+2) = 6 ÷ 2(1+2). (And that’s not an opinion. Just like 1/2π in not one half pi.)
2(1+2) is a monomial; therefore 2(1+2) is evaluated first. Returning (2+4). Distribution.
6 ÷ (2+4).
I don’t understand how this is an issue. I guess it’s only because Excel needs verbose.
PEMDAS order of operations is Parenthesis, Exponents (and roots), Multiplication AND division, Addition AND subtraction… You do anything in Parenthesis from inside to outside 1st… Then you do you exponents and roots (though roots aren’t mentioned, but they’re exponential and just the inverse function of an exponent) 2nd… Then you do your Multiplication or division 3rd, and you do it left to right… Note, you DO NOT do multiplication first, then division like PEMDAS might have you think you do because the M comes before the D… Because Multiplication and Division are the same type of math, just inverse to each other (like exponents and roots mentioned above are the same math, just inverse functions of each other) you treat them with the same priority… So you do them left to right… Same with the 4th step, Addition and Subtraction… Both have the same priority and are done left to right because both are the same function of math, just inversely done… INFACT, is 4-3 note the same as 4+-3?
So the order of operations is not exactly:
P
E
M
D
A
S
but rather, it is actually:
P
E
M or D, going from left to right
A or S, going from left to right
and again, wherever you see E, E also includes roots…
So with all that said… 6/2(1+2) ACTUALLY equals 9…
No one argues doing the parentheses 1st…. so 6/2(1+2) becomes 6/2(3) or 6/2*3… Everyone agrees this far… However it is the next step that is widely argued… do you divide 1st, or multiply? PEMDAS as mentioned above would have you thinking M before D, thus multiply 1st where 6/2*3 becomes 6/6 or 1 is the answer… BUT AGAIN, that is wrong… Because Multiplication is the same (but inverse) function as Division, it is treated as the same priority, and you do it left to right instead… So with that in mind, 6/2*3 is done left to right and becomes 3*3 or 9 as the answer… The answer is in fact 9…
I read somewhere that this confusion stemmed from teachings before 1917 but was clarified in 1917 and this has been how it has been taught since…
Again, Addition and Subtraction are grouped together as “co functions” or inverse functions of each other… IE 4+3=7 and 7-3 brings you back to 4… They are of the same means in moving across the number line, and only really specify in what direction you go… Again, is 4-3 not the same as saying 4+-3??? The same is true about Multiplication and Division… They both work the same way but inversely to each other….. 5*3=15 and 15/3 brings you back to 5…. You move across the number line the same way, just by difference in direction you go…
Same as exponents and roots…. 3^3=9 as 9 to the 3rd root is 3… Again, same function to move across the number line, but different direction as they are inverse to each other… This is why all these functions are treated with the same priority as their “co function” or inverse function… Otherwise explain why Multiplication should be done before Division? There is no reason… It is literally the same type of math…
So for these similar functions, you simply treat them with the same priority, and just do them left to right…
Because in PEMDAS, if Addition and Subtraction are considered, and you were to do A befor S just because A comes before S, is not 4-5+2 not the same as 4+-5+2? Now what do you do first? Originally you’d think the 5+2 as A before S, but the 2nd way it is written is the the same expression as the 1st, they are equal to one another, but in the 2nd expression you’d do the 4+-5 first because it is now an addition problem as opposed to subtraction… Even though it is no different than the subtraction problem of 4-5… This is why these functions are treated with the same priorities… Because they are the same functions… You just do them left to right…
In this example going by PEMDAS as commonly though or as taught before 1917:
4-5+2=4-7=-3
However using PEMDAS as it is taught now since 1917 with realization that addition and subtraction are the same function and are just done left to right:
4-5+2=-1+2=1
But lets revisit this converting the problem to 4+-5+2… Now it is ALL Addition, there is no subtraction:
4+-5+2=-1+2=1
And now you see why PEMDAS is really P/E/M or D (left to right)/A or S (left to right)… And for whatever reason roots are not included in the PEMDAS (or PERMDAS or PREMDAS as it would otherwise be with roots included into the acronym)… But Roots being exponential, are prioritized equally with E or Exponents and done again from left to right…
So to conclude…. 6/2(1+2) is in fact equal to 9…
You have just rehashed the argument present over and over (and over and over…) again on this thread, and it will come to the same result. Those of us who tend to read the answer as one are doing so because we read 2(1+2) not as simple multiplication equal to 2 times (1+2), but where 2 is the COEFFICIENT of the parenthetical expression, the same way we would read 6/2x as “six divided by 2x,” and not “six halves times x.” Which reading is correct? I do not think there is a correct because the statement as written could easily be read either way.
Ultimately it is the ambiguity of expressing mathematics in a single line format, instead of how this would normally be written with a horizontal line and clear as to what is in the numerator and what is in the denominator. Thus the debate rages on, with both sides demanding they are right, when it is really neither / both.
A lot of the misunderstanding is due to typography. My keyboard makes it hard to tap in an over/under fraction. Is the problem
6
_____
2(1+2)
Or is it (6/2) X (1+2).
Some problems should not be solved until clarified. It seems that mathematicians solve this as the over/under fraction, while many engineers and calculators solve it the second way.
Yes, certainly. Responsible problem solvers should refuse to give an answer until it is clarified what the intent is of the problem-expression, and this goes to education, teaching people to make sure their problems are not ambiguous. Is it (6/2)(1+2) or is it 6/[2(1+2)] ? In this way, there is a “grammar” of mathematics, similar to the grammar and punctuation or even spelling in language.
If the problem was written as
6÷2(a+b)
then I believe most would interpret it as
6÷(2a+2b)
therefore
6÷(2+4)=1
rather than
6÷2a+2b
replacing “a” and “b” above with “1” and “2” and applying pemdas results in
6÷2+4=7
Using the former method offers a far greater level of consistency between algebra and arithmetic, thus making mathematics simpler and more logical for all concerned.
Pemdas was introduced to make mathematics more consistent, yet in this instance the opposite is true. The original problem highlights the inadequacy of pemdas by exposing its oversimplification. i.e.: it fails to incorporate monomials and distribution, thus generating this seemingly endless debate … 1 or 9?
For 9 to be the correct answer I believe the problem should be stated as
(6÷2)(1+2)
Otherwise we are unnecessarily (and IMHO, wrongly) splitting a monomial and ignoring the law of distribution for the sake of placing pedmas on a pedestal which it does not deserve.
No seriously. Yes, the thing is made to mislead and that’s bad.
But no operator *always* implies multiplication. People who distribute the 2 into the brackets therefore just break the order of operations at that exact point – they evaluate the multiplication before the division, even though it comes after. and 1/2pi = 1/2 pi (I’ve just added a Space, which carries no mathematical meaning) which hopefully everyone already sees as half of pi. Without additional brackets there is no reason whatsoever to assume 2*pi is one term.
You obviously didn’t read any of my posts. I literally provided myriad of references proving you are completely incorrect.
1/2π is one over 2π. Only in excel is that one half pi.
Never have I seen that, and there dozens of supporting comments above that 1/2π is NOT one half pi.
I like the answer 7.
6 ÷ 2(1+2)
6 ÷ 2*1 + 2*2 (distribute the parenthesis)
6 ÷ 2 + 4 then left to right
3 + 4 = 7 division and multiplication before addition and subtraction
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People who rely on calculators or search engines will often (not always) get 9. In Wofram Alpha, for example, solving 3a/a(1+a) = 3 (a + 1). With a=2, then 9. I think this is wrong, but that’s the way they have it programmed.
6/2(1+2)
Question: is 2(1+2) a monomial ?
If not, when were monomials abandoned?
6/2(1+2) distribution first to evaluate the MONOMIAL
6/(2+4)
6/6=1
Question: if distribution isn’t done first, then when was distribution abandoned?
Are Dr Feynman’s physics books wrong? As he takes 1/2n as one over 2 times n, and explicitly not one half n.
Q: When did 1/2π (one over 2π) become one half π? Because that must be brand new math and existing math books will need to be burned.
PEMDAS is a tool to help avoid ambiguity. It was never intended to be a replacement for established laws governing, among other things, monomial expressions and distribution.
An unhealthy obsession that many in the 9 camp seem to display is to continually accuse us “oners” of confusing the correct order of the “M” and the “D” when applying the order of operation.
This has nothing to do with it! What the 9 people are doing is incorrectly interpreting the “P” part of PEMDAS. Us 1 people, on the other hand, are first resolving the “P” before moving on to the “M” and/or “D” – in total accordance with PEMDAS.
To correctly resolve what is inside the parentheses one must solve the WHOLE expression. The first “2” in 6÷2(1+2) is the coefficient in a monomial expression. It is an integral part of the expression, and not simply an isolated independent digit with an implied multiplication sign after it.
e.g.: 1/2π or x÷3ab
A monomial expression cannot be resolved correctly if you disregard any part of the expression, including the coefficient.
I think most of us agree that the “M and “D” in PEMDAS are interchangeable and should be solved left to right. The real difference of opinion centres on the correct method for solving the “P” part.
The 9 camp keeps endlessly repeating the “oners” supposed misinterpretation of the order of operation simply because they have no valid response to logical arguments relating to monomials and distribution.
For the handful of 7 followers, the correct distribution of x(a+b) is not xa+xb. It is (xa+xb). Distributing the expression does not magically dissolve the parentheses.
Be you a 9 or a 7 fan, you may choose to ignore the importance or even existence of
clear and unambiguous laws governing monomials and distribution , resorting instead to the parrot like repetition of a skewed misinterpretation of PEMDAS. But please note that in doing so you make a mockery of not only some fundamental mathematical principles, but also PEDMAS itself – the very tenet to which the 9 camp clings so passionately, albeit erroneously.
The answer is 9 why are you distributing like terms? Even if you did distribute correctly for no reason mind you the answer is still 9. Who doesn’t know 1+2 = 3. But ok let’s say u don’t if you are to distribute for whatever waste of time y’all have decided you have to distribute the whole 6/2 which is 3
3*1+3*2 =9
What a long way to do a simple problem that for some reason y’all are looking at as advanced math.
6/2(1+2)
6/2(3)=9
Kate,
In an earlier post you stated that “THIS ISN’T ALGEBRA ITS FOURTH GRADE ARITHMETIC”.
So, should algebra and arithmetic have two separate sets of rules governing the way problems are solved? Should x(a+b) be calculated differently to 2(1+2)?
Should knowing the values of “a” and “b” alter the way we solve the problem?
Are the laws governing monomials and distribution no longer relevant in modern mathematics? Or do they only apply in some branches of mathematics and not others?
May I suggest that what you are doing is inserting a non existent operator into the problem. Had it been written as 6÷2*(1+2) then the answer would be 9.
But it wasn’t!
In your most recent post you claimed that
“… if you are to distribute for whatever waste of time y’all have decided you have to distribute the whole 6/2 which is 3
3*1+3*2 =9”
6÷2 is not a monomial expression. It would be a monomial if written as (6÷2).
But it wasn’t!
It is quite unambiguously written as 6÷2(1+2). The 2(1+2) is all one, single monomial expression, and therefore must be solved first. This is not contrary to PEMDAS. It is actually using PEMDAS as intended – solving the “P” bit first.
Sadly, it’s unlikely that any of the above will convince a die hard 9 fan so let’s try to look at this another way:
Hopefully everyone (even fourth graders) will agree that
2a=a2
It is merely convention that we place the digit(s) ahead of the symbol(s). Either way they are both monomial expressions.
Similarly 2(a)=(a)2, just as 2(a+b)=(a+b)2, etc.
Therefore 2(1+2)=(1+2)2.
Applying your version of PEMDAS the problem 6÷(1+2)2 = 6÷3*2 = 2*2 = 4.
Applying the monomial rule, and PEMDAS correctly 6÷(1+2)2 = 6÷(2+4) = 6÷6 = 1.
Amazingly, we still arrive at 1.
You may substitute other values within and outside the parentheses, yet the consistency always remains when monomials are calculated correctly. And isn’t consistency and the absence of ambiguity the whole purpose of creating PEMDAS and other mathematical rules in the first place?
This isn’t really an issue with order of operations. But how and equation is written or converted from standard notation to line notation. Both of the equations below could be lazily rewritten as 6/2(1+2) however they are not the same. There is no clear way to tell which equation is intended. Hence both answers would be acceptable.
6
——— x (1+2)=
2
6
—————— =
2(1+2)
Dr Feynman textbook, every textbook I’ve seen, 1/2π is not one half pi. So that makes textbooks at large lazy.
My earlier copy and past from Wolfram Alpha got garbled.
If you go there and put in 3a/a(1+2), with a-2, the calculator will give you 9. I believe this is wrong, mathematically, but it is the way many calculators, hand-held and on-line, have been programmed. Even though resolving 6/2(1+2) should be easy and trivial, there is a deficiency of education with many people relying on rote application of (their understanding of) PEMDAS [BODMAS]. There is also an issue of — what to call it? integrity? ethics? — that some problems should not be solved until it is made clear, with extra brackets or otherwise, exactly what is intended by the expression. The extra brackets can be considered similar to grammar or punctuation in writing, and the ethical stance would be not to solve a problem that is characterized by sloppy writing. Sadly, this will require people to think.
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6/2(1+2) or 48/2(9+3)
The first answer is 1. The second answer is 2. Both are exactly the same with different numbers.
Because of the order of operations, anything in parenthesis is done first, and this includes the distributive property that needs to be done with both problems because in both, the 2 is directly outside and touching the parenthesis, and therefore included in the first step. First you do the arithmetic inside the parenthesis, then you multiply it by the 2 outside the parenthesis, then you’re left with 6/6=1 for the first problem and 48/24=2 in the second problem.
IF and only if the problem was written (6/2)(1+2) or (48/2)(9+3) would you get 9 for the first problem and 48 for the second problem. However, because there are no parenthesis in either problem separating the first two numbers from the numbers inside the parenthesis, you use the distributive property and multiply the 2’s outside of the parenthesis into the parenthesis.
The problem is purposely written this way to make people argue, but anyone who understands the fact that PEMDAS is a building block in which other properties can be added such as the distributive property, will get the right answer. Those who just move from left to right and ignore the distributive property or don’t understand that any number touching the parenthesis is included in the first “P” part of the order of operations will continue to get the wrong answer on paper or if they enter it into a calculator.
6/2(1+2) = 6/2(3) = 6/6=1.
48/2(9+3) = 48/2(12)= 48/24 =2.
For anyone who got 9 for the first answer and 288 for the 2nd answer, you’re wrong. It may come out that way if you enter it into a calculator but calculators don’t have brains and are only as good as they’re programmed.
This is still a problem to this day. Saw this 8 •/• 2(2+2) on FB a week ago and then on the news two days ago. I believe I was not taught all the rules to using DP. The masses were getting 16 because they didn’t even know what DP was. Lol. The scholars (myself included) were getting 1 because we used DP. After researching for a week i see that using DP with division is shunned upon for this reason. If using DP you must remove the division by converting it to a fraction. You would then distribute 6/2 and not just the 2. Keep in mind DP is a property, not a law. Meaning it is not needed to complete this problem correctly. I have changed my stance. I had a dumb moment. Lol! If you complete these problems using ‘straight’ math you will not get 1. Math should never change and I don’t believe it has. I was just misinformed on this earlier in life. If anyone knows any calculator that gets 1 without changing the way you enter the problem please let me know?
Also, another problem I see in a lot of your calculations. Mine also is that with DP it is a(b+c)=ab + ac
Hence, the parentheses are GONE. We cannot magically keep them there.
Also, the problem 6/2(1+2) is different from 6/(2(1+2)).
I looked in some old math books and it is actually called the “Distributive Property of Multiplication”. To stop the confusion i guess if we see a division symbol first, that should be a warning sign not to use DP.
Distributive property has not gone away.
a(b+c) is a monomial, also not gone away.
Apparently Excel rules are the new norm.
However, in what academic context is 1/2(pi) = one half pi? And explicitly not one over 2(pi)?
Answer that!
Excel cannot identity Monomials without verbose syntax. Computer code requires verbose syntax.
1÷2(pi) has never presented, in my experience, as one half pi. Ain’t no way.
Looks like I need to repost the following:
What are Quantities?
Intro to Algebra: Bonnycastle
pg 13 a and b are factors of ab
3abc is a composite quantity.
pg 25 simple quanities, examples on pg 26 such as 6ab÷2a=3b
https://books.google.ca/books/about/Bonnycastle_s_Introduction_to_Algebra.html?id=1YhTAAAAYAAJ&redir_esc=y
Introduction to Real Analysis by Bartle and Sherbert
http://iuuk.mff.cuni.cz/~andrew/bartle_introduction-to-real-analysis-new-edition.pdf
page 42: (x²-2)/2x
page 53: 1/2n = 1/(2n)
pg 350: e/2M
pg 363: 1/n(n+1) < 1/n² <= 1/n and n/(n-2)(n-1)
Measure and Integral: An Introduction to Real Analysis By Richard Wheeden, Antoni Zygmund 1977
https://www.scribd.com/doc/275407108/Richard-Wheeden-Antoni-Zygmund-Measure-and-Integral-Pure-and-Applied-Mathematics-1977
on page 31: 2Mε/4(k + 1)M = ε/2(k + 1)
Introduction to Real Analysis: William F. Trench
http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
pg 123: k(ε/2k) = ε/2 (I coudldn't find the correct font for ε here, so I used ε instead of the one in the book. I was accused of dishonesty in the past for doing this, so I'm getting that out of the way now)
The Everything Guide to Algebra: Christopher Monahan
Describes "PEMDAS" in detail, then later, on page 46 shows: 6x³÷3x=2x²
(That's because "3x" is a single quantity/operand, and is a product of both factors: 3 and x)
Basic Algebra I: Second Edition
By Nathan Jacobson
https://books.google.ca/books?id=JHFpv0tKiBAC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
pg 116: a/b + c/d = (ad +bc)/bd
Introduction to Algebra By Peter Jephson Cameron
https://books.google.ca/books?id=syYYl-NVM5IC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=mx&f=false
Page 17 "Since mx/nx = m/n" [Notice how it is NOT m * x / n * x = mx²/n ???]
FACTORS and COEFFICIENTS:
https://books.google.ca/books?id=yRRtCgAAQBAJ&printsec=frontcover#v=onepage&q&f=false
Otherlinks:
http://www.onlinemathlearning.com/multiply-divide-expressions.html
http://www.purplemath.com/modules/orderops2.htm (Elizabeth Stapel: http://www.purplemath.com/resume.htm)
Discussion on algebra: http://www.jstor.org/stable/pdf/2972726.pdf
It appears that the universal convention is that 6/2a = 6/(2a), and not otherwise. An author may choose to use any convention they like, but as for the last hundred-and-something years, this is the accepted notation.
Also, programmed calculators capable of handling equations like this have not been programmed to use juxtaposition properly; or it was just easier to convert juxtaposition to multiplication, ignoring monomials, etc, while stating in the instruction manual how to enter data properly. Wolfram Alpha has handled this very concept differently, changing it’s code a few years back. Because of the digital age, it appears people are learning from calculators.
What bothers me is that Google and other “sources” so easily exchanges a(b+c) with a * (b+c) instead of (a * (b+c)). The coefficient of the monomial should be treated as part of the parenthesis. Too bad in many Pemdas instructions they talk only about inside the parenthesis.
Google is also inconsistent.
It recognizes 1/2π correctly as 1 / (2 * π) but incorrectly interprets 1/2x as 0.5 * x.
It recognizes 2x/2x correctly as (2 * x) / (2 * x) but you might think there should be some information of the undefined part where x=0.
It recognizes 2π /2x correctly but for 6.28/2x it shows you the diagram för 3.14 * x. :O
If Google consistently exchanged the coefficient by also include a covering parenthesis it should all work out in the end, ie. 2x is not 2 * x it is (2 * x).
Mathematicians do it one way; programmers do it another way. PEMDAS (no matter how spelled) is useful, but too many have simply memorized it without understanding it.
And there you have the human mind versus computer software.
Computers need verbose syntax. And somehow excel syntax has become the new normal. And just electronic calculators damaged learning, computer software is damaging learning.
1/2pi 1/2(pi) 1÷2π
A computer needs some help. I only see one over 2 pi. Who how a computer will interpret these. It’s dependent on whomever wrote the code. And not all software is the same. Fortunately we brains and see context
Larry,
Math hasn’t changed. Technology has. There is no gold standard for coding calculators and this has become a huge crux of the problem. Not only that, the coding is inconsistent as Pontus have demonstrated. I have also witnessed the code change in Wolfram|Alpha back in 2013/2014, where something like 2n/2n used to be 1, but now it is = n², with explanation in the rules for entering equations. I even wrote to them, and they basically said that the user needs to know what their desired inputs and outputs *should* be, and use parenthesis as required.
Juxtaposition and implied parenthesis for quantities still exists:
cos2θ is not cos2 * θ. It is cos(2θ)
And 5 posts up, there are numerous more examples. 99.9% of reference material uses this notation, while I did find 1 book reference which does not, however, they did NOT blindly use something like 1/2a = (1/2)a … they actually explained and showed how they wanted their notation to be interpreted.
99% of the arguments in these debates are “my calculator says it’s (this)!! Calculators *can* be correct, but are not universally. The user has a responsibility to input the data according to the rules of the calculator.
It has to be one, because one is the only answer that is constant when reversing the equation
———————–
6 ÷ 2(1+2) = 1
1 × 2(1+2) = 6
———————–
6 ÷ 2(1+2) = 9
9 × 2(1+2) = 54
So the 9 doesnt work in this case.
Another thing to take into consideration for everyone is something called sideways formatting and Multiplication-by-Juxtaposition.
This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing. (It’s become an annoying popular game to post these to Facebook and other places.
Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
I don’t simplify in the usual way, and because of it, I usually match with most calculators:
16 ÷ 2[8 – 3(4 – 2)] + 1
16 ÷ 2[8 – 3(2)] + 1
16 ÷ 2[8 – 6] + 1
16 ÷ 2[2] + 1 (**)
8*2+1 = 17
But others will show this:
16 ÷ 4 + 1
4 + 1
5 in the denominator
The confusing part in the above calculation is how “16 divided by 2[2] + 1” (in the line marked with the double-star) becomes “16 divided by 4 + 1”, instead of “8 times 2 + 1”.
That’s because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses seems somehow to outrank division, at least according to some teachers, but most books teach it using the left to right rule. The first 2 in the starred line is often regarded as going with the [2] that follows it, in other words as distribution rather than with the “16 divided by” that precedes it. That is, the multiplication that is indicated by placement against parentheses (or brackets, etc) and is often regarded (by science-y folks) as being “stronger” somehow than “regular” multiplication which is indicated by a symbol of some sort, such as “×”.
Typesetting the entire problem in a graphing calculator verifies the existence of this hierarchy, at least in some software but not all, and as I said, I always teach it from left to right. However, what would happen when you get to upper math? I’ll show you at the end. Get ready, it’s quite interesting.
The actual answer is 3.2
Plug it into a calculator using parenthesis around the entire thing. Why use a parenthesis around the entire thing? If you don’t, your calculator reads the first number first, not the entire thing as a denominator. The mathematician is incorrect about the 9 too. It works the same exact way. Plug it into the calculator using the horizontal bar and you can see this take place automatically. 😉
I do not hesitate with some things like “16 ÷ 2[8 – 3(4 – 2)] + 1”, I´m a developer, so, the mathematics said me:
16 ÷ 2[8 – 3(4 – 2)] + 1 = 16 ÷ 2 x (8 ‒ 3 x (4 ‒ 2)) + 1 = 17
nothing else
to say…
Generally speaking the slash(/) and the discrete division symbol(÷) are used differently. Although they are both representations of a division process. 1/2π would be approximately 1.5708 and 1÷2π more akin to 0.15915. Similarly the viral problem of 8÷2(2+2) would first have the distributive nature of the parenthetical be resolved into 8÷(4+4) and then to 8/8=1. However the statement of 8/2(2+2) would go quite the other way, with 8/2 representing the coefficient of the parenthetical. This form would then result in 4(2+2)=16. The difference is in the separation of terms. In the first instance there is the division of one term discretely separate from the second, in the second case, it’s simply one term being evaluated.
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Has this or similar issues shown up in court cases?
Let’s say you have a half-dozen cupcakes. Sitting at each of the 2 tables in the room is 1 girl and 2 boys. You’re giving the cupcakes to the children. How many cupcakes does each child get?
6÷2(1+2)
2(1+2) is a term, specifically a monomial — like 2x. A monomial has a single value of the PRODUCT of the coefficient multiplied by the variable (factor). Since a monomial has a singular value (indicated by juxtaposition), no additional brackets are ever necessary around a monomial.
Now let’s say that the cupcakes came in 2 packages, with each pack containing 1 vanilla cupcake and 2 chocolate cupcakes. The statement of “cupcakes divided by
children” can now be written as:
2(1+2)÷2(1+2)
or with a slash…
2(1+2)/2(1+2)
Now replace what’s inside the parentheses [1+2] with the variable “x,” making the statement:
2x/2x
or that same fraction written vertically as…
2x
__
2x
No, the quotient is not “x squared” — because a monomial has a single value of the PRODUCT of the coefficient & the variable (factor). You don’t get to peel off the coefficient & do some other operation with it as if it had no bearing on the variable. There is never a need for additional brackets around a monomial because of its inherent singular value (indicated by juxtaposition).
David said:
“Generally speaking the slash(/) and the discrete division symbol(÷) are used differently.”
Incorrect.
from “Algebra I for Dummies” teaching site:
https://www.dummies.com/article/academics-the-arts/math/algebra/recognizing-operational-symbols-in-algebra-194538/
“The division, fraction line, and slash symbols all mean divide. The number to the left of the ÷ or / sign or the number on top of the fraction is the dividend (in this example, 6). The number to the right of the ÷ or / sign or the number on the bottom of the fraction is the divisor (in this example, 2). The result is the quotient (in this example, 3).
6÷2=3
6
_ =3
2
6/2=3”
————-
The fraction bar (vinculum), the slash (solidus) & the division sign (obelus) all mean “divided by,” making them all interchangeable.