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 = 
Also, according to this
http://creativepro.com/fine-tuning-your-type-setting-fractions/
is not the same 6/2(2 + 1)=1 and 6⁄2(2 + 1) = 9
That’s fine if you ONLY follow order of operations and ignore every other principle that was discussed, such as algebra, coefficients, product of factors, quantities, etc.
A bit of non-science
There are numerous threads on numerous blogs re: 6 ÷ 2(1+2) = ?
I always distribute 2(1+2) which = 6, of course. And 6 ÷ (6) = 1. And have seen much concurrence on that.
On most of the blogs there is no shortage of good and bad math. Including, that with or without symbolic variables versus numbers it’s reduced differently. And “my calculator returns …” as ‘proof’. And the bad math is too often supported by insistence, anger, and disrespect.
Over the past several months I’ve asked, in person, many people is it =1?, or =9?
Everyone that I’ve encountered in person supports =1.
Online is the only place I’ve seen any insistence for =9. And also Excel, which requires verbose syntax.
1/2π is one divided by the product, not one half pi.
In a physics textbook, UC Berkeley, Nobel laureate Dr Richard Feynman describes the inverse proportion to 2√N. And writes that as 1/2√N. Expressly returning one over the product of two times sq root of N. And specifically NOT one half sq root N.
Feynman physics lectures, UC Berkeley press
Section 6-7. (Page 64 of the online version.)
Six over two(one plus two).
Software documentation
https://qalculate.github.io/manual/qalculate-expressions.html
Implicit Multiplication and Parsing Modes
The evaluation of short/implicit multiplication, without any multiplication sign (ex. “5x”, “5(2+3)”), differs depending on the parsing mode. In the conventional mode implicit multiplication does not differ from explicit multiplication (“12/2(1+2) = 12/2*3 = 18”, “5x/5y = 5*x/5*y = xy”). In the
“parse implicit multiplication first” mode, implicit multiplication is parsed before explicit multiplication (“12/2(1+2) = 12/(2*3) = 2”, “5x/5y = (5*x)/(5*y) = x/y”). The default adaptive mode works as the “parse implicit multiplication first” mode, unless spaces are found (“1/5x = 1/(5*x)”, but “1/5 x = (1/5)*x”).
The original is poorly written because of how computers were originally programed with their limited capacities to work math problems.
You changed the equation.
from
6/2(1+2)
to
6/2*(1+2)
Please try and argue that there is some immutable law of mathematics that makes those two equal, because what made those two equal is how computers read math problems and not the rules of mathematics. In computer programing if that equation needed to be programed in, then it would have to be entered specifically as 6/(2*(1+2)) so the computer would properly understand what was being asked.
ab/cd = (ab)/(cd) != ab/c*d
http://www.jstor.org/stable/2972726?seq=2#page_scan_tab_contents
The only thing that has changed is how computers read the equations, not the rules on how people should read equations.
You get 9 putting it into a computer because it is poorly formated to get the computer to answer the question being asked. Junk in = Junk out.
Had a programing teacher predict this day, when people would forget the proper rule of mathematics because of how the computer solves equations like this one. Seems you are pretty poor “experts”.
2(2+1)/6= 1 , therefore 2(2+1)=6 , not 2/3 . 6 over 2(2+1) = 6/6 . 6 over 2/3=9
Math is a total lie. The truth of this equation is the correct answer. The fact that it requires a universal effort to see the problem the same way or you don’t get the same answer is proof of the lie.
@Beth Herman, that is just a cop-out answer. Everything we do in life is defined by a set grouping of rules. Language, biology, chemistry, driving, walking down the street, what you do at work. Same with mathematics. The arithmetic we do daily has a certain set of rules in place that define how it works. The truth with this problem, is that we are now calling out how computers are designed to answer problems versus how people have been intuitively (or being taught) to solve them in the past. For those that have been through the era of not having a calculator for everything, the 2(1+2) is telling me that it is a part of the same “item” and needs to be kept together (in the denominator of the system), thus I would get the (2+4) which makes the answer 1.
Personally, I read this equation as 6/(2*(1+2)). If it was written as 6/2*(1+2), I would read that as (6/2)*(1+2), two seperate items being multiplied together, and I would get the 9 as an answer.
So yes, this is all semantics, but a sort of arguement that brings to light the differences between the different experiences people have had when learning how to process these equations, and what should be done about correcting these issues.
Yes, =1.
2(a+b) is a denominator. But that’s coming into conflict with the current Excel style math that is taught. On this site alone there have been obstinate dismissal of 2(1+2) as a quantity. And across the internet the argument is so uncivilized.
Apparently x(a+b) = (xa+xb) is no longer the zeitgeist. Although it is still the distribution identity. And distribution is called for.
And, one over two pi is 1/2π.
But that too has been distorted to:
1/2π = one half pi.
Go figure.
Ok, Short and to the point.
-:- is an obelus- represents division should be done
/ is a solidus- Diaganol Slash used to seperate a numerator and a denominator in an in-line fraction.
Division:
6-:-2(1+2) Here we would address this with the order of operations. Parenthases first, then M&D from left to right.
Resulting in an answer of 9.
6-:-2(3) or 6 -:- 2 x 3
3×3
9
In-Line Fraction:
6/2(1+2) Now we have an inline fraction, remember anything left of the slash is the numerator and anything right of the slash is the denominator. Or think of it as a seperation.
Resulting in the answer of 1.
6/2(1+2)
6/2(3)
6/6 or 1
or…..
6 6 6
——- ——- ——- or 1
2(1+2) 2(3) 6
This is EXACTLY what the slash represents. If you argue the point, your arguing the very definition of a solidus at that point!!
So 2(1+2) is not a monomial?
So x(a+b), the x is not a coefficient of (…)?
When is x(a+b) = (xa+xb) ?
When did distribution get redefined?
Dan Wolfe, that’s ridiculous. Who on earth eats apple cookies or apple muffins?!
Larry Scott, the issue is you are not distributing the entire ratio. You can’t assume that the 2 and everything to the right is the denominator as that is not how these rules are defined. If you want to distribute you can do it two ways. (6/2)(1) + (6/2)(2) or (6)((1/2)1+(1/2)2).
I will point out that the problem itself is flawed and meant to mess up people who forget math rules or get involved with distribution far too soon in the problem.
So is 1/2π = one over 2π or one half pi?
Is 1/2(pi) the same?
I have a bunch of new references STILL showing 1/2a = 1/(2a) as a common convention and no books use 1/2a to mean (1/2)a. I’ll post when I get to my other computer and dig the titles and links out.
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.
Comments? Discussion?
So 1/2a ≠ 0.5(a) by “common convention.
1/2a = 1÷2a I’ve only seen / and ÷ to be treated identically.
If a = (x+y)
1/2a = 1/2(x+y) = 1/(2x+2y)
Therefore, 6/2(1+2) = 6/(2+4) = 1
Therefore 6 ÷ 2(1+2) = 6/2(1+2) = 1
Previous post said “gets involved with distribution far too soon in the problem”. And that’s not what I was taught. I don’t understand “Too soon”.
Computers require verbose syntax (e.g.; Excel) rules. Which has become a Zeitgeist. And somehow the new rule.
Larry, I agree wholeheartedly that computers and calculators are the biggest culprit in this discussion. As you can see by my last post, all published mathematics is on your side.
Thank you for the confirmation. I thought it was more than obvious. But in this thread there is vehement opposition. Full spectrum opposition in support of Excel format requirements as the new set of rules.
The fate of distribution and implicit multiplication is on the line. Granted, when in doubt, the extra and heretofore unnecessary parentheses may be cheap, but the concept of monomial is becoming archaic and not taught.
Distribution is currently being woefully taught and passed around as well.
Thank you.
Here is an interesting video on the topic:
https://youtu.be/S_W7gW0wDcI
And other for Hindi Speaking audience:
https://youtu.be/s0J2vrn0V7k
Short: The real point of contention is not that people don’t know how to solve this problem, but that we don’t know how to interpret the problem, because it is a poor expression.
Long after this beaten horse is dead, I have a comment! It’s interesting that so many here became exasperated trying to convince each other how to SOLVE the math problem; some going so far to proclaim their own mathematics degrees as some sort of reason their solutions must not be debated!! BAHHH!
The real contention with “6/2(1+2)” is not how to SOLVE it, but how to READ it (how to determine what it means)!! It should be a simple problem where advanced mathematics degrees are irrelevant. We can all change how the problem reads (re-write the problem) and try to push our methods to solve what we’ve re-written, but what’s the point of that? There should be little to no disagreement on the proper method(s) to solve clearly re-written problems, but can we not just leave the problem as written and all agree that the form of the problem, as written, is itself the problem? –that there is no clearly defined single way the problem, as written, should be interpreted? It’s funny that that is what is happening in this thread: self-proclaimed math experts disagreeing on how to solve simple math from a textual string of characters that is poorly written–that could/should be written more clearly. ….. blah blah …. “a leading term should first be distributed through the term(s) in parentheses before invoking PEMDAS, but should it be the leading term or group of multiplicative terms that is/are distributed?” It seems we don’t have a clear rule for this when it occurs in a serial string of characters …. what should be inferred to be in a numerator and what should be in a denominator … this is where all characters on the same line becomes the problem with forcing the language of mathematics into a representation that wasn’t designed for mathematics, but was designed to be read in series for languages like, English or Hebrew, etc.
So my two cents to the discussion is this:
There is a very good language for writing math problems that involves symbols not found on common computer keyboards and placements of symbols that do not follow along in a serial string, from left to right, on a single line. Math problems, written as serialized strings of characters, can be unclear, and this is a great example: 6÷2(1+2) could be 1 or it could be 9, depending on how the problem is interpreted, but since the problem, as it is written, is unclear, the solution is UNDETERMINED. Furthermore, these representations are a bit more clear: 6/(2(1+2)) and (6/2)(1+2) … but we do not know which of these the original problem is meant to represent.
I don’t imagine there should be much debate over that!
So 1/2π in doubt? Poorly written? By Excel rules it may be in doubt.
Kudos To Ro
The problem has been presented very often as 6 ÷ 2(2 + 1) = ? Is it 1 or 9?
And the answer of 9 has been vehemently argued for over the answer of 1.
The same equation has been frequently presented as 6/2(2+1) = 1 or 9?
So the question is really, is 6/2(2+1) = 6÷2(2+1)? In classroom, I’ve been taught that they are equivalent. Just as 1/2π = 1÷2π = 0.159… I’ve never seen 1/2π = 1.5708… (one half pi).
In this thread 1/2π (1 over 2π) = 1÷2π (1 divided by 2π) has not been in questioned. (I haven’t read every response, but there doesn’t seem to be much support for one half pi.)
The real question is, does x/2(a+b) =
x/(2a+2b), or
x/2 × (a+b)?
The latter is verbose/Excel, the former is what I find in math and physics textbooks.
And if the latter, when did verbose/Excel become superior?
Once you finish with the parenthetical expression, the parenthesis go away. If you don’t you create the anomaly based on the convenience you learn in 7th grade. Meaning ()=* so you can use them interchangeably if you look at as a multiplication. If you don’t you also have the other implied “1” all around the () (ie the divided by 1, the ^1, etc.) Meaning the proper conclusion is as follows. 6/2(1+2)= 6/2*3=(order of operations says M and D are on the same level and you do them in order from left to right)=9.
In 7th grade, and I still see today, x(a+b) = (xa +xb)
Please tegetence when that was abandoned.
Are you saying that 1/2π = one half pi? and not one over two pi?
Or 1/2(pi) = one half pi?
All this talk about pi is making me hungry as hell! APPLE PIE!
6÷2 (1+2) controversy solved!
The answer is 1.
I know alot of people have been struggling to understand different peoples’ mathematical strategies on this problem. Yes it will have you in a heated argument with a best friend or family member thinking your answer is the right one.
Let me help clarify this
Let’s start by forgetting an order of operations . Please forget excusing dear aun’t Sally.
Math is never just about numbers. These numbers are a representation of something more than a numerical value.
Let’s bring out our least favorite component.
Yes! Word problem
Ok so we have in a room 6 apples.
We have 2 groups of people
2 of those people in each group are female
1 is male.
How can we divide these apples amongst them?
Ok so 2 females and 1 male in each group
(2+1)
We have 3 people in each group.
If you were a part of the people who thought the answer was 1 :
You would now multiply the 2 (3) before dividing this is where you would go ahead and say there is 6 people here.
6 apples 6 people ….i would say they get 1 a piece.
Correct?
If you were one of the people who would say we divide before multiplying then let’s try it out your way.
Let’s divide those 6 apples by 2 . We only have 3 apples and the 3 in parenthesis .
If the 2 is used in division, then that just leaves your 3 people to divide those apples amongst.
Every one of those people will only receive 1 apple.
Correct?
Let me clarify there is no multiplication symbol in this problem. Just parenthesis.
Those parenthesis may be an indication for multiplication only if you were saying there are 2 sets of those 3 people.
If you chose to divide the number of apples instead of doubling the sets of people then that’s your decision.
It’s still only going to ever be 1 apple for each person. No matter which order you choose.
The parenthesis only clarify the sex of the people in the room ( 2 female 1 male)
If you like to say
6 apples divided by 2 ( 2 female + 1 male)
The parenthesis are only there to group the 2 and 1
( 2 females +1 male) = (3 people)
It’s your choice of how you divide them up….thats why there’s no order in pemdas for multiplication or division. They are on the same level of importance.
6 apples divided by 2 =3
between ( 3 people)
3 apples (3 people) = 9 what?
= 1 apple a piece.
Or 6 apples divided by 2 groups of (3 people)
6 apples divided between 6 people
= 1 apple a piece
1 is the answer order or no order. The question is how to divide something between something else.
The math is revealed when you remember numbers are not an entity in themselves. They represent a value of something.
Perception is key. I hope this helps dissolve some arguments and open a couple of minds. Not literally a couple like 2 but ….you know.
I would get rid of the division to solve this problem. By turning it into fractions and multiplying by the reciprocal of 2(1+2) you get 6 x 1/6 or 3 x 1/3 if you reduce first.
This is the argument I would use in academia to support my answer of 1.
Currently researching info on this same problem. It was introduced in a programming group, and I see it as a far better programming question than math problem. We can all agree that it is a poorly written math equation (more parentheses, if you please!). However, I would like to throw my lot in with the =9 group, and here’s why:
Using a word problem: Johnny has two recipes for apple pie. The first recipe says that for every 2 apples, you can make 2 pies. The second recipe is the same as the first, but yields one extra pie. Johnny has six apples. Using a single mathematical expression to determine the number of pies Johnny can make, we could write 6-:-2*2 for the first recipe, and 6-:-2(2+1) for the second. In this instance, we know that six apples must yield 6 and 9 pies, respectively. This is different than writing 6/2*2 and 6/2(2+1), which makes them into a numerator and denominator and would give an incorrect answer of 3/2 and 1, respectively. This distinction is necessary in order to distinguish between the two, primarily due to the invention and progression of technology.
This also means that 1/2pi = one over two pi but 1-:-2pi = one divided by 2 times pi, or one half pi.
That being said, it is a great example in programming in that what you think the output “should” be matters far less than what the system actually outputs. If you don’t adapt to what the system does, your program will always fail.
@WyoMonkey, I totally agree with your clear explanation. A big issue is a denominator. x ÷ 2y or x/2y —- we don’t use this anymore because it’s very confusing.
“We don’t use this anymore” ? Only because of the verbose syntax requirements of software driven environments like Excel. The question then is are Excel rules the law of land? If x/2y is unclear, then are older textbooks and original works relegated to be misinterpreted? Or left to academics for correct interpretation?
There should be no doubt in interpreting x/2π, or 1/2π
Or π/2θ
Or x/2Δ
Of course anything is confusing to someone who is not educated in that area. An electrical diagram for a car would be confusing to anyone unfamiliar with them. There is nothing confusing about 1/2π nor was it ever confusing. Wyo: You are wrong about your comparison of the obelus and the solidus. There is no difference and this isn’t even debatable.
As for programming, the programmers put the onus on the user to input their data using parenthesis that would not otherwise be required, and then they announce this convention in the user manuals. That’s perfectly fine, however, it doesn’t change the math rules that have been in place for hundreds of years.
My comment above needs to be proven incorrect (the one from Aug 9, 2017) located here: https://www.askamathematician.com/2011/04/q-how-do-you-calculate-6212-or-48293-whats-the-deal-with-this-orders-of-operation-business/comment-page-6/#comment-601585
No one has been able to provide a solid rebuttal to it to date.
The answer is simple. Simplify to remove any parentheses or exponents becore any arthimatic is performed. The juxtaposition expression of 2(2+1) must be simplified before any arithmetic can be performed. A juxtaposition expression is treated as the last step of exponents in PEMDAS or the last step in orders in BODMAS. NOW, as for as computers and calculators being programmed not to accept a juxtaposition expression is due to the humans that programmed the machine. Junk in will get junk out.
Um, Mathman…
Explain this-“Wyo: You are wrong about your comparison of the obelus and the solidus. There is no difference and this isn’t even debatable.”
Do they both divide? Yes, they do.
Do they both have the same meaning as to how to divide? No they dont.
Obelus-divide the number on the left by the number on the right.
Solidus-On the left of the / is the numerator and on the right is the denominator.
Other than they both divide, how are they the same. They clearly have 2 different meanings.
Your Logic is a Truck and a Car are the same, because you drive them. Yes they are both automobiles(division) but they clearly have different uses(4-:-2 3+1/2+4)
I would say that Larry Scott is correct.
a(b + c) is not the same as a * (b +c)
a(b + c) = ab + ac
And the problem in question clearly says 6/2(1+2)
In my book that would translate to:
6 / (2*1) + (2*2)
6 / 2 + 4
3 + 4 = 7
The question shouldn’t be whether the answer is 1 or 9 (I would agree with 9 if it said
6/2*(1+2) ). Because the answer to the question is clearly 7.
I totally agree with the first couple of comment. The distribute property is aggressively. Being a younger kid I know that you will alyways do the distributive propert first…
6/3(2+1) will equal 3(2+1) all over 6.
Everyone I believe is overthinking this it is a simple problem and as stated is just 3(2+1) over 6
Wayne C: My bad. I meant to say “There is no difference between the ‘division slash’ and the obelus.” (The solidus is a fraction line.)
The challenge of course, is using it in-line.
“This also means that 1/2pi = one over two pi but 1-:-2pi = one divided by 2 times pi, or one half pi.”
→ I was referring to the above statement. This is what is completely incorrect.
1/2π = 1÷2π
I already posted reference after reference showing the convention. 2π is a single quantity. Wyo is left with the responsibility of justifying his assertion, with at least a few references and usage. And even then, a ‘few’ references does not imply global convention. I have yet to see a single math/calculus text showing 1/2π = (1/2)π, or similar.
½ is showing a solidus, as it a fraction “one half”. Division statements can also be represented as fractions, as we well know. If you research a bit, the slash ‘/’ can also be interpreted as a solidus: abc/xyz can be interpreted as a fraction abc over xyz; which is the same as abc divided by xyz.
Someone did post once, an online reference for a physics lesson, showing the formula:
KE = 1/2 mv²
But this was a quote mine attempt. The person who showed me this example to support their argument, left out the fact that in the previous paragraph, the author explained the formula, and how it was to be interpreted. The author then used an unorthodox type setting technique, inserting a space between the 1/2 and the mv², to make the 1/2 appear as a fraction by itself. It was clear from the explanation that the 1/2 was the fraction “one half”, times mv². This is perfectly acceptable for authors of any text to do so: state their in-line convention. 2 issues arise when it isn’t done though:
1) if there is no explanation, we need to have a general convention to use
2) if incorrect convention is used, “loose mathematics” are required to ‘decipher’ the author’s intent. What I mean is this: I already showed the correct convention that has been around since the 1800s. 1/2a = 1/(2a). 1/2(a+b) = 1/(2a + 2b)
So if I write C = 5/9(F-32) we have contradicting information. Without explanation we shall interpret as 5/(9F – 288), but we can clearly see that this is a temperature conversion, and we know it is 5 ninths times (F – 32), which should be written (5/9)(F-32). I saw another (good) author write it as: (F – 32) x (5/9). So in this case, we have to do some mental interpretation of author intent, which is plain and simple: bad mathematics and bad communication to your audience.
To drive home my point a little more, how is cos2θ interpreted?
Is is cos(2θ) or cos2 x θ ?
The point is, quantities are quantities, and we can’t replace product quantities by simply inserting a multiplication sign between the factors/symbols. We can’t choose to be obtuse and ignore the other rules that have been in place for decades.
Kind regards.
waou.. still debating this… no wonder the world is so messed up..
12/2(1+2)=x is a similar problem.
This is an explanation I wrote in one of the Facebook “discussions.” One of the PEMDAS gang said it was “crap.”
Suppose you are a fruititarian and you want an apple and an orange for lunch. An apple costs $1, and an orange costs $2. How much does lunch cost? (1+2)=3. You pay $3. Now it is dinner time and you want an apple and an orange. How much do you pay? (1+2)=3. How much do you pay for the day? 2(1+2)=(2+4)=6; or 2(3)=6. You pay $6 per day. If you have $12, how many days can you have an apple and an orange for lunch and dinner? If someone can tell me how to stretch this out to nearly three weeks, I’d like to know.
There was so much disagreement that I turned to Mr. Google and found an engineer (?) who made al YouTube video to explain the 6/2(12) version. He said he would use WolframAlpha to verify his results. He first solved it to get 9. Then he solved it to get 1. He concluded by saying – and this is truly remarkable – that while 1 is correct mathematically, it is probable that 9 is the answer sought on tests, including the SAT!
Concerning WolframAlpha, when the problem is entered in letter-symbols you get it displayed in the traditional over and under format: (I tried to copy and paste but it didn’t work)
y/[a(1+a)]
Substituting numerical values, you get:
12/[a(1+2)]
But if you substitute in a different way, you get:
(12/2)(1+a)
So the on-line calculator decides for itself which problem to solve.
One bit of wiggle room for those getting 18 is that this could be the formula for the area of a trapezoid.
(1/2)h(a+b)=(h/2)(a+b)
division distributes over addition from the right but not from the left.
It’s Ok to follow “order of operations”, if you do it correctly.
When you follow the distribute laws of mathematics you’ll see that the answer 9 is false as it distributes with an answer that doesn’t exist. When using division with distributive properties the rule of mathematics explicit say to solve the distribution first. That said also if division is used you need to check distribution both ways. Left distribution and right distribution to spot false answers that doesn’t exist.
Therefore the answer 9 is a false.
only 1 is a true.
my 2 cents
We can agree that there is one correct answer (1), but what is intriguing, if not distressing, is that so many are being taught to find another answer; and that computer programmers are designing calculators, including on-line calculators, that will provide a different answer. I will refer back to WolframAlpha: you can get different results depending on how the problem is input. It would be nice to have a meeting of the minds between mathematicians and programmers. Also, it would be good if the use of brackets were emphasized to remove ambiguity.
is 2(a+b) a monomial or not?
If your answer is yes, the answer is 1.
PS I’m petty sure it is.
PPS excel etc require an operator, so you can’t say 2(1+2), you have to say 2x(1+2).
So you have to force into being a monomial by adding the extra brackets. Excel is not wrong, it just lacks the convention.
It almost seems like its a generational thing. Being taught maths prior to widespread use of computers and only minimal use of calculators, I came up with 1. To me the contents of the brackets are multiplied by 2.
When I asked my son, who has good grades in maths in his final year of high school, he came up with 9.
If I see an equation thus: 6 -:- 2y = x. I will always multiply y by 2 first. Same if I see 6 / 2y = x. If we Let y=3, the answer is 1, UNLESS you apply the “order of operations” as per the “electronic” version. I also agree with other commentators to 2(2+1) should be seen as 2 x 2 + 2 x 1 = 6 and not as two seperate numbers to have the “convention” applied to. As I said, are people being taught differently now, or have I had wrong for the last 50 years?
2(1+2) is a monomial; monomial is a lost concept.
Distribution applies.
Excel seems to be the new normal.
1/2π is NOT one half pi.
Larry,
Totally agreed. Every math text that I’ve read, and uses inline notation, agrees with you. Not a single use of this notation ever suggested (1/2)π or similar with other numbers/symbols. There is some guy over on youtube claiming that there are 2 sets of math rules: one for using digits to represent numbers, and another for when letters are used. Also, no one has been able to refute the interpretation of cos2θ or cos2π. That said, even when rules, publications, references galore, some people will still refuse to accept these facts.
The moral of the story is never write 6 ÷ 2(1+2) . If you’re at a limited keyboard write (6/2) (1+2) or 6/[2(1+2)] depending on which you mean; with a more powerful program or a blackboard , of course, you are more flexible as to what you can do with the division line.