**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

(((4^2) * 3) / 6) + 1 – 5 = 4.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.