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 = 
@Ray: If you could show me a few reliable sources where fractions written with a ‘/’ don’t use ( ), I would love to read about that. Those are the 2 convincing reasons to stand by 1.
1 – ( ) are required for (6/2) to be distributed (See below for the derivation)
2 – Grouping of terms like this {4 [2xz⁴ (2c³v ÷ 2cv²)³ ÷ 2]} ÷ c⁶x • v³ = 4z⁴ This is exactly how I was brought up to solve and simplify these equations. Do you get 4z⁴ as a solution? For example : 2c³v ÷ 2cv² = 2c³v ÷ (2cv²) = c²/v
Notice the grouping there, but no grouping for c⁶x • v³
6 ÷ 2n = 3/n where n = 2+1, therefore:
6 ÷ 2(2+1) = 3/(2+1) = 1
This was done with straight substitution, no extra ( ), no extra “times” symbols:
Here is the derivation of “9”:
3 * 3 = 9
6/2 * 3 = 9
6/2 * (2+1) = 9
Then, if you want to remove the “*”, you need to use ( ) since you have a fraction:
(6/2)( 2+1) = 9
Same goes for the “distribution” of 6/2, like this:
6 + 3 = 9
(6/2)2 + (6/2)1 = 9 Again, fractions with ‘/’ need the ( )
(6/2)(2+1) = 9
(6/2)(2+1) = 9
6(2+1) / 2 = 9
6/2(2+1) = 1
6/(2+1)2 = 1
One last time:
6 ÷ 2n = 3/n (the infamous wolfram even agrees on this)
6 ÷ 2 * n = 3n
Best Regards,
KM
@KM
In the question at hand, the parentheses used are for one purpose. It is to separate the values inside from the two to indicate that this operation should be done first. You, and others, are making an attempt that the 2(1+2) take the form of the basic Distributive Property, a(b+c)=ab+ac. You do this with no regard for the 6, which IS a factor of the values preceding the parentheses. There is no requirement to place 6/2 into () because the long ago agreed upon rule of Orders of Operation already state that if multiplication and division occur within the same expression, then one does not take precedence over the other and they are to be executed from left to right. There are no published exceptions to this rule. To say that this expression fits into the Distributive Property axiom is ludicrous. Whether anyone likes it or not, in the expression at hand, 6 and the first two are factors. All of the math texts I have provide this identity: a/b=ab^-1 and that this is where the identical precedence for multiplication and division gets its basis. In your explanations, you always place the last two factors in the denominator of the fraction which can only occur if you execute the multiplication on the right prior to the division on the left. The 2 and the (1+2) have no implied grouping as you have done and only applies as Distributive Property (which is the basis for your grouping of the two) if there were no factors preceding the first “2”. ab(c+d)=abc+abd and yes the “6” and the “2” are, by definition as provided above, factors.
but they are required when you write out the equation to the area of the rectangle and then attempt to divide a single number by that whole equation.
That is incorrect 6/2n where n=2+1 equals 9 inorder for it to be one the equation would have to read 6/(2n)
Agree caper26 and I am pretty sure that there are still people who will argue against the answer of 1..
the way I see the problem of 6/2(1+2) is ..
For it to be 9 it should be written like this (6/2)(1+2)
(6/2)(1+2) .. so that 6/2 comes to 3 and (1+2) sums to be also 3
then we get the outcome of 9 .
Further more we could do the (6/2)(1+2)
(3)(3) = 9 or we could say (6/2)(1) + (6/2)(2) which will also be 3 + 6 = 9 …
For it to be 1 there is no change need from the original equation 6/2(1+2)
I remember in algebra that the A(B+C) was always AB + AC and that the coefficient outside the brackets is directly factored from the whatever is in side the brackets 😀 … let me show you a demonstration let us take the number 6 and through the process of factoring it will look like this
6 = 2 + 4 .. there is no problem in there isn’t !! let us factor 2 from the right equation
6 = 2(1+2) … still seems legit i say wouldn’t you agree… so the 2 in front of (1+2) is actually the common factor and what ever affects the common factor should also directly affect the (1+2) .. if we want to divide it by 6 for example it will look like this
2(1+2)/6 .. but how about if we want to divide 6 with 2(1+2) then it should look like this 6 / 2(1+2) ..
this should not be something to be confused with … but what we could do !!!
Love Maths it is just lovely thing to do 😀
Sorry if my English language is some what under the bar 😀
Now that i am home from work I have time to post a proper response. @Ray, yes you are right this is probably futile and am wasting my time but I cannot help myself at this point.
Casper I do not mean to sound rude and I apoligize in advance but the problem is your just wrong your math is flawed, you state;”
3 * 3 = 9
6/2 * 3 = 9
6/2 * (2+1) = 9
Then, if you want to remove the “*”, you need to use ( ) since you have a fraction:
(6/2)( 2+1) = 9″
This is incorrect the * is elimated simply as notation and shorthand it does not change the problem 2*(2+1) is exactly the same as 2(2+1).
You also state”
6/2(2+1) = 1
6/(2+1)2 = 1″
This is incorrect they are not equivalent equations. The first one is the same as 3*(2+1)=9 and the second equation is 6/3*2=2*2=4. Neither of which is one.
However, you did say one thing that was right ”
One last time:
6 ÷ 2n = 3/n (the infamous wolfram even agrees on this)
6 ÷ 2 * n = 3n”
if n=2+1
then 3(2+1) =3*3=9
Again this whole argument should be closed … I don’t see the benefit of making such a big fuss over it …
6/2(1+2) what made it so debatable is the fact that the way it is written !
this is simple equation and we should be more thoughtful of working it out so simple and we agree on it … back in School whenever an equation like this was written on board it was straight and solvable and the way i see it is that we should try and write the equation on the whiteboard seriously ! so if i want to write that equation for an eighth grade student as a math teacher i should write it very clear i.e. Though when you try to read the equation and listen to it it actually could be interpreted either way … the only way to know which problem we are talking about then i guess we need the old basic way of writing and from then on it becomes clear which is which…
i.e.
6 ( I couldn’t write the division bar :D)
2(1+2)
or
6 (1+2) ( I couldn’t write the division bar :D)
2
And thus it would be very clear for an Eighth grade student to understand clearly the problem …
Peace
A.H.
@Ted:
6 ÷ 6 = 6 ÷ 4+2, but we have to use ( ) around the 4+2, for example.
Every text I read that uses 6/2, as such (not with a horizontal fraction line), and as a coefficient, uses ( ) . Could you show me 1 or 2 that do not?
You also have not addressed the Identity or Redundant Law where,
a = 1a No matter if you write the 1 or not, it is always there.
1a ÷ 1a = 1. I don’t need to use ( ) since a ÷ a = 1, and there are no ( ) there, but there is STILL a 1 in front of both a’s.
Lastly, you said nothing of this algebra lesson: http://cstl.syr.edu/fipse/Algebra/Unit2/parenth.htm
What would you do to simplify these:
1) (6y²z⁵ ÷ 2xyz² )²
2) {4 [2xz⁴ (2c³v ÷ 2cv²)³ ÷ 2]} ÷ c⁶x • v³
Regards,
KM
Why is it everytime you say ‘this is my last post on this’ the insults start? SMH…
There is nothing anal about parentheses, they are to be dealt with first, and until they are gone! Tis a math rule! That’s why I think the mathematicians need to go back to the drawing board, this little flip flop in rules should not exist in the perfection known as math! They are NOT gone once you have (3). Unless you are in 5th grade when it’s just multiplication. Algebra says otherwise. That’s what I’m talking about, you should not change rules in the middle of everything. Then it’s just make-up-your-own-rules math!
I have a solution to the dilemma. Maybe they should say on the problem “use 5th grade logic”…then we’ll all get 9! or not…
Wait, in terms of a fraction.
6 is the numerator, 2(2+1) is the denominator.
Then the answer has to be one.
6 6
——– = ——— = 1
2(2+1) 6
Patrick, the 2(1+2) is not grouped unless it looks like this, [2(1+2)]. In the original problem, 6÷2(1+2), so not incorrectly group by Distributive Property rules because this does not fit the form with the 6÷ behind it. The numerator contains the 6 and the (1+2) and the denominator contains the 2 because the ÷ between the 6 and the 2 comes, by rule, before the multiplication implied by the open parentheses between the 2 and the (1+2) grouping. This is actually in the form ab(c+d). Remember a ÷ b is the same as ab^-1. They are factors and cannot be disregarded when using Distributive Property.
Another real world example:
What is 100mm divided by 10 cm?
100mm ÷ 10cm = ?
@Sue: I get the same response everywhere “No it is not” but I never get any proof.
(6/2) will distribute to (2+1)
6/2(2+1), only the 2 distributes since 2(2+1) is the entire denominator.
Has anyone tried to take a step back and read the problem? Can anyone give me an example of how somebody would use this equation in a real life scenario? Let’s say that we have 6 apples to divide across 2 families were there are 2 adults and 1 child per family. So now let’s ask what is the most efficient way to write this equation to find the answer of how many apples each person will get regardless of their age. Let parenthesis only be used to represent a group. Also, only use operators when they cannot be implied. Every time I write this out I get: 6 / 2 ( 2 + 1 ) = 1
Why is there no multiplier or * between the “2 (“?
I cannot think of a scenario where we would want to solve for six halfs of a grouping, so for those of you whom believe the answer is 9 probably didn’t understand algebra either. You can be an engineer I suppose, if you rely on calculators. Show me how you would use this exact equation in a real life scenario, and I am not talking quantum physics Mr. Rocket Scientist.
Okay, Mr. Rocket Scientist, here we go. If I were to pay $6 per hour for a job done by two men and they worked 1 hour in the morning and 2 hours in the afternoon, how much would each of the men be owed. You see, anyone can find a situation to fit this expression.
$6÷2 men for (1+2) hours = $9.00 each
6÷2(1+2) = 9
If your logic is used, I would pay each of these men $1.00 and they would not be very happy. ‘Nuff said.
And very good on your part, Nonsense! I am an engineer as well as a math teacher with mathematics as only one of my degrees!
@Ray
While you can make your word problem fit the equation, I don’t think anyone who reads your scenario would write it the way of the original equation. Upon reading your formulation, I would write 6(1+2) / 2. And before you go on about how it equivalent mathematically to the original, it is the question of grouping and equivalency that is being debated (sometimes simply asserted).
The original problem is written to be purposely ambiguous, as attested by the fact that many people with math degrees and who teach the subject have all posted here with sincere effort, and yet fundamentally disagree on the interpretation.
First problem is you are probably a republican. $3.00 per hour? I said real life… Second, common sense dictates your grouping. Edward simplified your equation best. You try to solve for how much each person makes, so your grouping should be $ per hour times total hours worked divided by the number of workers. I was surprised though that you didn’t withhold your 15% gross margin for office and administrative costs. Wait you didn’t say you were a business man… My bad… Still, your scenario does work, you just wrote the equation backwards.
If I were to pay $6 per hour for a job done by two men and they worked 1 hour in the morning and 2 hours in the afternoon, how much would each of the men be owed. You see, anyone can find a situation to fit this expression.
$6/hr * (2+1)hrs / 2men
6*(2+1)/2 = $9/men or man. How did we call this in high school? Factor labeling?
100mm ÷ 10cm = ?
Also, yes, the grouping is implied. That was my whole proof
a ÷ 1a = 1
what is 3 ÷ 1(2+1) ?
Same idea, the 1 is Multiplicative Identity, or a redundant coefficient. The equation can be rewritten without the 1 there, period:
3 ÷ (2+1) = 1
It is NOT: 3÷1 * (2+1). Maybe that helps ?
Hi All,
Things for thought …
I think it is time to do some clear cut demonstration of what should an equation look like:
For those who argue that 6/2 is a fraction then they should have written it like the format of a fraction similar to this ½ half when we want to post it in the digital world e.g.
½(3 + 4) now every one clearly understands that the 1/2 is a half or 0.5 ..
so the problem with the 6/2(1+2) is the way it is written ….. if 6/2 is a fraction then we need a way to make it look like the ½ fraction format… So which one of you guys could contact the www and other program computing companies to inform them about this 😀 so that in the future whenever one a ‘mathematician’ wants to right 6/2 as a fraction it changes to ‘fraction format’ ..
Peace
A.H
OH .. MAY MISTAKE 6/2 CAN NOT BE WRITTEN AS A FRACTION … WHY BECAUSE IT IS NOT A FRACTION !! IT IS A WHOLE NUMBER e.g. 3 ………
yeah i was shouting there 😀 ..
Salaams
AH
All the identity property states is that anything times itself stays the same. It doesnt mean your allowed to add a one wherever you want. a ÷ 1a = 1 is wrong it is a ÷ 1 * a= a * a =a^2
But suit yourself you have mathematicians, engineers and computer scientists telling you your wrong, but you remember something you learned in high school and that trumps us. So Ray your right this is frustrating. For the record 3 ÷ 1(2+1) = 3÷ 1 * 3=3*3=9 mulitiplication and division go left to right with the same precedence.
Meant to say the ID property says anything times one stays itself
Um, I AM an engineer. I work with lots of engineering grads. I presented them all with this problem this week: Chem Eng, Comp Eng (Hardware), I am Comp Eng (Software), Electrical Eng, and Civil Eng. ALL… I repeat, ALL said it was 1. The only person who said it was 9 was an Arts grad. Maybe, perhaps, I could be wrong in using the ID law that way, … HOWEVER, I am definitely not wrong in that equation. It is the algebra assumption that “any variable with no coefficient, has a coefficient of 1” I think it was the Redundancy Law or something. Regardless of what it is called, it is definitely an axiom or some such accepted math principle. a = 1a . There is not doubt. a ÷ 1a = 1
a ÷ 1 * a = a^2.
I was not fond of how you wrote that a ÷ 1 * a = a^2 as a defense. I never said that was untrue. I DID say a ÷ 1a = 1.
I find it very unusual that every “rebuttal” is some sort of rearranging what I said in the first place??
3 ÷ 1(2+1) = ? anyone?
100mm ÷ 10cm = ?? Anyone ?
6 ÷ 2n = ? Anyone?
Hint: 2n = (n+n), like all multiplication (is really addition).
Keep in mind, when you calculate 6 ÷ 2n, you must multiply the equation in reverse and get 6 😉
Regards,
KM
The flaw in your argument and the disconnect is that you see a difference between 1a and 1*a. However, there is no difference they are 100% equivelent.
To respond to your questions:
1. 3 ÷ 1(2+1) = ? anyone?
2. 100mm ÷ 10cm = ?? Anyone ?
3. 6 ÷ 2n = ? Anyone?
1. =3*3=9
2 = 100÷100=1mm. This is because you have to first convert 10 cm into mm. So your wquation would have to me rewritten 100mm ÷ (10mm *10mm)
3. Simplified to 3n
BTW my apoligies for the hostility in my last post, it was uncalled for and it was very unprofessional of me.
I also, just wrote a very quick program in java
public class debate
{
public static void main(String[] args)
{
double x;
x=6/2*(2+1);
System.out.println(x);
}
}
When I ran it is gave the answer of 9. I had to use the / and the * as the complier doesnt understand ÷ and the inplied *, however, what I inputted is exactly the equivelent of 6 ÷ 2(2+1)
Most of the reasons for people of the 9 tribe is because of calculators and software, so they are changing their reasoning to “make it fit”. however,
1) 3 ÷ 1(2+1) = 3 ÷ (2+1) since the ‘1’ coefficient can be unwritten (omitted)
2) The answer 1, and not 1 mm. However you did something here, which I find interesting. you inserted ( ) that were not there, however for the 3rd question, you did not insert them.
3) 6 ÷ 2n = 3/n . Just as 6 ÷ 2 = 3, we must have 3 * 2 = 6
So, 6 ÷ 2n = 3/n, we must have 3/n * 2n = 6
If we do your solution in reverse, we get 3n * 2n = 6n² , which is incorrect.
This proves that 6 ÷ 2n = 3/n, AND 6 ÷ 2n = 6 ÷ (2n)
Here is another proof:
6n ÷ 2n = ?
Well, since multiplication is repeated addition, lets break it down into its raw form:
Ex: 4 ÷ 2 = (1+1+1+1) ÷ (1+1) = 2(1+1) ÷ (1+1) = 2
4 ÷ 2 = 4 ÷ (2) = 2. This doesn’t help us with the proof, so lets try a variable:
6n ÷ 2n =
(n+n+n+n+n+n) ÷ (n+n) =
[(n+n) + (n+n) + (n+n)] ÷ (n+n) =
3(n+n) ÷ (n+n) =3
6n ÷ 2n = 3
Proving, a second time, in addition to the division proof of ‘multiplying in reverse’, that
6n ÷ 2n = 6n ÷ (2n) = 3
I still don’t think you understood my point of a = 1a at all times, whether the 1 is written or not. a ÷ a is the same as 1a ÷ 1a.
In the end, it comes down to the “implied parentheses” around the divisor, like you did in question 2, but it holds valid for them all, as shown in both proofs.
So, if 6 ÷ 2n = 3/n, and n=2+1, we now have exactly
6 ÷ 2(2+1) = 3/(2+1) = 1
Once and for all.
If 2+1 was supposed to be in the numerator, it would be written there, no?
6(2+1) ÷ 2 = 9
6 ÷ 2(2+1) = 1
3(2+1) ÷ 1 = 9
3 ÷ 1(2+1) = 1
One more way of hopefully driving hope the implied ( ) [or, really just knowing what is the numerator and denominator] is to think about exponents and other grouping symbols. Your idea of 6 ÷ 2n = 3n is because you are saying 2n = 2 * n, thus breaking the coefficient away from its term, and using it as a constant.
Would you do the same for 6n ÷ 2n² . Since n² = n*n ?
If I were to follow this “strict method of left to right and ‘pemdas'”, I would do this as your student:
6n ÷ 2n² = 6n ÷ 2 * n * n = 3n^3 (which of course is incorrect)
6n ÷ 2n² = 6n ÷ (2n²)
Although there are no ( ), there is the idea of implied grouping, and a full understand of how this works for all areas of algebra is required.
The same would hold true for 4! ÷ 3! = 4! ÷ (3!) since
4! ÷ 3! = 4*3*2*1 ÷ (3*2*1) and not 4*3*2*1 ÷ 3 * 2 * 1
Hope this clears everything up
Regards,
~KM
@ Ted: (quoting)
>>> 100÷100=1mm. This is because you have to first convert 10 cm into mm. So your wquation would have to me rewritten 100mm ÷ (10mm *10mm)
This is wrong. There are no units in final answer. the problem can be written as 100mm / 100mm, or as 1ocm / 10cm, or even as .1m / .1m. The answer is one, but not one of whatever unit you converted to. I don’t really see what this problem has to do with the original problem, however.
Regarding the Java program; not a proof. the issue with most everyone’s arguments in these comments is that they all build their assumptions of the the “correct” way to interpret the problem right into the proof. You say ” I had to use the / and the * as the complier doesnt understand ÷ and the inplied *, however, what I inputted is exactly the equivelent of 6 ÷ 2(2+1),” but it is this equivalency that is the debate.
Ultimately, this whole debate is about INTERPRETATION of math language, and thus you cannot use math language to prove your interpretation. At heart there are two fundamental questions:
ONE:
a) The division symbol is localized, applying only to the adjacent numbers unless parenthesis intervene;
OR…
b) The division splits the entire equation into numerator and denominator unless parenthesis intervene.
TWO:
a) a constant in front of parentheses is merely a shorthand for multiplication, with no order or linkage implied;
OR…
b) a constant in front of parentheses is a direct coefficient of the term in parentheses, and is thus mathematically linked to that term.
People who believe a) get answer of 9, b) get answer of 1. Some people believe strongly in one of these and don’t debate the other. All of which leads to confusion, debate, and snarkyness.
I admit I come down squarely in the TWO, b) camp on the big question. I believe the 2 in the original problem to be a coefficient to the parenthetical expression, and thus its multiplication is linked. But I cannot prove this without using the assumption. I can only say that this linkage is what I was taught through many many math courses. Someone who was taught differently would believe differently. Short of an official ruling by some governing math body (joking) I don’t see any resolution to the problem.
I think your making this more complicated than it is. The rules of mathematics state solve everything in() first then * or ÷ letf to right after that + or –
Just using this rules makes it 9 and thats why software like java and C++ will spit out 9 as an answer. After you solve the parathesis the equation become 6 ÷ 2 * 3= 3*3=9
No where does the rule say multiply or divide left to right unless one of them is touching the outside of a parenthesis.
going back to your three questions
1)” 3 ÷ 1(2+1) = 3 ÷ (2+1) since the ’1′ coefficient can be unwritten (omitted)”
That is incorrect: The one is not being omitted, it vanishes because the 3 is divided by it. 3/1=3, then that 3 is multiplyed by (2+1) giving us 9.
2) “The answer 1, and not 1 mm. However you did something here, which I find interesting. you inserted ( ) that were not there, however for the 3rd question, you did not insert them.”
You are correct I mispoke it is in fact 1 not 1mm. The reason I added the parenthesis is because you are dividing the other number by the entire 10cm. So when you convert it to a different magnitude you have to keep it in parenthesis since that 10cm travels together no matter what format you express it in.
3) “6 ÷ 2n = 3/n”
This is the heart of our disagreement, I contend that 6÷2n=3n and not 3/n. You are giving higher precendence to the 2 being multiplyed by n. Why when the rules of math state precedence goes left to right and not right to left.
Another point you made is Since n² = n*n ? that is correct but if you were to break it out you would need to say n^2=(n*n) and include the parenthesis as you are taken one term already simpified and breaking it out. To do otherwise would be changing the equation. This is not the case with the original problem, apples and oranges as the saying goes.
“Another point you made is Since n² = n*n ? that is correct but if you were to break it out you would need to say n^2=(n*n) and include the parenthesis as you are taken one term already simplified and breaking it out”
Thank you. That is the point I have been trying to make the whole time. There are implied ( ) around the n² , the same as 2n is (n +n). Can you please comment on this proof:
4n ÷ 2n = 2
(n+n+n+n) ÷ (n+n) = 2
Actually, can everyone/anyone comment on the above proof, please? I would like some other input on it.
If you derive the original equation from scratch, you get:
6 ÷ 6 = 1
6 ÷ (4+2) = 1
6 ÷ 2(2+1) = 1 This is where the 2 cannot be broken away from 2+1 just like you stated about x^2.
Order of operations really doesn’t have to “left to right” either. Equations are collapsed according to the order of operations, plus there are associative laws that say right to left works when it applies.
Also, I totally disagree with the “1 vanishing because it is divided by the 3”
Everyone knows that a coefficient of 1 is redundant.
a ÷ a = 1a ÷ 1a = 1
Just because you write 1a ÷ 1a doesn’t change the equation to 1 * a ÷ 1 * a = a² !
No parentheses are required because both sides of that equality exist at all times, written or not.
Everyone is free to “believe” what they like, but there are things that just are not correct here.
@Edward: My point with the 10cm equation was to show that there are implied ( ), or grouping, with the 10cm.
According to Ted, I should be able to rewrite 10cm as
10cm =10 * 10mm
then,
100mm ÷ 10 * 10mm = 10 mm^2
However, he DID agree that 10cm = (10 * 10 mm), even though there were no ( ) around it to begin with. 10 is simply a coefficient, no?
Apply that to 6 ÷ 2n. The “2n” travels together(as you said), just like the 10 cm.
Care to address how the reverse of 6 ÷ 2n = 3n is 3n * 2n = 6n² ?
Here are a couple of references everyone may find interesting
http://i47.tinypic.com/34y7hxj.jpg
http://i50.tinypic.com/2mdnz34.jpg
I have yet to find a reliable reference that says otherwise to the above.
Have a great week all.
Regards,
KM
The simple question is do you believe that 2n = 2*n and I do. Since that is the case it is given the same priority as all other multiplication/division left to right.
“4n ÷ 2n = 2
(n+n+n+n) ÷ (n+n) = 2”
Not sure what that is trying to say but its not really a proof, 4n÷2n=2 is a statement and you can solve for n. I would change the ÷ into a multiplication by inverting the divisor. Then the equation would read 4n*(1/2)n=2 Then using the associative law of multiplication 2n^2=2 n=1 or =- 1
Placing 1 into the equation
4(1)÷2(1)=2; 4÷2(1);2(1)=2; 2=2
Placing -1 into the equation
4(-1)÷2(-1)=2; -4÷2(-1)=2; -2(-1)=2; 2=2
“If you derive the original equation from scratch, you get:
6 ÷ 6 = 1
6 ÷ (4+2) = 1
6 ÷ 2(2+1) = 1 This is where the 2 cannot be broken away from 2+1 just like you stated about x^2. ”
This incorrect if you were deriving it from scratch then the final statement would be
6÷(2(2+1)) and its not the same thing as the x^2 had a higher precedence already established on it by being an exponent.
“According to Ted, I should be able to rewrite 10cm as
10cm =10 * 10mm
then,
100mm ÷ 10 * 10mm = 10 mm^2”
Thats not what i said it would have to be 100mm÷(10*10mm) because the entire 10*10mm represent the entire 10cm. We are really talking about apples and oranges.
I contend the original problem is stated wrong it should have originally been written as 6/(2(2+1)) and then it would be 1.
And yes 3n * 2n = 6n² but for a different reason. The associative property of multiplication.
The bottomline is does implied multiplication have a higher priority than explicit multiplication, I contend it does not and that is why the answer is 9.
At first we solve the ( )
6/2(1+2)
=6/2(3)
=6/6
=1
its very simple
After some research I know now there is no correct answer for that.
http://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html
http://mathforum.org/library/drmath/view/57021.html
“I think this is far preferable to making detailed rules that are
likely to trick people. Sometimes one rule seems natural, and
sometimes another, so people will forget any rule we choose to teach
in this area. I’ve heard from too many students whose texts do “give
an example that really puts this rule to the test,” but do so by
having them evaluate an expression like:
6/2(3)
that is too ambiguous for any reasonable mathematician ever to write.
And no matter what the rule, we would still constantly see students
write things like “1/2x” meaning half of x, so we’d still have to make
reasonable guesses rather than stick to the rules.”
Okay, so you guys might be mathematicians, physicists, and computer programmers, but you really aren’t very good at getting down to the basics and actually proving a mathematical equation… Here is an extremely simple, back to the basics, step by step solution… my god people! I hope none of you are teachers!!!
First, let’s substitute the letters with numbers, that way we can carry out PEDMAS and then plug the numbers back in to check our answer.
let 6=a, let 2=b, and let 1=c (In algebra, we are taught to be able to interchange letters with numbers, and this is a very good example of exactly why we are given this skill… it’s called a PROOF when you can replace the numbers with letters and solve using only the rules of mathematics.)
we have 6/2(1+2), so replacing with letters we get
a/b(c+b), and because we can’t add the unlike terms (c & b), then we need to use the distributive property of multiplication over addition, which states a(b+c) = ab+ac (in other words, you distribute the a to the b via multiplication and the same with the c, in this manner we are able to eliminate the parenthesis), to simplify the expression… the question then becomes, do we distribute the b by itself? or do we distribute the a/b as a fraction? After we simplify the expression into individual terms, we can replace each term with the appropriate numbers and we will have our solution…
First let’s just distribute the b…
a/b(c+b) = a/b*c+b² and we can’t simplify anymore…
so now replace each term with the correct numbers…
a/b*c+b² = 6/2*1+2²
and by following PEDMAS
6/2*1+4 (no parenthesis, so exponents first, 2² = 4)
3*1+4 (multiplication and division from left to right, 6/2 = 3)
3+4 (multiplication and division from left to right, 3*1 = 3)
7 (addition and subtraction left to right, 3+4 = 7)
and I’m pretty certain that this is not the correct answer!
NOW, let’s distribute a/b in its fraction form…
a/b(c+b) = a/b*c+a/b*b and we can’t simplify it anymore…
so now replace each term with the correct numbers…
(remember, a=6, b=2, and c=1)
a/b*c+a/b*b = 6/2*1+6/2*2
and by following PEDMAS
3*1+6/2*2 (no parenthesis, no exponents, so begin with multiplication and division from left to right, 6/2 = 3)
3+6/2*2 (multiplication and division from left to right, 3*1 = 3)
3+3*2 (multiplication and division from left to right, 6/2 = 3)
3+6 (multiplication and division from left to right, 3*2 = 6)
9 (addition and subtraction left to right, 3+6 = 9)
And this very simple proof, my friends, is why the answer is 9 and not 1!!!!! Brought to you by a Kinesiology Master Student.
Val 🙂
Okay let’s look at this as if we are all from Denmark where the obelus “÷” is used for subtraction.
The original problem that is being presented is 6 ÷ 2(1+2), which becomes completely different from 6/2(1+2) or 6:2(1+2).
Translate this to an North American point of view would give us:
6 – 2(1+2) = 6 – 2(3) = 6 – 6 = 0 This is a new answer, and also correct using the Danish meaning of “÷”
It’s all in the interpretation people. The real question is which one is correct?
6 6
——– OR —–(1+2)
2(1+2) 2
Okay let’s go to Italy or Poland where “÷” means range.
So 10% ÷ 50% Reads ‘10% to 50%’
Now the problem is a range of 6 to 2 multiplied by (1+2) OR 6 ÷ 2(1+2) = 6 ÷ 2(3) = 18 ÷ 6 = ” A range of 18 to 6.” Would this now be considered a function, rather than an expression?
Okay now we have Four answers to argue about.
Enjoy.
Well I see that I failed to remember the fact WordPress uses “Regular Expressions” in its PHP code to delete white space and duplicate characters in order to conserve space in the SQL Database.
It should have been:
6
—–
2(1+2)
-OR-
6
— (1+2)
2
“Properties like the distributive property tell us how we can REWRITE an expression without changing its value.” <– A quote from a doctor.
If we ignore this property, and every other logical way of reading it, we get 9.
If we apply the property, we get 1. Applying the property DOES NOT change the value or meaning of the equation, therefore any other answer is incorrect.
@ Val: there are just way too many misuses of notation in your post, sweetheart.
Distribution:
6 ÷ 2(2+1) = 6 ÷ [2(2) + 2(1)] = 6 ÷ (4+2) = 1
Distributive Property NEVER has a "÷" in front of it. This is because the equations like this have ALWAYS been written as
6 _
2(2+1)
Only a plus (+) or minus (-) sign precedes expressions of distribution.
Distributive "property" is just that: A Property. Here is a quote from a person whose title is Dr.
"The order of operations tells us what an expression MEANS: if we follow the rules, we will correctly evaluate the expression. Properties like the distributive property tell us how we can REWRITE an expression without changing its value. So properties allow us to safely manipulate an expression to make it easier to evaluate, or to solve a problem, knowing that it still has the same value. That doesn't change its meaning, only how we actually calculate it."
Therefore, if distribution does not change the meaning, or its value, then any other answer obtained by not using the property, is wrong, since distribution is excluded from changing expression values. This is why we can use the distributive property before the order of operations even starts.
caper26,
i think what you are missing though is how the distributive property would be applied in this poorly written math problem.
As an example, if we use the values from outside of the parenthesis as a single value we would do the distribution as (6/2 * 1) + (6/2 *2), which would allow for the order of operations to be followed correctly applying multiplication and division with the same level of precedence.
When doing the other way as 6 / (2*1 + 2*2), we are giving precedence to multiplication, which is incorrect.
In order for the second example to be true, the formula would need to be written as 6/[2(1+3)], which it clearly isn’t.
I understand basic math properties quite well, thank you. This equation is clearly 6 over 2(2+1). Distribution is the opposite of factorising, so if you factorise 6 + 3, you would get (6/2)2 + (6/2)1. If you factorise 4+2 you get 2(2+1). 2(2+1) doesn’t require ( ) around it as they would be redundant, since it is very clear what should be happening here (distribution).
Q: Does 6 ÷ 2(1+2) really implies 6 ÷ 2 x (1+2)?
Would you get a true answer if we try to do this with algebra a=1 b=2 c=6
so it become c ÷b(a+b) then what would we arrive with?
Please I ll like to know I can be a member here,
So that if I see new things I can post them and
also learn new things.
The correct answer is 1
6/2(1+2) is 6 ÷ 2(3)
Its a fraction, 6 is d numerator, 2(3) is d denominator
So answer is 1
Just to the right of whatever post is at the top of the page are a handful of links to twitter and RSS and whatnot.
thank you so much angh, the links were positively fascinating!! and they definitively show that the problem is too ambiguous to solve, due to questions about the meaning of that particular notation. i also genuinely appreciate it whenever someone seeks the answer of a problem through research rather than simply arguing further. well done!!!
Depends how you were taught. In grade school it was easy brackets were the same as multiplications, and you followed PEDMAS (the forward slash does not denote fraction in these cases since division symbol not avail)
6 / 2(1+2)= ?
6 / 2(3)= ?
6 / 2 * 3= ?
3 * 3 = 9
then came high school and you were introduced to algebra
a(b+c) became ab+ac
2(1+2)=6 and
2*1+2*2 = 2+4 which =6
but it was easier to do 2(3)=6
All good but now use the same logic
6/ 2(1+2) = ?
6/ 2(3)=?
6/6= 1 using algebraic logic
Do the entire formula using algebraic logic you get this
6 / 2(1+2) = ?
6 / 2*1+ 2*2 = ?
6 / 2 + 4 = ? PEDMAS next
3 + 4 = 7
So there are 3 answers 9, 1, and 7
And they wonder why people hate math
As stated before, ambiguity.
Both answers, 1 and 9 are correct. Depends on the way you face the expression.
Perhaps you see the ambiguity better if you start with 6/6 going backwards:
1 = 6/6 = 6/(2+4) = 6/2(1+2) = 6/2*(1+2) = 3 * (3) = 9
1 = 6/6 = 6/(2+4) = 6/2(1+2) = 6/[2(2) + 2(1)] = 1
However,
6 + 3 = (6/2)2 + (6/2)1 = (6/2)(2+1)
Inline fractional coefficients require parentheses.
I have yet to see a text write 6/2x with the intention of meaning 6x/2.
6/2x = 3/x
x=2+1
The rest is simple substitution.
The way I see it, when you have a(b + c) you really have a * (b + c). The multiplication there is implicit. Then, the distributive property can be applied so you get a*b + a*c. If a = 2, b = 3, and c = 4, the result is clearly 14. Now, when I see something like:
6 / 2(1 + 2)
I translate that to:
6 / 2 * (1 + 2)
Now, ignoring the distributive property and only following the order of operations, we have:
6 / 2 * (3)
can lose the parenthesis:
6 / 2 * 3
3 * 3
9
Now, without ignoring the distributive property, we have to decide what a is. We can have:
a = 6 / 2
or
a = 2
I argue it is the first one. Why? Well, the division binds what is immediately to the left and what is immediately to the right. It doesn’t have “magical” property here to bind the remaining operand in the expression. So we have:
6
— * (some expression)
2
We now have a fraction multiplied by something. In our case, it is 3. So we get 18 in the numerator and 2 in the denominator and we are left with 9. I don’t really think it makes sense to use the distributive property here as we have numbers. In algebra, when you don’t have numbers, it’s a property that lets you proceed to simplify the expression (hopefully). And, I argue that in this example, everything outside the parenthesis is what a refers to.
Unfortunately, you took a coefficient of 2, and made it (1/2), which is totally illegal:
Example: 6 ÷ 2x
What is the coefficient of x? It is 2.
now, x = 2+1
so we have: 6 ÷ 2(1+2)
the coefficient of 1+2 is 2.
If you magically insert a multiplication symbol without containing the new expression in parentheses, then this is what you did:
6 ÷ 2(1+2) = 6 ÷ 2 * (1+2) = 6 * 1/2 * (2+1) = 6 * (2+1)/2
WHAT?? Now the same coefficient of 2 is now one-half ??
this is why distributive property must be used if there is any ambiguity. It helps remove it, as any property helps you solve equations easier.
Now, to address your interpretation of what a, b, and c equal.
Start with a(b + c)
Let a = 6/2, b=2, c=1. Substitute. What do you get?
Keep in mind substitutions must be contained in parentheses:
(6/2)[(2) + (1)].
Now, if you want to make a substitution in the original expression/equation, parentheses must be used as well:
if you want to say this:
2(2+1) = 2 * (2+1) and that is what to substitute, then it MUST be contained in parentheses:
2(2+1) = (2(2+1)) = (2 * (2+1))
If not, what you will end up having is something like this
Example:
6 ÷ 6
Now, we can clearly say 6 = 2 * 3, right ?
So does 6 ÷ 6 = 6 ÷ 2 * 3?
No. If we insert multiplication symbols and substitute expressions, no matter how subtle, we use parentheses.
================
This entire disagreement comes down to this:
What are the operands???
It is not order of operations. We all understand that very clearly I believe.
It comes down to: what are the operators and what are the operands.
Some people think “Parentheses mean multiplication”. This is very primitive thinking. Parentheses are a grouping tool. A coefficient of a parenthetical is a factor of all the terms contained within. This concept is called factorization.
So many tools in the tool-box to remove ambiguity, yet so many people solve it incorrectly, or claim there are 2 answers. I beg to differ 😉
Regards.
@mathman
This is where I say you are wrong. The expression 2x becomes 2*x. And yes, using your example: 6 * 1/2 * (2+1) is correct. As such, by the order of operations, everything I did is valid. On top of that, the distributive property is a tool. You are giving the implied multiplication between 2 and (1 + 2) a higher precedence than division, which is clearly wrong as they have the same precedence and are to be evaluated left to right.