6/2(1+2) or 6÷2(1+2) equals ? The truth.

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My previous video on the subject:
Music by Bertrand Laurence
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[Tsskyx]

I'd like to, after many years of doing math, make my own statement on this.

6/2(1+2) is the same as 6/2(3). This is true no matter what your interpretation of PEMDAS is, you have to do what's inside the brackets first, you cannot distribute just yet.

Now, some people would want to distribute (do the 2(3) first), while some would want to do the fraction first (the 6/2 part).
To be honest, I've never seen the definition of P in PEMDAS to include things outside the parentheses, if implicit multiplication is present. But some people argued that 1/2x is (2x)^-1, not 0.5x.

As for those that would do the 6/2 first, they do not believe that implicit multiplication extends the definition of the P in this way, which I agree on. What is weird though, is that some calculators, such as those made by Texas Instruments, show 1, since they do the distribution first. Why? I don't know. But I suspect it has something to do with the notation.

You see, the 9 camp states that for this to be 1, it would have to be written like this: 6/[2(1+2)].
While the 1 camp is saying that for this to be 9, it would have to look like this: (6/2)/(1+2).

This leads me to believe, that the problem truly comes from how people interpret implicit multiplication. To avoid this, people should always use fractions, to show how the problem is structured, or, include extra brackets.

However, I find it unbelievable, that in the 21st century, there's a schism among the mathematicians about something as simple as the order of operations. And I'm surprised no one is doing anything about this. But I guess this is because 6/2(1+2) is not the "standard" way of writing this problem. The standard way would be to use fractional notation, not an in-line notation. In-line notation works for YouTube comments, but it leads to confusion. If we want to fix this, we would have to define rules for this "in-line" notation.

So, my question is, did someone start working on such in-line notation rules yet?

[Syed_Institute]

Hey guys, I actually have the correct solution to this question on my channel - the correct answer. So I have to disagree with MindYourDecisions on this video for the reasons I discuss in my video for the solution of this problem.

[Cotronixco]

The answer is 1. In engineering in the US, we always treat a group as a singular item when there is no stated operation.
The solution for 6/2(1+2) is 1. This is also expressed as 6/(2+4).
The solution for 6/2*(1+2) is 9. This is also expressed as 6/2*3.

[TyYann]

You prefer attacking me personally, decide the validity of my responses instead of dealing with my arguments. And yes, "my" professor and I think there are 2 ways of interpreting those 2 equations. Not that they have 2 solutions. But you won't understand the difference. If you think I'm wrong, just thumb down the video, I won't feed the troll any more anyway.

[JamesZych]

this is like when two superheros are supposed to fight and they just make peace at the end without saying who won

[TyYann]

A friend of mine who is a mathematics professor in Cambridge university (UK) says the same as me: both solutions (1 and 9) are correct. Do you think he also need more schooling?

[Diabolousx]

Nah, its 1. Think of 'π' (pi)... 6 ➗ 2π is not 6 ➗ 2 ✖ π. It is 6 ➗ (2π). The very reason we don't write 2 ✖ π and instead write it 2π is to show that 2π is a single number and to not confuse the operation. The implied (but redundant) parentheses thus belong in the following manner: 6 ➗ (2(π)). This is how it has always been and ever will.

[SteveMcRae]

Also, "Wikipedia introduction to basic algebra" gives an example for an in-line expression of 6 / x + 3 where x=3. 6 / 3 + 3, 2 + 3= 5 while to make it 6 over x+3 it gives the example of 6/(x+3). Since this problem isn't 6/(2(1+2)) is it not the same as 6/2 * (1+2) since division and multiplication in standard have equal ranks and one does have make implied >explicit now to solve for 1?
Am I correct that you think standard = 9, and implied>explicit = 1?

[pvangastel]

You're absolutely right! The question makes no sense until you define the (order of) operations. Socrates's taught the same about philosophy: always define a concept (e.g. love) before discussing it.

[TyYann]

I need to check my sources? I'm telling what?
I said you DON'T HAVE TO distribute.
This is the way you understand any material? Twist it according to your needs?
But you won't admit you were wrong on this point.

[tonymarinov]

2(1/2) is the whole number which should be used to divide 6 on. Do you see multiplication between 6/2 and (2+1)? No In this situation we should consider 2(1+2) as one whole number. Also do not forget that multiplication and dividing is equal in rank. The same as subtraction and adding. So the answer is simple =1 

[TyYann]

I said you distribute if no operation of higher priority takes precedence over the multiplication, hence, the priority is the core of the problem, not distribution. That's why I work on the source of the problem, not on distributivity. How difficult is it to understand that? I explain it in the video, explained you several times in those comments, and still you come back with distributivity. *sigh*

[douglasward]

I personally do not believe 9 is a correct answer. To illustrate my point let us view this as a word problem that would generate said mathematical expression.

There are 6 people in a room, on a table there are 2 baskets each containing 1 apple and 2 oranges. How many pieces of fruit can you give to each person?

Regardless of which process you follow there is NO way possible to give 9 whole pieces of fruit to each person.

[samwell-ii]

This is what makes math fun, you can twist it however you want.

For example, if I say 2(1+2) you could distribute or calculate what's inside or multiply. But actually, this doesn't matter at all. Either you get
2(1+2)
(2+4)
6

Or you get
2(1+2)
2(3)
6

Either way, you get 6. This works because distributing is multiplying to things in parenthesis. We always distribute when we multiply into parenthesis but often to only one term (or thing for none math nerds).

So with that in mind (distribution being regular multiplication) lets look at the problem

6/2(1+2)
PEMDAS tells us to simplify (1+2) and then the rest. This would get us to this point
6/2(1+2)
(Put the (1+2) alone for a little bit because it is in the parentheses)
(1+2)
(3) (Now put it back)
6/2(3)

2(3)is the same as 2•3 because of thing stated earlier(distributing is simply a fancy way to multiply) So we can replace 2(3) with 2•3 This leaves us with
6/2•3
PEMDAS now tells us the multiply or divide numbers starting farthest to the left. This leaves us with
6/2•3
3•3
9

The problem comes when we start using terms such as distribution because they don't sound like multiplication but simply are and are still bound to multiplication in PEMDAS.


And also the part about 6/3x meaning two different things. Same problem here too, we start trying to simplify math by taking out "unnecessary" multiplication symbols or making new random words and terms like "coefficient" and "distribute", that we end up forgetting they were there.
3x is the same as 3•x
((2•1)+(2•2)) is the same as 2(1+2)

The solution is simple. When in doubt, just add any missing symbols in and think of how basic math is. We get so caught up in remembering rules that we forget why a rule works.

I just needed to put my input in

[Knightviper]

6÷2(1+2)
which is the same as writing:
6÷2(3)
which is the same as writing:
6÷2*3

The problem becomes 6÷2*3, PEMDAS states work left to right,

So 6÷2*3=
3*3=
9

[TyYann]

OMG sir. You really didn't just substitute an expression into another expression without using parentheses did you, and then expect me to not correct that?
a÷a is the same as... (1a)÷(1a).

[twkster]

LOL :0 I totally get the "social experiment"My two favorite quotes:"Most people spend 100 x more energy defending the way they think than ensuring the way they think is accurate!""Criticism requires no special talent or training, that's why so many people do it"Of course I just invited criticism :)Mathematics is supposed to be the universal language of science (nature) and therefore unassailable... However, math suffers from the same problems as all languages and or other forms of expression - individual interpretation!Equations in and of themselves cannot convey understanding unless we first ensure we understand what is intended by the author. Culture and individual experience (or variation in learning) can affect results!Speaking a common language does not import common understanding! So as you have said, the proper method is laying the foundation of notation to be used and ensure a common understanding of what solution is being sought.Regardless of the equation (communication), it is pointless to argue over the answer unless we certainly know what is intended to be communicated. CLARIFICATION - WHAT & WHY?LOGIC IS NOT UNIVERSALLY APPLIED THE SAME WAY - ESPECIALLY ACROSS CULTURES. So, let's make sure we understand the question before we argue over the answer!Cheers and best wishes!

[donmacqueen]

This is probably the clearest and most sensible presentation available on this topic.

[kolumdium]

6/2(1+2)
a/b(c+d)

a/(bc+bd)
6/(2*1+2*2) = 6/6 = 1
or
(a/b)*c + (a/b)*d
6/2 * 1 + 6/2 * 2 = 3 + 6 = 9

So both?

[phillipromo]

PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. Given two or more operations in a single expression, the order of the letters in PEMDAS tells you what to calculate first, second, third and so on, until the calculation is complete.May 12, 2015