After thinking a long time, I have decided to write about this blog post. I had no plan to create a blog post about this subject but the amount of conversation this one has created on my **Facebook **page, I decided to bring up a few of the question and concerns discussed on the Facebook page. There are more than **10,000** comments here so far.

There are lots of discussion about what should be the answer. Well, as far as I can tell there is a big debate going on on Facebook, for educational purpose you should go ahead and read some of the comments. They are very interesting and for sure teach some new stuff. Even though some of the comments are clearly wrong they have made some good points and I believe it for sure develops some logic. Here is my take on this subject. I believe the answer is **9 **as I follow** PEMDAS ** Order of Operation**. **PEMDAS stands for parentheses, exponents, multiplication, division, addition, subtraction. PEMDAS is commonly known as **BODMAS **in India. BODMAS stands for Brackets, Orders (ie Powers and Square Roots, etc), Division, Multiplication, Addition and Subtraction. PEMDAS and BODMAS are almost same and both of them follow the operation order from **LEFT to RIGHT**.

Let us try to simplify above statement using the PEMDAS or BODMAS (whatever you prefer to call).

**Step 1:** 6 ÷ 2 (1+2) (parentheses first)

**Step 2:** = 6 ÷ 2 * (1+2) (adding multiplication sign for further clarification)

**Step 3:** = 6 ÷ 2* (3) (single digit in parentheses – simplify using operator)

**Step 4:** = 6 ÷ 2 * 3 (Remember next Operation should be LEFT to RIGHT)

**Step 5:** = 3 * 3 (because 6 ÷ 2 = 3; remember LEFT to RIGHT)

**Step 6:** = 9 (final answer)

Some often find Step 4 confusing and often ended up multiplying 2 and 3 resulting Step 5 to be 6 ÷ 6, this is incorrect because in this case we did not follow the order of LEFT to RIGHT. When we do not follow the order of operation from LEFT to RIGHT we end up with the answer 1 which is incorrect.

Let us see what SQL Server returns as a result.

I executed following statement in SQL Server Management Studio

`SELECT 6/2*(1+2)`

It is clear that SQL Server also thinks that the answer should be 9.

Let us go ahead and ask Google what will be the answer of above question in Google I have searched for the following term: 6/2(1+2)

The result also says the answer should be 9.

If you want a further reference here is a great video which describes why the answer should be 9 and not 1.

And here is a fantastic conversation on Google Groups.

Well, now what is your take on this subject? You are welcome to share constructive feedback and your answer may be different from my answer.

**NOTE:** A healthy conversation about this subject is indeed encouraged but if there is a single bad word or comment is flaming it will be deleted without any notification (it does not matter how valuable information it contains).

Reference: **Pinal Dave (http://blog.SQLAuthority.com)**

thanks for reminding the basics!!!!

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Hi Pinal,

This is kiran i am new to SQL server and i need to list out all the tables used in stored procedure. collecting manually is very tedious job for such a long procedures so i am searching for alternative smart way. please suggest me how could i.

Thanks

Kiran

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Thanks pinal

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Could you please let me know how rule apply to resolve this ?

3+8-4+5*9/3-8/2*1-2+5 = ?

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3+8-4+((5*9)/3)-((8/2)*1)-2+5 =

7 + (45/3) – (4*1) -2+5 =

7 + 15 – 4 -2+5 = 21

http://www.google.com/search?q=3%2B8-4%2B5*9%2F3-8%2F2*1-2%2B5

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This SQL blog or basics of mathematics blog ??? Children that are 7 years old know all the things from this article.

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Great Point. It seems like Children know but many grown up do not. Please check the facebook link you will find more than 5000 comment with incorrect answer.

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You just got trolled hard, so hard that it consumed a lengthy blog post and much distress. Roughly 98% of the people that debate that are clearly trolling for the sake of rustling some jimmies. Congrats Pinal, you may have been their biggest catch with that bait.

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Occam’s Razor applies here:

“Never assume malice when stupidity is equally applicable”

It is fairly basic stuff, but I’m not surprised a lot of people don’t remember it.

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I’m slightly confused. PEMDAS: Parentheses [(2+1)], exponents [none], multiplication [2*(2+1)] = [2*3] = [6], division [6 / 2 * (2+1)] = [6 / 6], addition [none], subtraction [none]. 6 / 6 = 1.

Wouldn’t this mean the difference between PEMDAS and BODMAS is the reason for different results? Thus PEMDAS can’t be commonly known as BODMAS in India. Instead, BODMAS is used instead of PEMDAS in India?

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And to reply to myself, searched more of this and found out it is actually PE(MD)(AS) (left to right). Didn’t even remember such a thing. For some reason my mind turns 2(2+1) more important than 2 * 3. Anyway, glad to see this conversation as now I don’t need as much parenthesis to make things work for sure when scripting, I just need to remember the left-to-right rule of PEMDAS. :D

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Hi all. I researched this topic well for about 2 weeks now and have come to the following conclusions. I will summarize what I have said in other forums

with respect to the notations, then I will address other points.

First,

if you want to say 0.5x, then you HAVE to write (1/2)x with parentheses or, x “all over 2″ with a horiztonal fraction bar, or write x/2. I have never

seen (1/2)x before I researched this equation, but since searching online, I HAVE seen fractional coefficients written this way, only because computers

are limited to the horizontal typing space.

Therefore:

x/2 = (1/2)x = 0.5x

1/2n = 1/(2n) This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). It was simply 1/2pi, or 1/2e^2.

I have always used ab/cd to mean (ab)/(cd) and I topped almost all of my calculus classes since high school through university.(moot point, I know)

Just to re-iterate, to use 6/2 as a fraction, parentheses are REQUIRED. Every book will tell you this.

Now consider the Identity Law:

a = 1a = 1(a)

We know there is ALWAYS an ‘invisible’ 1 as a ceofficient of a variable if no other number is there. Therefore:

a/a = 1, and if a is also 1a, then a/1a = 1. Blindly using ‘pemdas’, some folks would do this:

a/1a = a/1*a = a*a = a^2. I hope this drives home the silliness of this calculation.

Now, on to my second point:

consider: factoring, simplifying equations, and the distributive property.

Lets start with the number 6.

6 = (4+2). There is a common factor here: 2. So let’s factor it out of both terms.

(4+2) = 2(2+1). The outside 2 remains a part of of the 2 inner terms at all times. It cannot be used in an operation by itself without the rest of (4+2).

The reverse of factoring is distribution, so, 2(2+1) = 6. This has to be true always. The argument I have seen to this is that (6/2) can be distributed.

This is true ONLY is 6/2 is in parentheses, otherwise, the 6 and 2 are separated by a division slash, and the 2 is a factor of 2+1.

So, let’s prove the initial equation:

6/6 = 1

6/(4+2) = 1

6/2(2+1) = 1

the same can be done for other factors:

6/6 = 1

6/(3+3) = 1

6/3(1+1) = 1

Distribution is actually a part of “Simplifying Equations” and is not bound to the order of operations as “multiplication”, since it is in fact “removing

parentheses by distributing”. This can be googled and several references found.

Simplifying 2(2+1) + 3(2+1) = 5(2+1). We “combined like terms” here, by adding, and did not perform the “parntheses” part of order of operations, nor did

we multiply, which is also higher priority than adding, because we only simplified.

Lastly, I hear the argument that “This is strictly numbers and you don’t use algebra rules since there are no variables”. That is the most asinine

arguement I have heard yet. All axioms, laws, and properties use variables, meaning that they hold true for “any number”, hence the proofs with

variables.

I welcome thoughts on this, in an intellectually formed response. I am tired of the ‘flaming’ that goes on by imbciles on some other forums with

rebuttals like “it is 9. go back to grade 3 you moron”, or “google says it is 9″, when google changes the equation to (6/2)*(2+1), and wolfram

contradicts itself with 2n/2n = 1, and 6/2n = 3/n, but then says 6/2(2+1) is 9. wolframs “terms” state that any answer should be verified with common

sense and accuracy should also be verified.

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Great reply mathman! Cleared a lot for me why I got 1 for the answer. Thanks!

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hey man….

6/2(1+2)=.?

first bracket

6/2(3)

again bracket

so want to solve the bracket..

2(3)=3

so,6/6

ie,,6/6=1

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The 2 in front of the parenthesis implies possession, doubling the parenthesis is the intent. Otherwise an alternative symbol would have been used to imply multiplication. In which case 9 would be correct.

6/2*(1 + 2) = 9

6/2(1+2)= 1

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I totally agree.

This equation is 6 over 2(2+1).

No one ever wrote a / bc when they meant ac / b

That is because they are different expressions.

Distributive property cannot change the final value,

and, when used, is 6 / (4+2).

One more thing, do the factoring for 6 + 3 = 9

Now factor 6/2 from 6 + 3 and we get:

(6/2)2 + (6/2)1 = 9

Notice how 6/2 MUST be in parentheses?

Now final step:

(6/2)(2+1)

The same simple idea applies to “Foiling” binomials:

x+1(x+2) = x + ((1)x + (1)2)

(x+1)(x+2) = (x+1)x + (x+1)2

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Pingback: SQL SERVER – Weekly Series – Memory Lane – #048 | Journey to SQL Authority with Pinal Dave

This is wrong the answer is 1 You can’t just change 2 (3) into 2*3 therefore parentheses would come first 2 (3) = 6 ..

.

6÷6=1..

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Mathman, i quite follow your explanations and found it interesting. I’m a proponent that 6/2(2+1) is 9.

Your explanation from the point view of 6/6 overlooked one math rule; Introduction of parenthesis or bracket.

While 6/(2(2+1)) = 1,

6/2(+1) = 9.

What helps our communications in math is the parenthesis or bracket.

6/6 = 6/(4+2) = 6/(2(2+1))! Here, as a math student one knows that the 2(2+1) is one as a denominator. However,

6/2(2+1) = 6/2x(2+1), which when either BODMAS or PEMDAS with LEFT TO RIGHT is applied you’ll get 9.

6/2x(3) = 3×3=9..

Now consider, 1/2n was shown in your comment.

This could be interpreted as (1/2) x n or 1 /(2n)

You’ll agree with me that both would give different answers. But in simple math operation, the expression if written the way it’s stated, i.e. 1/2n, any procedure or system of calculating it would take (1/2) x n. If the other is meant, it must surely be communicated with parenthesis or bracket, i.e. 1/(2n) any no mathematical method or procedure of calculation would mistake its answer.

Therefore beloved mathman, the expression 6/2(2+1) would simply be solved as 6/2*3=3*3=9.

Thank you.

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Ifeoluwa,

“any procedure or system of calculating it”

And any machine’s solution needs to be verified by a human for correctness. I am not talking about “how a machine would parse this equation”. I am talking about how humans have solved this for the last couple hundred years. Not sure how this will turn out on this forum, but let’s try:

Intro to Real Analysis by Bartle and Sherbert:

—————-QUOTE——-

X :=

1

— : n ∈ ℕ

2n

or more simply: X = 1/2n

———-End Quote———

You can verify that passage by googling the eBook.

The introduction to machines and programs have very much clouded the way things have been done historically. Wolfram|Alpha changed its code in 2013 and was solving it as above, before the change.

One of your arguments relies on : 2(2+1) = 2 x (2+1)

But doesn’t 2(2+1) also equal (2x(2+1)) ??

This is why the equation is ambiguous.

Since you also used machines as an argument, I strongly welcome your comment the following: Use wolfram|alpha and type in:

First example: cos2a

Second: cos2a/cos2a

Third: cos2a/b/cos2a/b

You can also find examples of algebraic division on a mathematician’s math page at Syracuse University:

http://cstl.syr.edu/fipse/algebra/unit2/parenth.htm

Eg: 6 / 2n = 3/n

Now let n=2+1 and solve both sides…

Remeber how we were taught to simplify algebraic division?

1 – Divide the coefficients of the like terms

2 – Subtraction the powers (exponents) of the like terms

Eg: 4a^2 / 2a = ??

Divide coefficients: 4 / 2 = 2

Subtract powers: 2-1 = 1

result: 2a

I stand by 1 as how I would solve it and how my colleagues would accept and understand it, however, after researching the topic, it is very clear how ambiguous it is :)

I look forward to your comments on these points.

King Regards,

your beloved mathman :)

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