Hey, this is Presh Talwalkar.
What is the value of this mathematical expression?
This math problem has gone viral
and it has received millions of comments on
Facebook, Twitter, Youtube, and other social media sites.
In this video, I'm going to present the correct answer.
The problem is an example of the order of operations.
These are commonly refered to as PEMDAS or BODMAS.
This refers to evaluating the parentheses/brackets,
then the exponents/orders,
Then multiplication and division
And finally, addition-subtraction.
You have two operations of the same precedence,
you want to evaluate them from left to right.
The first of the problem has no controversy.
This expression contains a parenthetical expression which must be evaluated first.
1+2 is inside the parentheses,
so we'll evaluate 1+2 to get 3.
Now, the question is what to do next.
If you this into google, wolfram alpha, or pretty much any scientific calculator,
the thing that's going to happen next is
all of these will interpret the parentheses as an implicit multiplication.
So this two parenthesis three will be converted into 2 times 3.
Now we continue the order of operations.
This expression only contains multiplication and division,
These are operators of equal precedence, so we'll evaluate them from left to right.
Starting on the left, we have 6 divided by 2
6 divided by 2 is equal to 3
We then have 3 multiplied by 3 - one final multiplication.
And that gets us to the correct answer of 9.
This is, without a doubt, the correct answer to
this expression as written according to
the modern usage of the order of operations.
So why did this problem cause so much controversy?
Well, there is another answer
that you could argue from a historical
perspective, so I actually found some
documentation that the order of
operations did have a slightly different
understanding in certain texts in 1917 or before.
So the first part of the
equation is the same as before: we have a
parenthetical expression and this should
be evaluated first we have 1 plus 2 and
that becomes 3. The debate then centers
around this division symbol.
So what does it mean that we have 6 divided by 2
parentheses 3? While there were textbooks
and there was a lot of usage that if you
had this division symbol where you had
something on the left divided by something on the right,
this was understood to mean
you want to divide the entire product on the left
by the entire product on the right.
So, for example, if a textbook wrote
"x divided by 2y" with this division symbol
they actually did mean x divided by
parentheses 2y. You wanted to take
2y as the entire product and have that as your denominator.
So under this historical usage - which is a special
exception to the order of operations (and we don't use it anymore) -
you would want to take this product on the right
as your divisor. So applying this rule would
then lead to the expression 6 over 2 x 3.
We will now convert the multiplication
in the denominator so that 2 times 3 is
equal to 6 and we now have one division
which is 6/6 and that's equal to 1.
So, many people argue that one is a correct
answer, and there is some historical
justification of this because of the way
that texts used to use the division symbol.
I would suggest this is probably
because of some historical artifact about typesetting:
it would have been much easier
to write the division symbol and have
the understanding you want to divide
everything on the left by everything on
the right; you wouldn't need to have an
expression where you write a numerator
over a denominator that would take a lot
more vertical space and you also would
need to keep putting parentheses everywhere.
This would be just something that would be understood.
Today we don't use this practice, because it can be confusing.
Instead, we follow the order of operations.
If we want to have a fraction,
we will put it as an expression like 6 over 6
which is written here.
So the correct answer to this problem is 9, but there is
some historical justification for the answer 1, but it's not how we'd interpret the problem today.
Did you get to the correct answer of 9?
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