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I hold a masters in computer science from one of the worlds top universities and until today I thought I more or less know basic math.

I'm sure you guys all know these click-bait simple "90% of people can't solve this equation" posts on facebook where everyone starts to argue over a simple equation (and I DEFINITELY don't want to kick off one of those - but I'd like to discuss the roots of this confusion).

So today there was another one of those:

6/2(1+2)

Based on what I learned and applied throughout all my years in university, this equals 9, since brackets are evaluated first and then it's left to right, since division and multiplication have the same operator precedence.

Google agrees with that:

Wolfram Alpha agrees with that:

and my texas instruments agrees with that too. So it's 9, right?

Well today I came across a claim I hadn't heard before, which is "implied multiplication takes precedence over both explicit multiplication and division" - so by that rule it would not be left to right in the above example, but the implied multiplication would be evaluated before the division, which would mean that

6/2(1+2) == 1 != 6/2*(1+2)

So, are google, wolfram alpha and my calculator all wrong (they by the way also yield 9 if ÷ is used instead of /)?

The only thing i found on the issue so far is this statement on Wikipedia ():

Mixed division and multiplication: Similarly, there can be ambiguity in the use of the slash symbol / in expressions such as 1/2x.[5] If one rewrites this expression as 1 ÷ 2x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:

1 ÷ 2 × x = 1 × 1/2 × x = 1/2 × x.

With this interpretation 1 ÷ 2x is equal to (1 ÷ 2)x.1[6] However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[7] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[a]

So one thing this tells me is clearly AVOID IMPLIED MULTIPLICATION

but what is internationally actually 'more correct and less wrong'? Also, I don't fully see how ÷ vs. / is relevant to this question?

By the way, some calculators do seem to evaluate to 1:

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3 Answers

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While the rules of mathematics are usually very precise, this is more of a grammar issue. Once the written form is correctly parsed, then you can apply precise and well-known rules to solve it. The problem is that there's no consensus on the right way to parse implied multiplication in the context of a larger expression.

There's a good analysis here, in which the author basically asserts that both answers are right, and the question is wrong.

My personal theory is that because many of us see polynomial terms as being discrete things, we tend to treat similar structures in other places as being discrete. When I see something that looks like a polynomial term, my inclination is to treat it as if it has parentheses around it. So, I see 2/4x as 2/(4*x), while others may see (2/4)*x. Both interpretations can be found in different textbooks.

You could avoid the ambiguity by rewriting the question without implied multiplication. An even better approach would be to change the way you represent division, and put the numerator above the denominator.

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In your picture of the $2$ calculators the one with the $9$ answer was only programed for order of operations. The calculator with the answer of $1$ was also programed to consider the distributive property. A calculator is only as good as its programming and only as intelligent as its user.

When I went to school we were taught implied multiplication took precidence because of the distributive property. And the order of operations was a suggestion not concrete.

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@Robert w Shull is correct,

Implied multiplication in this instance comes from the Distributive Law.

Distributive law, in mathematics, the law relating the operations of multiplication and addition, stated symbolically, a(b + c) = ab + ac; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product ab + ac.

–Encyclopedia Britannica

Because Parenthesis have the highest priority in the order of operations, the equation is correctly solved as follows:

6÷2(1+2)=?
6÷(2*1 + 2*2)=?
6÷(2 + 4)=?
6÷(6)=?
6÷6=1

Because there is only one "explicit" operator (division) outside of the parenthetical term, the equation is a simple fraction...

$$ \frac{6}{2(1+2)}$$

which we can then reduce...$$ \frac{3}{(1+2)}$$

finally...$$ \frac{3}{3}=1$$

Updated: Implied multiplication is everywhere, 1x = x and x/1=x. Any number times 1 is that number and any number divided by 1 is that number. We can use this fact to maintain that the given equation is an implicit explicit division problem.

6÷2(1+2)=?
3÷1(1+2)=?
3÷1(3)=?
3÷3=1

More support here:

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