This is a problem that’s been thrown around the Facebook recently it seems. Man, it seems so simple:
\[6÷2(1+2)=?\]Simple until you ask a group of people what the answer is, and find out half of us think it’s nine, and the other half think it’s one.
So what IS the answer?
Short answer: it’s a terribly written problem, or more accurately, it’s a problem written solely to make people argue over its order of operations. It touches on the ambiguity of parentheses, multiplication, and division in the “Please Excuse My Dear Aunt Sally / Parentheses Exponents Multiplication Division Addition Subtraction” mnemonic.
The issue stems from the incorrect (but often subconscious) belief that there exists a Bible of Mathematics somewhere, which decrees every detail of how arithmetic should be expressed and simplified. As simple as the problem seems, there’s no prevailing convention on whether $6÷2$ or $2(1+2)$ gets precedence here. Particularly when half of school teachers say multiplication has precedence over division, and half say multiplication and division have equal precedence and are evaluated left-to-right. (And hey does implied multiplication by parentheses fall into the P or M of PEMDAS?)
In the real world of mathematics, this is as unanswerable a question as me asking “Is there a mathematical size bigger than the amount of integers but less than the amount of real numbers?” [1] because you can’t address it without first specifying the rules you are allowed to use on the problem.
Thus I’m forced to settle for what we call a “consistency result”:
If ÷ division takes precedence over implied multiplication located further to the right, then $6÷2(1+2)=9$. If not, then $6÷2(1+2)=1$.
That’s the best that can be done, mathematically. If you wanted a definitive answer, you should have asked me to evaluate $(6÷2)(1+2)$ or $6÷(2(1+2))$. That’s what parentheses are for, anyway.
[1] This is something called The Continuum Hypothesis. We use the word “hypothesis” here because once mathematicians formalized the rules of set theory (well, one set of rules for set theory), it couldn’t be proven true or false. We later proved that it couldn’t be proven true or false, basically because we didn’t provide enough rules to handle the question. So just like $6÷2(1+2)=9$ or $6÷2(1+2)=1$ depending on what rules we use, we can assume the answer to the Continuum Hypothesis is either yes or no.
Or we can completely ignore it and go on with our lives, just like we do with other silly questions like $6÷2(1+2)=?$.</small>
Addendum: Someone rightfully pointed out on Hacker News that calling the Continuum Hypothesis a “silly” question doesn’t do it justice. Rather, I mean that posing the Continuum Hypothesis as a “yes or no” question is silly.