The Autodidacts

Exploring the universe from the inside out

Disorder of Operations

Yesterday morning I saw, for the first time1, a math problem trending on Twitter:

$$8 ÷ 2(2+2) = \color{#990000} \textbf{?}$$

As the co-founder of algebrarules.com, I have received several emails about this stupidly-simple looking math problem in the past few days.

The correct solution23 is $\color{#990000} \mathbf{16}$. But many people (the majority of people, according to unreliable social media polls) and even some cheap Casio calculators, get $\color{#990000} \mathbf{1}$. Why?

There are a number of reasons why this problem creates so much confusion — and was, most likely, created to create such confusion — all of them involving Order of Operations. Most of them, fortunately, are the result of confused people, rather than actual mathematical ambiguity. But there’s a bit of that as well.

  1. Acronyms: Mashable says the confusion is caused by the fact that order of operations acronyms used in different parts of the world — PEMDAS (USA), BEDMAS (CA & NZ), BODMAS/BIDMAS (UK & many other English-speaking countries) — put multiplication and division in different order.

    But the order of multiplication and division in the acronym doesn’t matter, because multiplication and division have equal priority, and are done left to right.4 (Same goes for addition and subtraction. ) That’s how they can be reversed in some acronyms. PEMDAS, BODMAS, BEDMAS all mean the same thing, despite the difference in order.

    Confusion arises because the acronyms don’t make the groupings clear. Follow the acronyms to the “letter”, without knowing this, you will get the wrong result.

  2. Notational conventions are a work-in-progress. Further confusion arises because as late as the early 20th century, some textbooks had multiplication take precedence over division.

  3. Divisive division. Another source of confusion arises from the different ways of denoting division: using an Obelus (8 ÷ 2) , slash (8/2), fraction slash (8⁄2) or fraction line ($\frac{8}{2}$). These will all evaluate to the same thing in a modern calculator. However, they each lead to a different intuitive grouping of elements. Historically, the Obelus was sometimes used to mean you should divide by the entire product on the right of the symbol.5 In addition, the fraction line forces you to group elements in a less ambiguous way than the other symbols.

  4. Implied Multiplication. In at least a handful of respected academic journals6, textbooks7, and lectures8, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division. This makes sense intuitively, but most decent calculators have no truck for it, and doggedly follow the left-to-right order for division and multiplication.

Step by step solution

First, do the stuff in parentheses.

$$8 ÷ 2(\color{#990000}2+2\color{#444444}) = 8 ÷ 2(\color{#990000}4\color{#444444})$$

There’s no argument about that.

Some people get confused here by the fact that there’s still a set of parentheses, and think that they affect the multiplication. They don’t. Everything that is in the parentheses has already been done, so we can remove them.

Now we have:

$$\color{#dd4814}8 ÷ 2\color{#444444} \times \color{#990000}4$$

Apply PEM/DAS, starting from the left, with the division, gives:

$$\color{#dd4814}4 \color{#444444} \times \color{#990000}4$$

And we have: $$\mathbf{16}$$

All the confusion would be solved by coming up with a better mnemonic for order of operations and using more parentheses. (Hey, people say Lisp is a beautiful language!)

$$8 ÷ 2(2+2) = 16 + \text{confusion}$$

$$(8 ÷ 2)(2+2) = 16$$

$$8 ÷ (2(2+2)) = 1$$

But, lest we forget, this is a problem of mathematical notation convention, not mathematical truth. Whoever cooked up this little doozy and loosed it on the internet (again) is probably howling with laughter.