Math Gone Viral: 8÷2(2+2)

Last week was an exciting one for the math community. Why, you ask? A math problem went viral, sparking a heated debate on the internet about which solution was “correct” and why. What an exciting time to be a mathematician!

Only, you don’t need to be a mathematician to join in on the conversation. In fact, you just need to be familiar with order of operations and know a little bit of algebra.

Remind me again, what’s “order of operations?”

Order of operations provides a consistent framework for simplifying algebraic equations. The name itself reveals its purpose: it tells you in which order to perform your operations (defined as addition, subtraction, multiplication, division). This may have been taught to you in school as PEMDAS, or “Please Excuse My Dear Aunt Sally.” 

PEMDAS stands for:

Parentheses

Exponents

Multiplication/Division

Addition/Subtraction

To use PEMDAS, first simplify any parentheses in the equation, then any exponents, then any multiplication/division (from left to right), and finally, any addition/subtraction (from left to right). All algebraic equations are written under the assumption that PEMDAS will be followed while solving or simplifying. It’s part of the universally agreed mathematical ruleset.

So, why is this problem causing so much commotion?

Math is, for the most part, a straightforward subject. We’re taught that there’s only one “right” answer. We’re also told that by following the rules, we’ll be automatically led to that correct solution. So when a math problem isn’t as clear-cut as promised, we’re thrown for a loop.  Let’s take a look.

The viral math problem was presented as the below:

6.1.2.png

Following PEMDAS, we simplify the addition inside the parentheses:

6.2.2.png

But then what happens next? Do we allow the parentheses to continue taking priority? If we simplify based on that interpretation, we get the following:

6.3.2.png

However, we can also interpret the parentheses as multiplication, in which case we follow the left-to-right division/multiplication rule. If we simplify based on that interpretation, we get the following:

6.4.2.png

So, which is the “right” solution? Is it one, or sixteen? Well, technically both can be correct. How is that possible? There was interpretation involved in solutioning, which is not a best practice in mathematics. The problem should have been written more clearly from the start to avoid confusion.

How should the problem have been written?

That depends on what the expression is representing! We learned that in algebra, relationships between variables come together in an equation to convey some larger concept or meaning. In this example, the equation should be rewritten to better express the underlying value (no interpretation required). 

So, if the equation is supposed to equal one, it should be rewritten as:

6.5.2.png

And if the equation is supposed to equal sixteen, it should be rewritten as:

6.6.2.png

Creating clearly written, unambiguous equations is part of being a good mathematician. It’s a standard that mathematicians at any level must adhere to, in order to communicate their work to each other without causing confusion. Sometimes, it’s as simple as adding in an extra set of parentheses or an additional multiplication sign.

So, what does this whole exercise show? That math is truly the universal language… if you follow the universally agreed communication rules!

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