What divides the Internet is not math
They swear by their PhD in maths or by their experience as a software developer. They invoke PEMDAS or BODMAS as the correct order of operations. Some of them admit that the mathematical expression (because it’s not an equation) might be ambiguously written, but still consider everyone else stupid if they don’t say the correct answer is 16. Or 1. Or… what is it?
Here’s the so-called equation:
It’s been the source of countless comments on Twitter, and of many other comments (hundreds and hundreds) on each and every article featuring it on the Internet–of which I’ve selected a few:
- Same on Yahoo News, for the sake of the comments
- Popular Mechanics: This Simple Math Problem Drove Our Entire Staff Insane. Can You Solve It?
- INSIDER: This equation has 2 wildly different answers depending on what you learned in school, and it’s dividing the internet
Here’s the truth: this has nothing to do with what you’ve been taught in school. In only has to do with whether you’re using your brain or not. Most likely, you’re not using it properly.
Before even try to evaluate the “correct” answer, let’s demystify the “laws” invoked by the collective
§1. PEMDAS vs. BEDMAS. Even Wikipedia, in Order of Operations, where it lists both acronyms, notes, based on John A. Ball’s Algorithms for RPN calculators (1 ed., John Wiley & Sons, 1978), p. 31:
These mnemonics may be misleading when written this way. For example, misinterpreting any of the above rules to mean “addition first, subtraction afterward” would incorrectly evaluate the expression
10 − 3 + 2.
The correct value is 9 (and not 5, as if the addition would be carried out first and the result used with the subtraction afterwards).
This means that the famous mnemonics
- Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
- Brackets, Exponents, Division/Multiplication, Addition/Subtraction
don’t have to been interpreted literally, you dumb asses. Multiplication and Division have the same priority and they have to be interpreted from left to right! PEMDAS and BEDMAS are the exact same thing, and rightfully so: there’s only one math on Earth!
§2. Different calculators give different results. As I have noted in my previous post on calculators (under “UPDATE 1”), there is a problem with implied multiplication. For instance, our expression would evaluate:
- on Casio fx-991ES: 16
- on Casio fx-991ES PLUS: 1
To quote myself:
The first device, fx-991ES, uses the correct order of the operations (which, BTW, is PEMDAS), performing a strict left-to-right interpretation of the explicit and implicit multiplication and division operators. The second one, fx-991ES PLUS, chose by design to give a higher priority to the implied multiplication (“multiplication where the multiplication sign is omitted”). I can understand that some people might consider that omitting the multiplication sign somewhat “creates a tighter bond” between the operands, but there is no rule to state that. I can only think of CASIO as of a company that gave way to the idiots when designing newer devices.
But in the case of such calculators, the user manual specifically lists the order of operations:
This being said, there’s no reason to give implied multiplication a special status; we should then trust calculators that give the answer “16”–and beyond “some Casio models” (as opposed to “some other Casio models”), it looks like TI-30X Plus and TI-30X Pro are using the correct order of operations. (America’s greatness has been restored. Boo, Casio!)
§3. It’s ambiguously written. We can probably agree with that, on the following grounds:
- It includes implied multiplication, and some people are either mimicking their calculators or trusting their calculators so much that they compute the implied multiplication before other multiplications or divisions. Other people do not, hence the conflict.
- It’s a fraction that’s written on a single line (this is the easiest way of doing it while using a computer instead of pen and paper), and (regardless of any PEMDAS) in such cases parenthesis/brackets are usually added to separate the nominator and the denominator. There nothing has been added, hence the conflict.
- To add insult to injury, or rather to prove that the aforementioned parenthesis were needed, some people take the mnemonics PEMDAS and BEDMAS literally, i.e. they give multiplication and division different priorities (while they shouldn’t), hence the conflict.
A perfectly unambiguous arithmetic expression would have been:
8 / (2*(2+2))
Such an expression is what was probably intended, and all calculators would evaluate it to… 1.
- People are throwing hard words at each other on the Internet because of an arithmetic expression that has been concocted specifically to create conflict, chaos, and dissent.
- There are people who don’t know how to interpret PEMDAS and BEDMAS.
- There are calculators (and people too!) that give special priority to implied multiplication, for no good reason.
- Using the correct precedence rules (PEMDAS, correctly interpreted, and no privileges to implied multiplication), it evaluates to 16.
- Adding parentheses to transform it to what the original intention probably was, it then evaluates to 1, but only then.
Here. It’s 16, but it’s sneaky. It should have been 1, but not as it’s written. And most Americans don’t understand PEMDAS. These are all facts. No need to read 3,000 comments of people screaming at each other. What divides the Internet is not math, but people’s conceit and lack of a systematic approach.
BTW, my approach is consistent with this other “problem” from 2017 where the answer is 288. Just saying.
Oh, and don’t blindly trust your pocket calculator until you read their order of operations rules.
Curand se va vota rezultatul!
Steven Strogatz, professor of mathematics at Cornell: The Math Equation That Tried to Stump the Internet:
A version of this article appears in print on Aug. 3, 2019, Section A, Page 10 of the New York edition with the headline: 8 ÷ 2 (2 + 2) =.
So I fixed it before the NYT 🙂
Also, the NYT article only convers “BODMAS is just PEMDAS, multiplication and division have equal priority, we work from left to right” and “we should write unambiguous math expressions and add parentheses as needed,” but it fails to notice that the expression being 8 ÷ 2 (2 + 2) and not 8 ÷ 2 × (2 + 2), there’s also the issue of the implied multiplication, which has a higher priority on some calculators.
Some calc apps now ask the user about how to treat the implied multiplication: