> ## Content Index > Fetch the complete content index at: https://www.autodidacts.io/llms.txt > Use this file to discover other available public pages before exploring further. # Disorder of Operations - URL: https://www.autodidacts.io/disorder-of-operations/ - Published: 2019-08-03T15:12:00.000Z - Updated: 2023-10-18T22:21:23.000Z - Author: Curiositry - Tags: Essay, Math, Algebrarules.com, Order of Operations Yesterday morning I saw, for the first time[\[1\]](#fn1), a math problem trending on Twitter: $$8 ÷ 2(2+2) = \\color{#990000} \\textbf{?}$$ As the co-founder of [algebrarules.com](https://algebrarules.com/?ref=autodidacts.io), I have received several emails about this stupidly-simple looking math problem in the past few days. The correct solution[\[2\]](#fn2)[\[3\]](#fn3) is $\\color{#990000} \\mathbf{16}$. But many people (the *majority* of people, according to [unreliable social media polls](https://twitter.com/neworIean/status/1156297121118605312?ref=autodidacts.io)) and even some [cheap Casio calculators](https://pbs.twimg.com/media/EAv9QjeUYAAPfRa?format=jpg&name=4096x4096&ref=autodidacts.io), 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](https://mashable.com/article/math-equation-divides-twitter/?ref=autodidacts.io) 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*.**[\[4\]](#fn4) (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.[\[5\]](#fn5) 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 journals[\[6\]](#fn6), textbooks[\[7\]](#fn7), and lectures[\[8\]](#fn8), multiplication denoted by juxtaposition (also known as [implied multiplication](https://en.wikipedia.org/wiki/Multiplication?ref=autodidacts.io#Implicit)) 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 PE**M/D**AS, **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. --- 1. It turns out that an almost identical problem — $48/2(9+3)$ — made the rounds of the internet eight years ago. [↩︎](#fnref1) 2. [wolframalpha.com/input/?i=8 ÷ 2(2+2)](https://www.wolframalpha.com/input/?i=8+%C3%B7+2%282%2B2%29&ref=autodidacts.io) [↩︎](#fnref2) 3. [google.com/search?q=8 ÷ 2(2+2)](https://www.google.com/search?q=8%20%C3%B7%202%282%2B2%29&ref=autodidacts.io) [↩︎](#fnref3) 4. [*Order of Operations* on Wikipedia](https://en.wikipedia.org/wiki/Order%5Fof%5Foperations?ref=autodidacts.io#Mnemonics) [↩︎](#fnref4) 5. [https://www.jstor.org/stable/2972726?seq=1#page\_scan\_tab\_contents](https://www.jstor.org/stable/2972726?seq=1&ref=autodidacts.io#page%5Fscan%5Ftab%5Fcontents) [↩︎](#fnref5) 6. [https://cdn.journals.aps.org/files/styleguide-pr.pdf](https://cdn.journals.aps.org/files/styleguide-pr.pdf?ref=autodidacts.io) [↩︎](#fnref6) 7. [https://en.wikipedia.org/wiki/Course\_of\_Theoretical\_Physics](https://en.wikipedia.org/wiki/Course%5Fof%5FTheoretical%5FPhysics?ref=autodidacts.io) [↩︎](#fnref7) 8. [http://www.feynmanlectures.caltech.edu/I\_06.html](http://www.feynmanlectures.caltech.edu/I%5F06.html?ref=autodidacts.io) [↩︎](#fnref8)