As many people have opinions on this problem, I want to share a bit about myself. I run the MindYourDecisions channel on YouTube, which has over 1.5 million subscribers and 245 million views. I studied Economics and Mathematics at Stanford University, and my work has received coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.
I also have covered similar problems before, including the following videos:
What is 6÷2(1+2) = ? The Correct Answer Explained (over 12 million views)
9 – 3 ÷ (1/3) + 1 = ? The Correct Answer (Viral Problem In Japan) (over 9 million views)
Since there is another problem that’s going viral right now, it’s time for the order of operations to save the day!
What is the correct answer to the following expression?
8÷2(2 + 2) =
(Note: some people write 8/2(2 + 2) = but this has the same answer.)
Watch the video where I explain the correct answer.
What is 8÷2(2 + 2) = ? The Correct Answer Explained
Or keep reading. "All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
.
.
.
.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
.
.
.
Answer To 8÷2(2 + 2) = ?
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
The correct answer is 16 according to the modern interpretation of the order of operations.
The order of operations
The expression can be simplified by the order of operations, often remembered by the acronyms PEMDAS/BODMAS.
First evaluate Parentheses/Brackets, then evaluate Exponents/Orders, then evaluate Multiplication-Division, and finally evaluate Addition-Subtraction.
Everyone is in agreement about the first step: simplify the addition inside the parentheses.
8÷2(2 + 2)
= 8÷2(4)
This is where the debate starts.
The answer is 16
If you type 8÷2(4) into a calculator, the input has to be parsed and then computed. Most calculators will convert the parentheses into an implied multiplication, so we get
8÷2(4)
= 8÷2×4
According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right. First take 8 and divide it by 2, and then multiply by 4.
8÷2×4
= 4×4
= 16
This gets to the correct answer of 16.
This is without argument the correct answer of how to evaluate this expression according to current usage.
Some people have a different interpretation. And while it’s not the correct answer today, it would have been regarded as the correct answer 100 years ago. Some people may have learned this other interpretation more recently too, but this is not the way calculators would evaluate the expression today.
The other result of 1
Suppose it was 1917 and you saw 8÷2(4) in a textbook. What would you think the author was trying to write?
Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below).
Under that interpretation:
8÷2(4)
= 8÷(2(4))
(Important: this is outdated usage!)
From this stage, the rest of the calculation works by the order of operations. First we evaluate the multiplication inside the parentheses. So we multiply 2 by 4 to get 8. And then we divide 8 by 8.
8÷(2(4))
= 8÷8
= 1
This gives the result of 1. This is not the correct answer that calculators will evaluate; rather it is what someone might have interpreted the expression according to older usage.
Binary expression trees
Since some people think the answer is 16, and others think it is 1, many people argue this problem is ambiguous: it is a poorly written expression with no single correct answer.
But here’s my counter-point: a calculator is not going to say “it’s an ambiguous expression.” Just as courts rule about ambiguous legal sentences, calculators evaluate seemingly ambiguous numerical expressions. So if we take the expression as written, what would a calculator evaluate it as?
There are two possible binary expression trees.
I suggested the binary expression tree on the left is consistent with PEMDAS/BODMAS. But what does a calculator actually do?
If you try Google (see it evaluate 8÷2(2+2)) you’ll get an answer of 16. Furthermore, the Google output even inserts parentheses to indicate it is using the binary tree on the left of (8/2)*(2 + 2).
Most popular calculators evaluate the expression the same way, and I would argue that is NOT a coincidence, but rather a reflection that calculators are programmed to the same PEMDAS/BODMAS rules we learn in school.
Common topics of discussion
I’m so happy people think of me for these kinds of questions. And I’m proud of everyone that takes the time to explain PEMDAS/BODMAS and why 16 is the correct answer. Along the way I have had the chance to help people clear up common sources of confusion.
—“I learned it a different way.”
Please do let us know a textbook or printed reference. Many people remember learning the topic a different way, but in 5 years no one has presented proof of this other way.
—“What about the distributive property?”
This is irrelevant to the answer. The distributive property is about how to multiply over a grouped sum, not about a precedence of operations. It is definitely true that:
8÷2(2+2) = 8÷2(4)
The issue is whether to do 8÷2 first or 2(4) first. PEMDAS says to go from left to right.
—“What about implied multiplication?”
Most calculators treat it the same way as regular multiplication. Grouped terms are typically grouped with parentheses if they are meant to be evaluated first.
—“The problem is not well-defined.”
To someone that says that, I would ask, “what is the sum of angles in a triangle?” If they say 180 degrees, I would point out that answer is only true in plane geometry (Euclidean geometry). In other geometries the answer can be different from 180 degrees. But no one would say “what is the sum of angles in a triangle” is not a well-defined question–we most often work in the plane, or we would specify otherwise.
Similarly you can ask if 0 is a “positive” number. In America, the convention is that 0 is neither positive nor negative. But in France 0 I am told 0 is considered to be positive. You’d have to re-write a lot of math tests in America if you say that “positive” is not a well-defined word.
Ultimately we say things like “a triangles angles sum to 180 degrees, according to the axioms of plane geometry,” and “0 is not positive, according to the definition in America.” Similarly we can say “8÷2(2+2) = 16, according to the modern interpretation of the order of operations.”
Isn’t the answer ambiguous?
Some mathematicians believe the expression is incorrectly written, and therefore can have multiple interpretations. I strongly disagree with this point. The main cause of confusion is the order of operations!
For example, consider the problem 9 – 3 ÷ (1/3) + 1 (over 9 million views). This is an unambiguous expression and has only a single answer. But the problem went viral in Japan after a study found 60 percent of 20 somethings could get the correct answer, down from a rate of 90 percent in the 1980s. It is clear the problem is students do not learn the order of operations.
Mathematicians who say “the answer is ambiguous” overlook that students get unambiguous expressions wrong at an alarming rate. It is our duty as mathematicians to emphasize the order of operations in its modern form so that we can write proper expressions and interpret them correctly. Not a single person who disagrees with me has considered why students get the wrong answer to 9 – 3 ÷ (1/3) + 1.
The symbol ÷ historical use
Textbooks often used ÷ to denote the divisor was the whole expression to the right of the symbol. For example, a textbook would have written:
9a2÷3a
= 3a
This indicates that the divisor is the entire product on the right of the symbol. In other words, the problem is evaluated:
9a2÷3a
= 9a2÷(3a)
(Important: this is outdated usage!)
I suspect the custom was out of practical considerations. The in-line expression would have been easier to typeset, and it takes up less space compared to writing a fraction as a numerator over a denominator:
The in-line expression also omits the parentheses of the divisor. This is like how trigonometry books commonly write sin 2θ to mean sin (2θ) because the argument of the function is understood, and writing parentheses every time would be cumbersome.
However, that practice of the division symbol was confusing, and it went against the order of operations. It was something of a well-accepted exception to the rule.
Today this practice is discouraged, and I have never seen a mathematician write an ambiguous expression using the division symbol. Textbooks always have proper parentheses, or they explain what is to be divided. Because mathematical typesetting is much easier today, we almost never see ÷ as a symbol, and instead fractions are written with the numerator vertically above the denominator.
*Note: I get many, many emails arguing with me about these order of operations problems, and most of the time people have misunderstood my point, not read the post fully, or not read the sources. If you send an email on this problem, I may not have time to reply.
Sources
0. Google evaluation
https://www.google.com/#q=8÷2(2%2B2)
1. Web archive of Matthew Compher’s Arguing Semantics: the obelus, or division symbol: ÷
2. In 2013, Slate explained this problem and provided a bit about the history of the division symbol.
3. The historical usage of ÷ is documented the following journal article from 1917. Notice the author points out this was an “exception” to the order of operations which did cause confusion. With modern typesetting we can avoid confusing expressions altogether.
Lennes, N. J. “Discussions: Relating to the Order of Operations in Algebra.” The American Mathematical Monthly 24.2 (1917): 93-95. Web. http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents
4. In Plus magazine, David Linkletter writes a differing perspective that the problem is not well-defined (and see his longer article too). I do not agree with the portrayal of what “mathematicians” say, as many mathematicians are happy for the articles I have written. The article also does not address why students incorrectly answer the unambiguous problem 9 – 3 ÷ (1/3) + 1.
PEMDAS Paradox (Plus magazine)
PEMDAS Paradox (longer article)
5. Harvard mathematician Oliver Knill also has a differing perspective that the only wrong answer is saying there is a single correct answer. I strongly disagree and the article does not address why students incorrectly answer the unambiguous problem 9 – 3 ÷ (1/3) + 1.
6. I have also read many articles from people who disagree with me and allude to my video, but then they do not link to my work. Academic disagreements should be kind spirited and fair minded. If you see an article, please let them know to link to my video or blog post.
MY BOOKS
If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.
Book ratings are from January 2023.
(US and worldwide links)
https://mindyourdecisions.com/blog/my-books
Mind Your Decisions is a compilation of 5 books:
(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines
The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)
40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)
The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)
The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)
Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)
Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.
Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.
Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.2/5 stars on 33 reviews)
Math Puzzles Volume 3 is the third in the series. (rated 4.2/5 stars on 29 reviews)
KINDLE UNLIMITED
Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.
Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription you will get access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.
US, list of my books (US)
UK, list of my books (UK)
Canada, book results (CA)
Germany, list of my books (DE)
France, list of my books (FR)
India, list of my books (IN)
Australia, book results (AU)
Italy, list of my books (IT)
Spain, list of my books (ES)
Japan, list of my books (JP)
Brazil, book results (BR)
Mexico, book results (MX)
MERCHANDISE
Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.
