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You are watching: What is the answer to 2

Posted July 31, 2019 through Presh Talwalkar. Read about me, or email me.

As many people have opinions on this problem, I desire to share a bit about myself. I operation the jajalger2018.org channel top top YouTube, which has over 1.5 million subscribers and also 245 million views. Ns studied Economics and Mathematics at Stanford University, and my work has actually received coverage in the press, consisting of the Shorty Awards, The Telegraph, Freakonomics, and also many other famous outlets.

I also have covered similar problems before, consisting of the following videos:

What is 6÷2(1+2) = ? The exactly Answer described (over 12 million views)

9 – 3 ÷ (1/3) + 1 = ? The exactly Answer (Viral difficulty In Japan) (over 9 million views)

Since there is an additional problem that’s going viral right now, it’s time because that the bespeak of to work to conserve the day!

What is the exactly answer come the adhering to expression?

8÷2(2 + 2) =

(Note: some civilization write 8/2(2 + 2) = but this has the same answer.)

Watch the video clip where I define the exactly answer.

What is 8÷2(2 + 2) = ? The exactly Answer Explained

Or save reading. . ."All will certainly be well if you use your mind for your decisions, and mind just your decisions." due to the fact that 2007, i have committed my life to share the pleasure of game theory and also mathematics. jajalger2018.org now has actually over 1,000 cost-free articles v no ads many thanks to ar support! aid out and also get early access to posts with a pledge top top Patreon.

. . . . . . M ns N D . Y O U R . D E C i S ns O N S . . . . Answer come 8÷2(2 + 2) = ?

(Pretty lot all posts are transcribed conveniently after ns make the videos for them–please let me understand if there are any type of typos/errors and I will certainly correct them, thanks).

The exactly answer is 16 according to the modern interpretation the the order of operations.

The bespeak of operations

The expression have the right to be simplified by the stimulate of operations, frequently remembered through the abbreviation PEMDAS/BODMAS.

First evaluate Parentheses/Brackets, climate evaluate Exponents/Orders, climate evaluate Multiplication-Division, and finally evaluate Addition-Subtraction.

Everyone is in agreement about the an initial step: simplify the enhancement inside the parentheses.

8÷2(2 + 2) = 8÷2(4)

This is where the debate starts.

The prize is 16

If you kind 8÷2(4) right into a calculator, the input needs to be parsed and also then computed. Many calculators will transform the parentheses right into an implied multiplication, so we get

8÷2(4) = 8÷2×4

According to the stimulate of operations, department and multiplication have the exact same precedence, therefore the exactly order is to advice from left to right. An initial take 8 and divide the by 2, and also then multiply by 2.

8÷2×4 = 4×4 = 16

This it s okay to the correct answer of 16.

This is without debate the correct answer of exactly how to advice this expression according to present usage.

Some civilization have a different interpretation. And while it’s not the correct answer today, it would have been related to as the exactly answer 100 years ago. Some civilization may have learned this other interpretation an ext recently too, but this is no the means calculators would evaluate the expression today.

The other an outcome of 1

Suppose it to be 1917 and also you witnessed 8÷2(4) in a textbook. What would you think the writer was trying to write?

Historically the symbol ÷ was offered to mean you need to divide by the entire product top top the right of the price (see much longer explanation below).

Under the interpretation:

8÷2(4) = 8÷(2(4)) (Important: this is outdated usage!)

From this stage, the rest of the calculation works by the bespeak of operations. Very first we evaluate the multiplication inside the parentheses. So we multiply 2 by 4 to obtain 8. And then we division 8 through 8.

8÷(2(4)) = 8÷8 = 1

This offers the an outcome of 1. This is not the correct answer that calculators will certainly evaluate; rather it is what someone could have construed the expression follow to enlarge usage.

Binary expression trees

Since some people think the price is 16, and also others think that is 1, many people argue this difficulty is ambiguous: that is a poorly created expression through no solitary correct answer.

But here’s mine counter-point: a calculator is not going to say “it’s an pass out expression.” just as courts rule around ambiguous legitimate sentences, calculators evaluate seemingly ambiguous number expressions. For this reason if we take the expression together written, what would a calculator evaluate it as?

There space two feasible binary expression trees.


The in-line expression also omits the clip of the divisor. This is like how trigonometry publications commonly create sin 2θ to typical sin (2θ) due to the fact that the dispute of the role is understood, and writing bracket every time would certainly be cumbersome.

However, that exercise of the division symbol was confusing, and it went against the stimulate of operations. It was something the a well-accepted exemption to the rule.

Today this exercise is discouraged, and also I have actually never viewed a mathematician create an pass out expression utilizing the department symbol. Textbooks constantly have proper parentheses, or they define what is to be divided. Due to the fact that mathematical typesetting is much simpler today, we virtually never see ÷ as a symbol, and also instead fractions space written through the numerator vertically over the denominator.

*Note: I get many, numerous emails arguing with me about these order of to work problems, and also most the the time human being have misunderstood mine point, not check out the short article fully, or not review the sources. If you send an e-mail on this problem, I might not have time to reply.


0. Google evaluation https://www.google.com/#q=8÷2(2%2B2)

1. Web archive the Matthew Compher’s saying Semantics: the obelus, or division symbol: ÷

2. In 2013, Slate explained this trouble and noted a bit about the background of the department symbol.


3. The historical consumption of ÷ is documented the adhering to journal article from 1917. An alert the author points out this was an “exception” to the order of operations i m sorry did cause confusion. With contemporary typesetting we have the right to avoid confusing expression altogether.

Lennes, N. J. “Discussions: Relating come the bespeak of work in Algebra.” The American mathematics Monthly 24.2 (1917): 93-95. Web. Http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents

4. In to add magazine, David Linkletter write a different perspective that the problem is no well-defined (and see his longer article too). I carry out not agree through the portrayal that what “mathematicians” say, as countless mathematicians are happy because that the write-ups I have actually written. The article additionally does not address why students mistakenly answer the unambiguous difficulty 9 – 3 ÷ (1/3) + 1.

PEMDAS Paradox (Plus magazine)

PEMDAS Paradox (longer article)

5. Harvard mathematician Oliver Knill also has a differing perspective the the just wrong answer is saying over there is a single correct answer. I strongly disagree and also the post does not resolve why students wrongly answer the unambiguous trouble 9 – 3 ÷ (1/3) + 1.

See more: What Is The Sine Of 45 Degrees, Sin 45 Degree Value

Ambiguous PEMDAS

6. Ns have additionally read many articles from civilization who disagree through me and allude to my video, yet then they execute not connect to my work. Academic disagreements must be kind spirited and also fair minded. If you view an article, you re welcome let them know to connect to my video clip or blog post.