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Bạn đang xem: 8 ÷ 2(2 + 2) the viral equation has only one answer and that is 1 not 16

Posted July 31, 2019 By Presh Talwalkar. Read about me, or e-mail me.


As many people have opinions on this problem, I want to mô tả a bit about myself. I run the randy-rhoads-online.com channel on YouTube, which has over 1.5 million subscribers & 245 million views. I studied Economics & Mathematics at Stanford University, and my work has received coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, và 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 lớn the following expression?

8÷2(2 + 2) =

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

Watch the video clip 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, và mind only your decisions." Since 2007, I have devoted my life to sharing the joy of trò chơi theory và mathematics. randy-rhoads-online.com now has over 1,000 không tính phí articles with no ads thanks to lớn community support! Help out and get early access khổng lồ posts with a pledge on Patreon.

. . . . . . M I N D . Y O U R . D E C I S I O N S . . . . Answer khổng lồ 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 lớn 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 đầu vào has lớn 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 khổng lồ evaluate from left khổng lồ 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 khổng lồ current usage.

Some people have a different interpretation. & 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 & you saw 8÷2(4) in a textbook. What would you think the tác giả was trying to lớn write?

Historically the symbol ÷ was used to mean you should divide by the entire sản phẩm 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. & 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 lớn older usage.

Binary expression trees

Since some people think the answer is 16, và 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 khổng lồ 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.

*

The in-line expression also omits the parentheses of the divisor. This is like how trigonometry books commonly write sin 2θ lớn mean sin (2θ) because the argument of the function is understood, & writing parentheses every time would be cumbersome.

However, that practice of the division symbol was confusing, & it went against the order of operations. It was something of a well-accepted exception lớn 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 khổng lồ 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, và most of the time people have misunderstood my point, not read the post fully, or not read the sources. If you send an thư điện tử 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 & provided a bit about the history of the division symbol.

http://www.slate.com/articles/health_and_science/science/2013/03/facebook_math_problem_why_pemdas_doesn_t_always_give_a_clear_answer.html

3. The historical usage of ÷ is documented the following journal article from 1917. Notice the author points out this was an “exception” lớn the order of operations which did cause confusion. With modern typesetting we can avoid confusing expressions altogether.

Lennes, N. J. “Discussions: Relating lớn 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 bởi vì 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 & the article does not address why students incorrectly answer the unambiguous problem 9 – 3 ÷ (1/3) + 1.

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Ambiguous PEMDAS

6. I have also read many articles from people who disagree with me và allude to my video, but then they vày not link to my work. Academic disagreements should be kind spirited & fair minded. If you see an article, please let them know to links to my clip or blog post.