whats the correct answer to this 2+2×0+2=
As many people accept opinions on this trouble, I desire to share a fleck about myself. I run the MindYourDecisions channel on YouTube, which has over 1.5 meg subscribers and 245 million views. I studied Economic science and Mathematics at Stanford University, and my piece of work has received coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.
I too have covered similar problems before, including the following videos:
What is half dozen÷ii(ane+2) = ? The Correct Answer Explained (over 12 million views)
9 – 3 ÷ (1/three) + i = ? The Correct Answer (Viral Problem In Nihon) (over 9 million views)
Since there is another problem that'southward going viral right at present, it's time for the guild of operations to save the twenty-four hours!
What is the correct answer to the following expression?
8÷2(two + 2) =
(Notation: some people write 8/ii(2 + 2) = but this has the same answer.)
Watch the video where I explicate the correct answer.
What is viii÷2(2 + 2) = ? The Correct Answer Explained
Or proceed reading.
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"All will be well if you lot utilize your listen 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 ane,000 free articles with no ads thank you to customs support! Assist out and go early access to posts with a pledge on Patreon.
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Answer To viii÷2(2 + ii) = ?
(Pretty much all posts are transcribed rapidly after I brand the videos for them–please let me know if there are whatever typos/errors and I will correct them, thanks).
The correct reply is xvi co-ordinate to the modern interpretation of the order of operations.
The club of operations
The expression tin exist 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-Due southubtraction.
Everyone is in agreement about the first step: simplify the improver inside the parentheses.
eight÷2(2 + 2)
= viii÷2(4)
This is where the debate starts.
The answer is 16
If you type viii÷2(4) into a calculator, the input has to be parsed and and then computed. Virtually calculators will convert the parentheses into an implied multiplication, so we become
8÷2(4)
= viii÷2×4
According to the lodge of operations, division and multiplication have the same precedence, so the right order is to evaluate from left to correct. First take viii and divide it by 2, and and so multiply by 4.
8÷2×iv
= iv×iv
= xvi
This gets to the correct reply of 16.
This is without argument the correct answer of how to evaluate this expression according to current usage.
Some people accept a different interpretation. And while it'southward not the correct answer today, it would accept been regarded as the correct answer 100 years ago. Some people may accept learned this other estimation more recently too, but this is not the fashion calculators would evaluate the expression today.
The other result of i
Suppose information technology was 1917 and you saw 8÷2(4) in a textbook. What would y'all think the author was trying to write?
Historically the symbol ÷ was used to mean you should divide past the entire product on the right of the symbol (see longer explanation below).
Under that estimation:
8÷ii(4)
= 8÷(2(4))
(Important: this is outdated usage!)
From this stage, the remainder of the calculation works by the club of operations. Start nosotros evaluate the multiplication inside the parentheses. And then we multiply 2 past 4 to become 8. And and so we divide 8 by 8.
8÷(2(4))
= eight÷eight
= 1
This gives the result of ane. This is not the correct reply that calculators will evaluate; rather information technology is what someone might accept interpreted the expression according to older usage.
Binary expression trees
Since some people think the answer is 16, and others call up it is one, many people argue this problem is ambiguous: it is a poorly written expression with no unmarried correct answer.
But here's my counter-bespeak: a calculator is not going to say "it'south an cryptic expression." Just every bit courts dominion about cryptic legal sentences, calculators evaluate seemingly cryptic numerical expressions. So if we take the expression as written, what would a calculator evaluate it as?
There are ii possible binary expression trees.
I suggested the binary expression tree on the left is consistent with PEMDAS/BODMAS. Merely what does a calculator really do?
If you try Google (see information technology evaluate eight÷two(ii+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)*(two + 2).
Nearly popular calculators evaluate the expression the same manner, and I would fence that is NOT a coincidence, just rather a reflection that calculators are programmed to the same PEMDAS/BODMAS rules we larn in school.
Mutual topics of discussion
I'yard so happy people call back of me for these kinds of questions. And I'm proud of anybody that takes the time to explicate PEMDAS/BODMAS and why 16 is the correct answer. Along the way I have had the risk to assistance people articulate upwards common sources of confusion.
—"I learned it a different way."
Please practise permit us know a textbook or printed reference. Many people remember learning the topic a different mode, only in 5 years no one has presented proof of this other mode.
—"What about the distributive property?"
This is irrelevant to the respond. The distributive holding is about how to multiply over a grouped sum, not virtually a precedence of operations. It is definitely true that:
viii÷2(2+2) = eight÷2(4)
The issue is whether to do 8÷2 first or 2(iv) first. PEMDAS says to get from left to right.
—"What virtually implied multiplication?"
Most calculators care for it the same way every bit regular multiplication. Grouped terms are typically grouped with parentheses if they are meant to be evaluated offset.
—"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 reply tin can be different from 180 degrees. But no ane 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 tin ask if 0 is a "positive" number. In America, the convention is that 0 is neither positive nor negative. Only in France 0 I am told 0 is considered to be positive. Y'all'd have to re-write a lot of math tests in America if you say that "positive" is not a well-divers word.
Ultimately we say things like "a triangles angles sum to 180 degrees, co-ordinate to the axioms of plane geometry," and "0 is not positive, according to the definition in America." Similarly we can say "8÷ii(2+ii) = sixteen, according to the modern estimation 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 primary crusade of confusion is the order of operations!
For instance, consider the trouble 9 – 3 ÷ (one/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 become the correct answer, down from a charge per unit of ninety percent in the 1980s. It is clear the problem is students do not learn the gild of operations.
Mathematicians who say "the answer is ambiguous" overlook that students get unambiguous expressions wrong at an alarming rate. Information technology is our duty every bit mathematicians to emphasize the lodge of operations in its modern grade so that we can write proper expressions and translate them correctly. Non a single person who disagrees with me has considered why students get the wrong answer to 9 – 3 ÷ (one/3) + ane.
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 accept written:
ixa 2÷iiia
= iiia
This indicates that the divisor is the entire product on the right of the symbol. In other words, the trouble is evaluated:
9a 2÷threea
= 9a 2÷(3a)
(Of import: this is outdated usage!)
I doubtable the custom was out of practical considerations. The in-line expression would have been easier to typeset, and information technology takes up less space compared to writing a fraction every bit a numerator over a denominator:
The in-line expression also omits the parentheses of the divisor. This is similar how trigonometry books commonly write sin 2θ to hateful sin (2θ) considering the statement of the part is understood, and writing parentheses every time would be cumbersome.
However, that practice of the division symbol was confusing, and it went confronting the lodge of operations. It was something of a well-accepted exception to the rule.
Today this do is discouraged, and I have never seen a mathematician write an ambiguous expression using the partitioning symbol. Textbooks ever have proper parentheses, or they explain what is to be divided. Because mathematical typesetting is much easier today, we about never meet ÷ as a symbol, and instead fractions are written with the numerator vertically above the denominator.
*Note: I go many, many emails arguing with me about these social club of operations bug, and most of the time people have misunderstood my indicate, not read the post fully, or not read the sources. If you send an electronic mail on this problem, I may non take fourth dimension to answer.
Sources
0. Google evaluation
https://www.google.com/#q=8÷two(ii%2B2)
1. Web archive of Matthew Compher's Arguing Semantics: the obelus, or division symbol: ÷
two. In 2013, Slate explained this trouble and provided a bit virtually 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
three. The historical usage of ÷ is documented the post-obit 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 tin 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. Spider web. http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents
four. In Plus magazine, David Linkletter writes a differing perspective that the problem is not well-defined (and see his longer article as well). 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 non address why students incorrectly reply the unambiguous problem 9 – iii ÷ (1/three) + 1.
PEMDAS Paradox (Plus magazine)
PEMDAS Paradox (longer commodity)
5. Harvard mathematician Oliver Knill besides has a differing perspective that the just incorrect answer is saying there is a single correct respond. I strongly disagree and the article does not address why students incorrectly answer the unambiguous problem ix – three ÷ (ane/three) + ane.
Ambiguous PEMDAS
vi. I have also read many manufactures from people who disagree with me and allude to my video, just then they do non 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 web log post.
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Source: https://mindyourdecisions.com/blog/2019/07/31/what-is-8-%C3%B7-22-2-the-correct-answer-explained/
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