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You are watching: What's 8 * 8
Posted July 31, 2019 By Presh Talwalkar. Read around me, or email me.
As many world have actually opinions on this trouble, I want to share a bit around myself. I run the chrischona2015.org channel on YouTube, which has over 1.5 million subscribers and 245 million views. I studied Economics and also Mathematics at Stanford University, and my work-related has actually received coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many type of various other famous outallows.
I also have spanned comparable troubles prior to, consisting of the complying with 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 tright here is an additional difficulty that’s going viral appropriate now, it’s time for the order of operations to conserve the day!
What is the correct answer to the adhering to expression?
8÷2(2 + 2) =
(Note: some people write 8/2(2 + 2) = but this has actually the very same answer.)
Watch the video where I define the correct answer.
What is 8÷2(2 + 2) = ? The Correct Answer Explained
Or save reading. . ."All will certainly be well if you usage your mind for your decisions, and also mind only your decisions." Due to the fact that 2007, I have actually devoted my life to sharing the joy of game concept and math. chrischona2015.org now has over 1,000 cost-free write-ups via no ads many thanks to community support! Assistance out and also obtain early accessibility 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 write-ups are transcribed easily after I make the videos for them–please let me recognize if there are any type of typos/errors and also I will certainly correct them, thanks).
The correct answer is 16 according to the modern-day 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 also lastly evaluate Addition-Subtractivity.
Everyone is in agreement around the first step: simplify the addition inside the parentheses.
8÷2(2 + 2) = 8÷2(4)
This is wright here the dispute starts.
The answer is 16
If you kind 8÷2(4) into a calculator, the input hregarding be parsed and then computed. Many calculators will certainly transform the parentheses into an implied multiplication, so we get
8÷2(4) = 8÷2×4
According to the order of operations, division and also multiplication have the very same precedence, so the correct order is to evaluate from left to best. First take 8 and also divide it by 2, and also then multiply by 2.
8÷2×4 = 4×4 = 16
This gets to the correct answer of 16.
This is without discussion the correct answer of how to evaluate this expression according to current intake.
Some people have a various interpretation. And while it’s not the correct answer today, it would certainly have actually been regarded as the correct answer 100 years earlier. Some human being might have actually learned this other interpretation even more recently as well, however this is not the means calculators would evaluate the expression now.
The various other result of 1
Suppose it was 1917 and also you experienced 8÷2(4) in a textbook. What would certainly you think the author was trying to write?
Historically the symbol ÷ was supplied to suppose you should divide by the entire product on the ideal of the symbol (check out longer explanation below).
Under that interpretation:
8÷2(4) = 8÷(2(4)) (Important: this is outdated usage!)
From this stage, the remainder of the calculation works by the order of operations. First we evaluate the multiplication inside the parentheses. So we multiply 2 by 4 to obtain 8. And then we divide 8 by 8.
8÷(2(4)) = 8÷8 = 1
This gives the outcome of 1. This is not the correct answer that calculators will certainly evaluate; quite it is what someone might have actually interpreted the expression according to older consumption.
Binary expression trees
Since some world think the answer is 16, and also others think it is 1, many people argue this trouble is ambiguous: it is a poorly composed expression via 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 ascendancy around ambiguous legal sentences, calculators evaluate seemingly ambiguous numerical expressions. So if we take the expression as composed, what would a calculator evaluate it as?
There are two feasible binary expression trees.
The in-line expression additionally omits the parentheses of the divisor. This is favor just how trigonomeattempt books typically create sin 2θ to mean sin (2θ) because the argument of the function is interpreted, and writing parentheses eincredibly time would certainly be cumbersome.
However before, that practice of the division symbol was confutilizing, and it went against the order of operations. It was something of a well-welcomed exemption to the rule.
Today this exercise is discouraged, and I have never viewed a mathematician write an ambiguous expression making use of the division symbol. Textbooks constantly have correct parentheses, or they define what is to be divided. Due to the fact that mathematical typesetting is much less complicated this day, we practically never before watch ÷ as a symbol, and rather fractions are created through the numerator vertically over the denominator.
*Note: I obtain many, many type of emails saying with me about these order of operations difficulties, and also the majority of of the time world have actually misconstrued my suggest, not review the write-up totally, or not read the resources. If you sfinish an e-mail on this trouble, I might not have time to reply.
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 trouble and provided a bit around the history of the department symbol.
3. The historic usage of ÷ is recorded the following journal short article from 1917. Notice the writer points out this was an “exception” to the order of operations which did cause confusion. With contemporary typesetting we deserve to protect against confusing expressions altogether.
Lennes, N. J. “Discussions: Relating to the Order of Operations in Algebra.” The Amerideserve to Mathematical Monthly 24.2 (1917): 93-95. Internet. http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents
4. In Plus magazine, David Linkletter writes a differing perspective that the trouble is not well-defined (and check out his much longer write-up too). I execute not agree with the portrayal of what “mathematicians” say, as many kind of mathematicians are happy for the write-ups I have actually written. The short article also does not address why students erroneously answer the unambiguous difficulty 9 – 3 ÷ (1/3) + 1.
PEMDAS Paradox (Plus magazine)
PEMDAS Paradox (much longer article)
5. Harvard mathematician Oliver Knill additionally has actually a differing perspective that the only wrong answer is saying tbelow is a solitary correct answer. I strongly disagree and the article does not deal with why students mistakenly answer the unambiguous difficulty 9 – 3 ÷ (1/3) + 1.
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6. I have actually additionally read many kind of write-ups from people who disagree through me and point to my video, however then they perform not connect to my work-related. Academic debates must be kind spirited and also fair minded. If you see an article, please let them understand to attach to my video or blog short article.