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You are watching: What is the answer to 2

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

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

I likewise have covered similar problems before, including the adhering to videos:

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

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

Since over there is an additional problem that’s walking viral ideal now, it’s time because that the order of work to conserve the day!

What is the exactly answer come the following expression?

8÷2(2 + 2) =

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

Watch the video where I describe the correct answer.

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

Or store reading. . ."All will certainly be well if you usage your mental for her decisions, and also mind just your decisions." because 2007, i have specialized my life to share the happiness of game theory and mathematics. rebab.net now has actually over 1,000 cost-free articles with no ads many thanks to ar support! assist out and also 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 write-ups are transcribed easily after i make the videos because that them–please permit me understand if over there are any kind of typos/errors and also I will certainly correct them, thanks).

The correct answer is 16 follow to the contemporary interpretation of the order of operations.

The stimulate of operations

The expression can be simplified by the bespeak of operations, regularly remembered through the abbreviation PEMDAS/BODMAS.

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

Everyone is in agreement around the an initial step: leveling the enhancement inside the parentheses.

8÷2(2 + 2) = 8÷2(4)

This is where the conflict starts.

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

8÷2(4) = 8÷2×4

According come the order of operations, department and multiplication have the same precedence, so the exactly order is to evaluate from left to right. An initial take 8 and also divide that by 2, and also then main point by 2.

8÷2×4 = 4×4 = 16

This gets to the correct answer the 16.

This is without discussion the correct answer of how to advice this expression according to existing usage.

Some civilization have a different interpretation. And while it’s not the exactly answer today, the would have been pertained to as the correct answer 100 year ago. Some civilization may have learned this other interpretation much more recently too, but this is not the means calculators would evaluate the expression today.

The other result of 1

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

Historically the symbol ÷ was provided to median you have to divide by the whole product on the best of the symbol (see longer explanation below).

Under the 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. An initial we evaluate the multiplication within the parentheses. So we multiply 2 through 4 to acquire 8. And then we division 8 by 8.

8÷(2(4)) = 8÷8 = 1

This offers the result of 1. This is not the exactly answer the calculators will evaluate; fairly it is what someone can have construed the expression follow to older usage.

Binary expression trees

Since some people think the answer is 16, and also others think it is 1, many world argue this problem is ambiguous: that is a poorly created expression v no single correct answer.

But here’s my counter-point: a calculator is no going to say “it’s an faint expression.” simply as courts rule about ambiguous legitimate sentences, calculators evaluate seemingly ambiguous numerical expressions. Therefore if us take the expression as written, what would a calculator evaluate it as?

There room two feasible binary expression trees. The in-line expression likewise omits the parentheses of the divisor. This is like exactly how trigonometry books commonly create sin 2θ to average sin (2θ) because the debate of the function is understood, and writing parentheses every time would certainly be cumbersome.

However, that exercise of the department symbol to be confusing, and also it went against the bespeak of operations. It to be something that a well-accepted exemption to the rule.

Today this exercise is discouraged, and I have never viewed a mathematician create an pass out expression making use of the department symbol. Textbooks constantly have suitable parentheses, or they define what is to be divided. Since mathematical typesetting is much simpler today, we almost never check out ÷ as a symbol, and also instead fractions room written through the molecule vertically above the denominator.

*Note: I acquire many, countless emails suggesting with me about these stimulate of to work problems, and also most the the time human being have misunderstood mine point, not check out the article fully, or not check out the sources. If girlfriend send an e-mail on this problem, I may not have time to reply.

Sources

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

2. In 2013, Slate defined this difficulty and listed a bit around the background of the division symbol.

3. The historical usage of ÷ is recorded the complying with journal post from 1917. Notification the author points the end this to be an “exception” come the stimulate of operations i beg your pardon did cause confusion. With modern typesetting we can avoid confusing expression altogether.

Lennes, N. J. “Discussions: Relating to the bespeak of to 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 plus magazine, David Linkletter writes a different perspective the the trouble is no well-defined (and see his longer article too). I perform not agree through the portrayal that what “mathematicians” say, as plenty of mathematicians are happy for the write-ups I have written. The article additionally does not resolve why students wrongly answer the unambiguous difficulty 9 – 3 ÷ (1/3) + 1.