8÷2(2+2) = ? Mathematician Explains The Correct Answer

What is 8÷2(2+2) = ? Everyone is arguing about this problem, so let's figure out the correct answer! The problem involves the order of operations, historical math notation, and binary expression trees. Glad to see the world is passionate about mathematics! (Note: some people write 8/2(2 + 2) = but this has the same answer.)

0:00 Order of operations
1:12 Historical usage
2:11 Common questions
4:08 Binary trees

I studied Mathematics and Economics at Stanford. Press coverage of my work:

*I get many, many emails about this problem and am unable to reply to them.

Here is a 1917 article from "The American Mathematical Monthly" that explains the usage of the division symbol as an exception to the order of operations

Google evaluation

What is 6÷2(1+2) = ? The Correct Answer Explained

9 - 3 ÷ 1/3 + 1 = ? The Correct Answer (Viral Problem In Japan)

Plus Magazine David Linkletter suggests the problem is not well-defined

Harvard's Oliver Knill also says PEMDAS does not resolve the ambiguous problem

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[nolongerhooman]

1980: I bet there will be flying cars in the future
2020: *arguing over 16 and 1*

[starflame]

Either my memory from high school is collecting dust somewhere in my brain, or my teachers were using 100 year old textbooks.

[gcoffey]

I am sticking with 1, screw the new math

[mikekelly]

Any engineer or physicist (or mathematician for that matter) would rewrite the expression in unambiguous fractional format (much the same way as a calculator does) and come up with an answer of 1. Three of my own calclators give 1 as the answer. My phone, on the other hand gives 16 as the answer, which probably says something about using the correct tool for the job.

[bacul]

I got 1, probably cause my teachers drilled us "brackets first" endlessly, to the point where my brain interpreted that as "anything that involves a bracket takes always precedence".

[fl]

To ignore confusing situations, we should use more brackets to make it clear.

[OMDolton]

I studied mathematics from 2004-2017, and I would always solve this as 1. My calculator (CASIO fx-85GT PLUS) also does so.
I was taught something called "multiplication by juxtaposition", and that it takes precedence over division as a separate entity to explicit multiplication which has the same level of priority to division.

But realistically the equation itself is just written terribly. For any equation, a/b(c+d), I would interpret it as (a)/(b(c+d)).

[dennismood]

What happens when you rewrite the equation into a long division format or making it a fraction? In the long division format, the division sign would cause the equation to be separated as 2(2+2) ) 8. Following PEMDOS, it would be 2(2+3) = 2(4) = 8. 8)8 = 1. As a fraction, it would read as 8/2(2+2). Simplifying the denominator would be 2(2+2) = 2(4) = 8. The fraction would then be 8/8 =1.

[kateadams]

The bottom line: don't write ambiguous equations! Scientific publishers screen for this and will send it back to rewrite it more clearly (you don't get to use those division signs, for starters). You would have to write it 8/(2(2+2)) or (8/2)(2+2). There are writing conventions that enforce this.

[partyfish]

This is why I just put parentheses everywhere so that there is no ambiguity in solving.

[chakotaandrews]

I feel like a really easy way to fix this would be 8:2x(2+2)=16 and 8/2(2+2)=1 I graduated high school 2 years ago and every math class I had since 7th grade would have 1 as the correct answer.

[tangoangel]

I studied in Europe, graduated accounting, and was taught the answer is 1.

[DerekDominoes]

I must've actually gone to school 100 years ago.

[redbud]

I graduated in 1991, and I learned math in way that gives answer 1. Our math teacher always told us not to use a calculator to try to cheat. Now I think I understand why she said that.

[DavidCookeZ]

I know I'm late to this, but this is a regional difference in the treatmeant of the precedence of implicit multiplication. In North America there is no higher precedence, so that's the way expression trees for programming languages developed there work. In other parts of the world there is, and calculators for those markets will evaluate this expression differently, getting the answer 1, with modern ones introducing additional brackets (removing the ambiguity) to the displayed equations differently. This is why schools have lists of approved models of calculator that correspond to their region.

Being from the UK, I was taught the higher precedence of implied multiplation. Thus when you get to 8/2(4) you still haven't fully dealt with the brackets; they're still present! So the next step is 8/8.

[drinkchan]

When I input 8 / 2(2+2)
it automatically adds a parenthesis around 2(2+2), making it 8/(2(2+2)). Im using an fx-50FHII calc

[ihatebritain]

I saw a TikTok which said “please restore my faith in humanity” with this equation 51% got 1

[ProCoolBlue]

I’m 23 and was taught to solve this as 1, I won 2 awards for top math grades in high school too, I guess my life is a lie...

[ahmedhaaqil]

I am from Sri Lanka.

There was somewhat a vague memory in my younger years, in grade 3-7 where the "left to right" was used.
(Plus we use BODMAS.)

However, as soon as I reached Secondary grades. (Grade 7 to grade 13.)

This symbol is almost always emitted: *÷*

So this expression was first in my mind when written as 8/2(2+2) . In this way, the only reasonable answer is 1.

[johanfagerstromjarlenfors]

Our math books said that the number next to the parenthesis is a part of it (and if there is no number there it is the number 1).
And with that it also said that to solve the parenthesis you also make the multiplication with the number next to the parenthesis.

It can as simple as a poorly made sentence in the math books, or they are actually based of a different interpretation of the problem🤷‍♂️

But what you said in the video kinda proves the “this is not how maths were taught when I was in school” to some extent. I mean, what if the materials that were used actually were based on the older method even if it wasn’t 100 years ago? What if the texts were misinterpreted or poorly written? Information lost in translation etc…


(And the answer books to our math books had the answer that in “modern interpretation” say is wrong)