News Brief by Kip Hansen
There are two recent stories in the NY Times that bring up a curious seemingly inconsequential oddity of mathematical computing. They are both written by Steven Strogatz — in time order they were: “The Math Equation That Tried to Stump the Internet” and then, two days later, “That Vexing Math Equation? Here’s an Addition”. Steven Strogatz is a professor of mathematics at Cornell and the author of “Infinite Powers: How Calculus Reveals the Secrets of the Universe.”
So what’s this all about? A Tweet — that’s right — a Tweet on what Strogatz calls “Mathematical Twitter”. The tweet was this:
oomfies solve this
— em ♥︎ (@pjmdolI) July 28, 2019
That’s easy! The correct answer is:
Yes, that’s right, the correct answer is either 16 or 1, depending on an interesting point of mathematics. The featured image gives us some insight into what’s going on here. Strogatz explains it this way:
“The question above has a clear and definite answer, provided we all agree to play by the same rules governing “the order of operations.” When, as in this case, we are faced with several mathematical operations to perform — to evaluate expressions in parentheses, carry out multiplications or divisions, or do additions or subtractions — the order in which we do them can make a huge difference.”
When we resort to our handy electronic scientific calculators, we find that my answer is absolutely right!
(This image was supplied by a twitter participant…see the twitter thread).
The Texas Instruments TI-84Plus C returns an answer of “16” while our Casio fx-115MS returns “1”.
A quick survey of online scientific calculators returns mixed results as well:
And maybe a bit more accurate:
Math guys and gals know that the problem is order of operations and there are conventions for which operations come first, second, third and so on. In high school we learn the convention as one of the following (depending on where you went to school):
BODMAS is an acronym and it stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. In certain regions, PEDMAS (Parentheses, Exponents, Division, Multiplication, Addition and Subtraction) is the synonym of BODMAS.
PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. Given two or more operations in a single expression, the order of the letters in PEMDAS tells you what to calculate first, second, third and so on, until the calculation is complete. If there are grouping symbols in the expression, PEMDAS tells you to calculate within the grouping symbols first.
Strogatz says: “Strict adherence to this elementary PEMDAS convention, I argued, leads to only one answer: 16.” Ah, but his editor ( and a slew of readers ) “…strenuously insisted the right answer was 1.”
To get Strogatz’s “16” one has to do this: 8/2 = 4 then do 4 x (2+2) or 4 x 4 = 16.
How to get “1” is explained in this quote from Strogatz:
“What was going on? After reading through the many comments on the article, I realized most of these respondents were using a different (and more sophisticated) convention than the elementary PEMDAS convention I had described in the article.
In this more sophisticated convention, which is often used in algebra, implicit multiplication is given higher priority than explicit multiplication or explicit division, in which those operations are written explicitly with symbols like × * / or ÷. Under this more sophisticated convention, the implicit multiplication in 2(2 + 2) is given higher priority than the explicit division in 8÷2(2 + 2). In other words, 2(2+2) should be evaluated first. Doing so yields 8÷2(2 + 2) = 8÷8 = 1. By the same rule, many commenters argued that the expression 8÷2(4) was not synonymous with 8÷2×4, because the parentheses demanded immediate resolution, thus giving 8÷8 = 1 again.”
So, if everyone followed exactly the same conventions, both when writing equations and in solving them, all would be well and we’d all get the answer we expected.
But…..
This [more sophisticated] convention is very reasonable, and I agree that the answer is 1 if we adhere to it. But it is not universally adopted. The calculators built into Google and WolframAlpha use the more elementary convention; they make no distinction between implicit and explicit multiplication when instructed to evaluate simple arithmetic expressions.
Moreover, after Google and WolframAlpha evaluate whatever is inside a set of parentheses, they effectively delete the parentheses and no longer prioritize the contents. In particular, they interpret 8÷2(2 + 2) as 8÷2×(2 + 2) = 8÷2×(4), and treat this synonymously with 8÷2×4. Then, according to elementary PEMDAS, the division and multiplication have equal priority, so we work from left to right and obtain 8÷2×4 = 4×4 and arrive at an answer of 16. For my article, I chose to focus on this simpler convention.
Our dear mathematician concludes:
“Likewise, it’s essential that everyone writing software for computers, spreadsheets and calculators knows the rules for the order of operations and follows them.”
But I have already shown that writers software do not all follow the same conventions….Strogatz points out that even sophisticated software like WolframAlpha and Google’s built-in calculator in GoogleSearch don’t follow the sophisticated rules and get “16”.
The final statement by Strogatz is: “Some spreadsheets and software systems flatly refuse to answer the question — they balk at its garbled structure. That’s my instinct, too, and that of most mathematicians I’ve spoken with. If you want a clearer answer, ask a clearer question.”
Update Before Publication: The NY Times’ Kenneth Change waded into the fray in today’s (Aug 7) Science section with “Essay: Why Mathematicians Hate That Viral Equation“.
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I hope that you have found this essay either instructive or amusing. The real basic on this issue is that original problem written as “8 ÷ 2(2+2)” is intentionally badly formed so as to be ambiguous.
It does bring up a very serious question: If simple mathematical equations can be interpreted and solved to different answers, depending on the order of operations and given that even serious mathematical software differs in conventions followed, what of very sophisticated mathematical models, in which variables are all inter-dependent and must be solved iteratively?
In CliSci, do we get different projected future climates if one changes the order of calculation? I mean this not in the simple sense of the viral twitty equation, but in a much more serious sense: Should a climate model, a General Circulation Model, first solve for temperature? Or air pressure? Or first consider incoming radiation? Here’s the IPCC diagram:
I attempted to count up the number of variables acknowledged on this simplified diagram, getting to a couple of dozen before realizing that it was too simplified to give a real count. Each variable affects at least some of the other variables in real time. Where does the model start each iteration? Does it matter which variable it starts with? Does the order of solving the simplified versions of the non-linear equations make a difference in the outcomes?
It really must — I would think.
Do all of the western world’s GCMs use the same order? What about the mostly independent Russian models (INM-CM4 and 5)? Do the Russian models produce more realistic results because they use a different order of operations? Do they calculate in a different order?
I certainly don’t know — but it is a terrific question!
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Author’s Comment Policy:
There is always yet another really great question to be asked. Don’t ask me the one above, I don’t know the answer but I’d love to read your ideas. If you are involved in a deep way with GCMs, please try to give us all a better understanding of the order of operations/order of calculation issue.
Start you comment with “Kip…” if you’re speaking to me. I do read every comment that you post under any essay I write. I try to reply when appropriate and try to answer questions when I can.
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Kip —
May not be correctly computed. (RPN: 2¦2+2*8/ ==>1)
Not all computer languages do strict left-to-right sums (or products); if there is a desired order of operations, they must be indicated by the programer.
a+b+c may be evaluated as
(a+b)+c
a+(b+c)
(a+c)+b
(b+a)+c
…
Changing the order of evaluation of floating point chain sums or chained product operations can produce different results. [sum is add or subtract, product is multiply or divide.]
(There has been at least one compiler that shuffled the order of evaluation of (say) a+b+c from one compilation to the next in an attempt to catch such errors; it left notes saying which order was used.)
Errors of that type are usually due to rounding. Consider 1,000,000,000+2,000,000,000+0.000000001+0.000000002+0.000000002+0.0000000002… repeated 10,000,000 times. If the first two terms are added first, and then the rest, you need very high precision arithmetic to get the right answer: without the answer is 3,000,000,000. If the 10,000,000 0.000000002 terms are added first the error will likely be the 0.000000001 term in the final sum.
I think Kip Hansen deserves 16 lashes for destroying my morning.
Ric ==> Please, blame em@pjmdoll — the originator of the Twit….
someone mentioned above the spaces, and looking at that image yes there are spaces between the 8 the division symbol and the 2
8 space / space 2(2+2)
to me the answer will always be 1. like i stated above, i was taught multiplication first and to me the 2(2+2) is one term of the equation.
I read the linked PDF a little, and I noticed they mentioned multidecadal oscillations. I think other climate models ignore multidecadal oscillations while hindcasting the past, and are tuned to hindcast the past (especially 1975-2005) without consideration of multidecadal oscillations, and so they model excessively positive feedbacks to hindcast the warming from 1975-2005.
I think the O stands for “order”, not “Of”.
I saw the reference you used when I Binged BODMAS definition, but more authoritative sources and my innate sense of “Huh?” indicate that is a errant source.
Kip..
Of course the right order is pivotal. Climate models are derived from the understanding of GHGs, the role of GHGs is derived from the GHE, and the GHE is based on assumptions. Checking these assumptions will be the starting point for anybody interested in “climate change”.
1. Assumption: Surface emissivity of Earth = 1.
This is funny since no real surface has a perfect emissivity. However, with most of Earth covered by water, we should know its hemispheric emissivity. It turns out to be rather 0.91 than 1. We can get this from text books on physics, or try to calculate it ourselves with Fresnel equations knowing the refractive index of water in the range of terrestial LWIR (N2 = 1.27).
This makes a huge difference, since it cuts away about 35W/m2 of an assumed GHE of 150W/m2.
2. Assumption: Clouds are cooling Earth
This assumption comes in 2 flavors.
a) Clouds would reflect about 70W/m2 of solar radiation, but do nothing to hold back LWIR emissions, so that all the original 150W/m2 can be attributed to GHGs.
b) Clouds would reflect only about 44W/m2 of solar radiation and reduce LWIR emissions by about 30W/m2. So at least they would have a share in the GHE thus reducing the role of GHGs to at least 120W/m2. However even this “reduction” of the GHG induced GHE would usually be ignored..
Either way, whatever part of reality threatens the mantra of a 150W/m2 or 33K GHE is getting denied. Given this strange and inconsistent behaviour, one should take a serious look on what clouds actually do to climate. And what we see is jaw-dropping (if you understand the issue..) at least.
Not just clouds are NOT negatively correlated to temperatures, allowing for rain chill bias, we even see a strong positive correlation. In other words, clouds are warming Earth rather than cooling it. Here a simple illustration..
So if clouds realistically reflect about 70W/m2 of solar radiation, and their overall effect is definitely positive, their contribution to an assumed “GHE” will be >70W/m2, maybe like 90W/m2. This would leave us with a GHG induced GHE of 150 – 35 – 90 = 25W/m2, or about 5K. Which is certainly a bit less than a 33K.
Leitwolf ==> But why start there? why start with GHE? Why not start with Air Pressure? Surface Water Temperature and its effect on Atmospheric Water Vapor?
???
Leitwolf ==> GCMs have variable that are interdependent — air pressure affects temperature, temperature affect air pressure, air and sea surface temperature affect atmospheric water vapor — the results of the first calculation cause changes in the subsequent calculations. reverse the order of calculation and the effects are different….
Poorly written expression. If you want the answer 1, use brackets to clarify:
8/[2(2 +2)] = 8/8 = 1
…if you want the answer 16, don’t use brackets:
8/2(2+2) = 8/2 *4 = 4 * 4 = 16
If you enter “8/2(2+2)” into wolframalpha it then replies with “(8/2)(2+2)” which by order of operation is actually 4(2+2) which becomes 16, but it clearly changed the problem to do so and it’s response of what it is solving is NOT the same as the fraction of 8 over the denominator of 2(2+2).
Remember that both / and ÷ mean [enumerator]÷[denominator]
https://www.wolframalpha.com/input/?i=8%2F(2(2%2B2))
See, knowing how to correctly *program* a computer will get you a correct answer.
“8/2(2+2) = 8/2 *4 = 4 * 4 = 16” is wrong
8/2(2+2) = 8/2(4) = 8/8 = 1
You must finish processing the parenthesis before proceeding.
8/2(2+2)=x solve for x
To isolate the 8 you multiply both sides by (2(2+2))
which gives 8= x(2(2+2)),
therefore x=1
Isn’t it fun to manipulate numbers?
This idea of ambiguity won’t fly with me. There are rules defined. Follow the rules. There isn’t some special rule for some higher sophisticated convention. That’s a politician’s excuse. This is arithmetic. There are rules. Follow them.
Also this isn’t about computers or mathematicians. Again, there are rules. Follow them.
Order of operations is well defined. Multiplication and division have equal priority. If no groupings by parentheses or brackets, go from left to right. That’s the rule. Follow it.
8/2(2+2) = 8/2(4) = 4(4) = 16.
coaldust: “8/2(2+2) = 8/2(4) = 4(4) = 16.
Why do you leave out the spaces that exist in the original expression?
It appears to me that the issue here is whether you solve this equation manually using arithmetic/algebra rules or whether you try to solve it by programming a computer.
The spaces in the original expression *DO* have meaning in the expression. When you eliminate them in order to simply programming a computer then you actually lose part of the information included in the original expression.
No programming I have ever done recognizes spaces in an expression, the first thing they do is eliminate all spaces. Nor do they recognize manual algebra expressions like 2(2). Now admittedly I have not used all programming languages but I’ve used a lot.
When you see something like 8 + 2(2+2) it should be immediately recognized that this expression must be evaluated using manual methods and not programming rules. In doing this 2(2+2) is a separate part of the expression from the 8. Doing this gives an answer of 1.
If you are going to convert this into an expression to be solved by a computer then first you have to replace the spaces with parenthesis or brackets, you can’t just ignore the spaces!
The rule is spaces don’t matter. Follow the rules.
p.s.
Unless it is a large space separating expressions, then you might have:
8/2 (2+2)
=4 4
two expressions -> two numbers.
But 8 / 2(2+2) != 8/8. The spaces don’t matter. Follow the rules.
Spaces *do* matter. They have always mattered in written expressions. Spaces are only ignored by program compilers such as for the C language.
coaldust ==> The whole point of the essay, and Strogatz’ two NY Times articles is that there are rules — more than one set — and not everyone follows the same rules. The essay shows two scientific calculators giving differing results, and three online scientific calculators each giving different results (one returns “error”).
“The rules I learned in school” are not necessarily the same as a greater truth.
I agree. There have to be rules, and the rules have to be known, and then followed. And if there are different rules then you need to specify what rules you are following.
However, there really aren’t different rules for arithmetic. They are well defined and have been for a long time. It does appear that some people don’t know the rules.
And calculators? So some might do it wrong? Or maybe you have to follow the calculator’s programming instructions or else it says “error”. So what? This isn’t a calculator or machine problem. It’s an arithmetic problem.
Kip ==> While what you say has some truth to it, I don’t think it’s that big an issue in practice. I think the best reference on how to screw up or not screw up your digital math is probably Richard Hamming’s 1973 treatise “Numerical Methods for Scientists and Engineers”. I recall it as being surprisingly readable considering the subject matter. I looked for a free download to link to, but couldn’t find one.
As I recall, the biggie isn’t order of operations per se. It’s math that inadvertently subtracts two large similar numbers. The problem is that if you subtract say 2099999.0 from 2100002.0, your answer of 3.0 probably has about six fewer significant digits than you intended. Trouble is that it’s very hard to see you’ve done this unless you are “lucky” enough to generate a truly absurd answer (and have enough sense to realize that absurd is bad).
Don ==> In simpler maths, you are quite right. In GCMs, the instantaneous result of one calculation is the input for another, at the same time step — so order of calculation (not order of operations, which is a different fish) might wellbe very important.
“8/2(2+2) = 8/2(4) = 4(4) = 16” is wrong.
You have to finish with the parenthesis before you proceed.
8/2(2+2) = 8/2(4) = 8/8 = 1
This ambiguity seems an elementary matter of specifying parentheses, which accordingly determine order. For example, where 8 / (2x(2+2)), the answer is indubitably 8 / (2×4) = 1; writing 8/2 x (2×2) returns 4 x 4 = 16. Rather than waffle-and-fuss about some subjective “order of operations” [anathema in math], best simply remove the ambiguity by adequate notation.
“” *Strogatz says: “Strict adherence to this elementary PEMDAS convention, I argued, leads to only one answer: 16.”* “”
PEMDAS says to do the goddamned parenthesis FIRST and will always provide an answer of 1 on the subject.
Incompetence. When you use a computational language format you MUST follow that computational language format’s rules. Using parenthesis as a (multiplythisvalue) requires that PEMDAS be followed.
You want a REAL laugh about incompetence try comparing https://www.wolframalpha.com/input/?i=1%2F1.000.000 , https://www.wolframalpha.com/input/?i=1%2F1,000,000 , and https://www.wolframalpha.com/input/?i=1%2F1.000 then get back to me on whether you think most of the world are morons who fall for advertisers who use . to separate phone numbers because they’re too incompetent to properly format a spreadsheet.
mike TECHNICALLY your 8/2*4 is not correct
it is Actually 8/2(4) just because you do the math inside the paranthesis…does not mean you move said parenthesis and do 8/2*4 if you do that you CAN get 16 or 1 depending if you do division first or multiplication first.
if you leave the parenthesis there, you have to solve the term 2(4) before you can do the division.
answer is 1 🙂 i can see how you get 16 but to me the term 2(2+2) always has to be solved completely first before doing the division.
Whether true or not, my brother who holds a degree in Pure Mathematics from Cal explained to me that there really is no division or subtraction per se, that was just shorthand for multiplying by the inverse, or adding a negative number. Which is what is actually happening. So 8 / 2(2+2) is really 8 * 1/2(2+2)—> 8 * 1/4+4 —>8 * 1/8=1. By some logic you could do 8/ 2(2+2) –> 8/ 4+4–> 2+4=6
My answer to the correct order of mathematical operations is not based on math expertise , but rather on the basis that math is a language.
To apply a mathematical operation to two ‘things’, the ‘things’ must be in equivalent language.
8 and 4 are both equivalent language, so 8 divided by 4 can be operated on to produce 2.
8 and 2(2+2) are not equivalent language, so 8 cannot be operated on by division until 2(2+2) is put into the same language as 8.
So let’s rewrite the “8” into another form such as “2(2+2)”
We then have
2(2+2)/2(2+2)
which of course is 1
Occam’s Razor says not to make things more complicated than they need to be, so “2(2+2)” is just a complicated unnecessary way of expressing “8”. Therefore the answer is “1”.
Tom
+1
One should simplify an expression before completely evaluating it!
Clyde:
-1
The only simplification to be performed is evaluating the parenthetical expression, that’s it. Then you go back to the beginning. No soup for you.
That is correct, but it has nothing to do with Occam Razor, it is the way it is supposed to be done.
Like others, you have grouped those numbers without justification. You evaluate the whole string by the rules, not little pieces. 2(2+2) is not a sub-part to be separately evaluated.
D.J.
-1
“Like others, you have grouped those numbers without justification. You evaluate the whole string by the rules, not little pieces. 2(2+2) is not a sub-part to be separately evaluated.”
Of course it is a sub-part to be separately evaluated. That is why is separated by a space in the expression. You are, like so many others, trying to evaluate the expression as if it were written to be compiled by a computer language compiler.
While you probably don’t realize it this is probably giving away your age. Those of us who grew up on analog computers evaluate written expressions as just that, not as entries in a computer program.
The answer must be one. BODMAS has the answer.
Brackets (2×2) =4
Multiplication 2x(2×2) = 8
Division 8/8 = 1
Wrong Mike. The rule says operations go LEFT to RIGHT. You clearly violated the rule by going RIGHT to LEFT after you did the parenthetical operation.
Kip
“Start you (your)? comment with “Kip…” if you’re speaking to me. ?
First. Simplify the denominator.
As a programmer, I see the problem as simply “it depends on the rules of evaluation.” In the programming languages I’ve used, the expression would be flagged as a syntax error. None of them used implicit multiplication. In the absence of rules, an expression is meaningless. In this case somebody dropped an expression on the internet with no rules of evaluation attached.
Much of the trouble here is due to people being invested in the rules they’re familiar with colliding with people who have different assumptions.
If I propose that 2+2=11, some of you might think I’m math challenged. At least until I mention that this is correct in base 3. Most people assume base 10 if not otherwise specified.
The moral to the story is that if something is ambiguous or seems wrong, it simply needs further inspection to see what is going on and what, if anything, needs to be clarified.
“In the programming languages I’ve used, the expression would be flagged as a syntax error. None of them used implicit multiplication. ”
Where was it stated that this was a line of program code in the original example? There was no assignment operator shown at all.
Is the expression mx+b a syntax error as well? It isn’t in any math book I have.
Tim Gorman
+1
I never thought I would see the day when 8/2(2+2) = 1 could be written and many people agreeing!!!!
The rules of BODMAS tell us the order to perform the operations, in case there are order sensitive steps.
It is an accepted fact that 2() means 2 times the contents of the bracket.
It is nicer to see in the examples in this thread to have 2 times (2 plus 2) but if the times sign is missing, you mentally insert it and carry on.
Look at this example from the premier UK text book on engineering maths used in colleges and uni for the first two year of maths.:
https://jpmccarthymaths.files.wordpress.com/2012/09/john_bird_engineering_mathematics_0750685557.pdf
This is an old copy..
Have a look at page 43 problem 26.
Good god! we have [3{2(4a what to do!! also 5( what does that mean?
If there is a number butted up to a lhs bracket insert a multiplication sign and follow BODMAS.
BODMAS and the laws of precedence start on page 44. If you want to get closer to the example in this thread then see problem 40 on page 45 where we see the dreaded divide sign!!!
For clarity, Bird uses ÷ instead of a / for division.
Note that the ÷ only affects the 4a term.
If you want to force someone or a computer program to calculate a result using a formula you have created, you must use brackets to force the order you want.
Any formula you wish to input into a calculator or computer program must be tested by hand, on paper and the result compared with the computer/calculator output. If they do not agree the computing device did not obey the rules.
It was always compulsory at the first maths lesson at college to get the students to enter a calculation from the white board. A number of answers were given by the students. Once the random answers were removed, it always turned out that those with incorrect answers had a non standard calculator that required lots of extra brackets to be inserted for every calculation if you wanted the correct answer.
So the answer is 16 because 8/2(2+2) = 8/2*(2+2) and either form is okay for use on paper and calculators if you have a good calculator like mine Casio fx-83ES.
If the standard states that 2*( and 2( are the same, who are we to argue?
Steve ==> Read the two Strogatz pieces and see if you can figure out why he took one position then switched to the other.
I forgot to add that some say spaces are important in maths. Not that I am aware. Each number or term 3 or 3c is separated followed by an operator such as *+-/^() etc.
You can remove all of the spaces from the Bird maths book and the maths and answers will not change.
x = 8 ÷ 2(2 + 2)
x = 8 ÷ (2 * 2) + (2 * 2)
x = 8 ÷ 4 + 4
x = 2 + 4
x = 6
😀
(Just to stir the pot…)
Jonas ==> Cute…that.
x = 8 ÷ 2(2 + 2)
x = 8 ÷ ((2 * 2) + (2 * 2))
x = 8 ÷ (4 + 4)
x = 8 ÷ 8
x = 1
😀
Fixed it.
As my son says to people he codes with, “quit being a whiny little bitch and use more parentheses”.
Put the expression in C code and compile and run it. That will give definitive answer, otherwise the code would not be portable to different compilers which would cause so many problems someone would have noticed the problem already.
Stevek ==> Oh, they noticed it alright — thus this whole kerfuffle.
I doubt very much the C code is the arbiter of all things mathematical.
A) It won’t compile, C and most other programming languages don’t have the concept of multiplication by juxtaposition, and that’s the heart of much of the controversy.
B) Adding a C multiplication operator makes it legal. Multiplication and division have the same precedence and are computed left-right.
C) It’s not the same problem as it’s no longer ambiguous.
Ok,
simply put, the equation in question, in proper sterile math approach seems to offer two outcomes,
both considered same in validity.
Where the actual outcome happens to be either 1 or 16 or both,
not actually good, as there has to be only one proper mathematically valid outcome or answer.
But trying a add logic in that application may just sort the problem.
So, moving it from;
“8/2(2+2)=?”
to the logical approach;
8/2(2+2)=X then a logical separation could be used and employed as per the given condition, giving out the:
X/8/2=(2+2) or X/(2+2)=8/2, where in both cases:
X/4=4 which points out that X=4*4=16 happens to be the only favored and possibly the correct answer from this angle… no way you get a 1 as an answer from this approach.
Eh, well, all depends in consideration of the value and acceptance of the approach…
Standing to be corrected. 🙂
(sorry, have not read or checked all of the comments, so if same shown in any earlier comment there, I do apologize.)
cheers
Whiten ==> Skip the comments,first read Strogatz’ two piece in the NY Times (links in essay)…then check back with me.
Checking back….!
Sorry, but that approach is just wrong.
8/2(2+2) = x written visually is 8
—— = x
2(2+2)
which becomes 8
——- = x
2(4)
which becomes 8
——- = x
8
which becomes 1 = x
John Dilks
August 8, 2019 at 9:14 pm
John for your understanding, and any one else there interested.
The method applied in my comment that you replied to, as far as I can tell is a very valid mathematical and algebra one.
Unless you can prove it otherwise it stands as a very correct validation method in the consideration of the given equation.
It actually validates the result as only the 16.
Falsifies the result of 1.
There is no way out of this, as far as I can tell, unless the approach of separation offered is proven as invalid.
And there is only one place mathematically valid for separation there, unless one seriously considers breaking the basic mathematical rules of precedence there, with no much regard.
That approach validates the 16 as the only proper result of the:
8/2(2+2)=?
and clearly falsifies the 1…. regardless of all else.
Ambiguity, interpolation, innuendos, in terminology or otherwise do not change much there, or hold any value.
The only thing that matters is whether the method is correct-valid or not.
My attempt there was to offer a validation in that case.
It seems to me that that method quite immune to ambiguity and interpolation.
You can spoon with it at your heart’s content, but unless you can prove that the method not mathematically correct or valid, then I think you got to accept its result… and there happens to be only one valid correct result, in accordance.
thanks
cheers