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“.
# # # # #
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.
# # # # #
Notice that they have to use a program like Wolfram to actually enter the equation as shown. It cannot be entered as it is in almost any language I know of and as several of you have shown above. Properly, it can be entered as 8 / 2 * (2 + 2).
If you want to get the value 1 then you need 8 / (2 * (2+2)).
All programming languages demonstrate that the equation shown can be considered ambiguous. However, once it is established explicitly there is no ambiguity. There will be only one answer.
Larry ==> if we only had to solve this one little ambiguous equation, we’d find a pretty easy answer.
The bigger question is: what if we change the order of calculation of GCMs?
Kip Hansen
August 8, 2019 at 1:01 pm
If the ratios in the base parametrization maintained , still when a GCM simulation “locks in” to the warming, there would not be a way to either
slowdown-stop or speedup-increase of either the warming or CO2 concentration, including the use of modification of the “”artificial knob’s “power””, I think.
Maybe I think wrong!
But that is how I see it so far.
According to above, the “lock in” considered to be established somewhere between 0.4C to 0.8C warming mark, with proper irreversible “lock in” at ~0.8C.
(the way I understand it, which does not necessary has to be considered as valid or correct, but in the case that it may “spice” a bit more the GCM consideration)
cheers
The Answer is 1.
Kip – When I was at school, many decades ago, I don’t think they had invented BODMAS or PEMDAS. We were simply taught that the priorities were 1. brackets, 2. exponents, 3. multiplication and division, 4. addition and subtraction. It might now be called BO(DM)(AS) or PE(MD)(AS) perhaps. The way it worked is that for example a – b + c – d + … is always read as a + (- b) + c + (- d) + … and a ÷ b * c ÷ d * … is similarly read as a * (1/b) * c * (1/d) * …
You will note that there is no need to distinguish between multiplication and implicit multiplication. Also that a string of multiplies and divides can be given and calculated in any order – thus for example 8 * 4 ÷ 2 ÷ 1 = 8 ÷ 1 ÷ 2 * 4 = 4 ÷ 2 ÷ 1 * 8 = 16. I haven’t checked carefully, but I think that BODMAS gives the same answer provided that the divisions are done left-to-right. Anyway, the way I describe is superior because it is so much easier to remember and to apply. PEMDAS seems just wrong.
Using this old-fashioned but superior (!) method, the correct answer to your challenge is:
8 ÷ 2(2+2) = 8 * (1/2) * (2+2) = 16
Having said all that, though, it is a really good idea to present any sequence of multiplications and divisions completely unambiguously.
“8 ÷ 2(2+2) = 8 * (1/2) * (2+2) = 16” is wrong.
8 ÷ 2(2+2) = 8 * 1/(2(2+2)) = 8 * 1/(2(4)) = 8 * 1/(8) = 8 * 1/8 = 8/8 = 1
Yes, I was wrong about implicit multiplication. It does have to be done first. But I think we can all agree that being unambiguous is best.
Mike:
If I gave you the expression
ʃ x2 / x(x + 1) dx
exactly how would you evaluate it manually? Is this ambiguous? It’s right out of my 1968 copy of “The Calculus with Analytic Geometry” by Leithold.
Or we could try my 1940 “Differential and Integral Calculus” by Middlemiss:
differentiate y = 4 / 3x-1
These were written long before computer programming tainted so many today for reading manual expressions and evaluating them.
1. Saying that one should write expressions unambiguously doesn’t mean that any particular expression is ambiguous.
2. Your example expression is ambiguous, but probably not in the way you meant. The expression reads
ʃ x*2 / (x*(x + 1)) dx
but I suspect you intended
ʃ (x exponent 2) / (x*(x + 1)) dx
3. y = 4 / 3x-1 is definitely ambiguous, because it can be very difficult in some fonts to see spaces. As shown on my iPhone screen I read it as very clearly saying y = (4/3x)-1, and it was only because you made it clear that there was a possibility of ambiguity that I then worked out where you had put the spaces.
4. I’m slightly bemused at the wording of your last comment – I had conceded that you were right.
5. No matter, I think this whole page shows that it is important to be unambiguous. The designers of the algorithms that found “syntax error” are the ones that got it right.
Mike;
Implicit multiplication does NOT have priority. No where is that part of any prior or current rule.
TL;DR
Answer is = 1.
Long answer.
There is no problem with how the mathematical equation is stated. The problem lies with several factors.
1) The problem is a numerator and denominator problem, as others have mention and alluded too. If one reads the equation and translates it into English. You will end up with this statement. Reading left to right we get “8 divided by 2 lots of (2 plus 2)”. The 8 is the numerator and the denominator is the “2 lots of (2 plus 2)”.
Which reads
8
————–
2(2+2)
Which will yield the following
8
—
2(4) (the denominator is then worked out fully)
8
—
8
Therefore the answer is 1.
2) What the BODMAS OR PEMDAS do not state is the implicit multiplication ie: 2(2 plus 2) as stated by Strogatz “implicit multiplication is given higher priority than explicit multiplication”, which needs to be evaluated first, that means the distributive law takes precedence. Yielding the value 8 and finally 8/8 giving the answer 1.
3) Mathematicians are lazy and they leave out certain symbols. For example 2(2 + 2) fully expanded is 2 x (2 + 2) or even 2.(2 + 2). So brackets are left out of the original equation and should be read as 8 / ( 2(2 + 2)), this is the explicit form of the equation. Which when worked out yields the answer 1.
So Wolfram and calculator manufactures need to implement the explicit form of an equation. That is the original question must be clearly defined.
Regards
Climate Heretic
>TL;DR?
I assume you aren’t expecting anyone to read the rest of your reply.
Bah, Num Bug.
Kip:
TL;DR
Answer is = 1.
Long answer.
There is no problem with how the mathematical equation is stated. The problem lies with several factors.
1) The problem is a numerator and denominator problem, as others have mention and alluded too. If one reads the equation and translates it into English. You will end up with this statement. Reading left to right we get “8 divided by 2 lots of (2 plus 2)”. The 8 is the numerator and the denominator is the “2 lots of (2 plus 2)”.
Which reads
8
————–
2(2+2)
Which will yield the following
8
—
2(4) (the denominator is then worked out fully)
8
—
8
Therefore the answer is 1.
2) What the BODMAS OR PEMDAS do not state is the implicit multiplication ie: 2(2 plus 2) as stated by Strogatz “implicit multiplication is given higher priority than explicit multiplication”, which needs to be evaluated first, that means the distributive law takes precedence. Yielding the value 8 and finally 8/8 giving the answer 1.
3) Mathematicians are lazy and they leave out certain symbols. For example 2(2 + 2) fully expanded is 2 x (2 + 2) or even 2.(2 + 2). So brackets are left out of the original equation and should be read as 8 / ( 2(2 + 2)), this is the explicit form of the equation. Which when worked out yields the answer 1.
So Wolfram and calculator manufactures need to implement the explicit form of an equation. That is the original question must be clearly defined.
Regards
Climate Heretic
Climate Heretic
August 8, 2019 at 3:02 pm
Heretic,
There is a logical reason why multiplication takes precedence in Algebra.
You have just clearly shown what happens when that rule openly broken,,,
a wrong non valid result.
In your try you clearly have broken that rule.
If the multiplication has precedence, then that is where the only valid place of separation could be in a given equation…something you have clearly infringed above.
So it should be like:
8/2
*
(2+2)
ha, vertical math seems easier….
cheers
“If the multiplication has precedence, then that is where the only valid place of separation could be in a given equation”
The spaces in the original expression specifically distinguish where the valid place of separation should be.
I do not know that algebra takes account of spaces.
Is that a proper “game”rule or a playing rule, how to play the game rule?
Sorry, simply missing the point you making there.
cheers
In written expressions spaces indicate groupings. Every calculus book I have writes the expression of a line as:
y = mx + b.
Not y=m*x+b
Implicit multiplication is part of every math book I have from basic algebra on up. The expression given is a *written* expression just like mx + b. It is not until programmers get involved that things get confusing. There is nothing in the example that indicates this expression is written in a programming language.
Tim Gorman
August 8, 2019 at 4:26 pm
Tim,
I still do not get your point.
These
y = mx + b.
y=m*x+b
both he same
In both, the multiplication rule means the same result, some like:
m+b=y/x or m=y/x-b,
and not:
m=(y-b)/x or m=y-b/x
It simply serves the purpose of guarding against such an error, I think.
It does not seem like spaces matter much or hold any real value, but the multiplication rule does, don’t you think?
cheers
I used the mx + b as an example of spacing defining groupings not as an example of precedence.
Again, if you read the 8 / 2(2+2) as it is written, the 2(2+2) forms its own grouping to be calculated on its own. If it had been written 8/2 (2+2) or 8 / 2 (2+2) then it would make no written sense since the implicit multiplication operator is lost. Actually no language compiler would be able to make sense of it either.
In computer-speak, if you will, spaces are ignored and the expression would be written 8/2*(2+2) and getting an answer of 16 would make sense. In written expressions, however, spaces (especially coupled with non-spaces) must not be ignored. In written form spaces carry information. Ignoring spaces causes that information to be lost.
Tim, you could save a lot of people a fair amount of time if you just posted a reference that supports your claim that spaces are algebraic operators.
I stumbled across no mention whatsoever when I was looking for a definitive description of multiplication by juxtaposition .
Ric: “Tim, you could save a lot of people a fair amount of time if you just posted a reference that supports your claim that spaces are algebraic operators.”
I never said that spaces are algebraic operators. I said they work as grouping indicators. When you ignore the spaces in a written expression meant to be solved manually then you are ignoring the information the spaces are meant to provide.
In a written expression you can substitute parenthesis for spaces (usually) but you simply cannot ignore the spaces like the computer programmers on here want to do.
Well, it sounds like we need to hear your definition of “algebraic operator”. Mine includes grouping symbols (including null strings).
Of course, some of that comes from formal descriptions of expressions in programming languages, but it all applies to things like diagramming sentences or expressions.
@whiten
“In your try you clearly have broken that rule.”
Absolutely not. I have not broken any rule. It is you that is wrong, multiplication and division have equal standing in the BODMAS OR PEMDAS. These rules do not subtly convey the nuances of in depth mathematics.
The problem is a numerator over a denominator which is the key. That is “The key is that the line separating the numerator from the denominator of a fraction (called the vinculum or solidus or obelus) is itself a grouping symbol.
Definition: Fraction Line (Vinculum or Solidus or obelus) The fraction line also acts as a grouping symbol. Everything above the line in the numerator is grouped together, and everything below the line in the denominator is grouped together.
Definition:A vinculum is a horizontal line used in mathematical notation for a specific purpose. It may be placed as an overline (or underline) over (or under) a mathematical expression to indicate that the expression is to be considered grouped together.
Definition: Obelus is a symbol consisting of a short horizontal line with a dot above and another dot below. In mathematics it is mainly used to represent the mathematical operation of division.
Definition: Solidus is a symbol representing a slash “/” used as the bar between numerator and denominator of an inline fraction. The solidus is also called a diagonal line.
Special care is needed when interpreting the meaning of division, because of the ambiguity in expressions such as a/bc. Whereas in many mathematical textbooks, “a/bc” is intended to denote a/(bc). For clarity, parentheses (brackets) should therefore always be used when delineating compound denominators.
Regards
Climate Heretic
@kid for your perusal
Heretic ==> well, your final answer is the same as Strogatz’. He gives the caveat that if we are to accept that, then we must all use the “more sophisticated” rules, which deal with implicit and explicit operations.
Unfortunately, much software does not use the rules needed to arrive at that answer.
Thus, lazy programmers, writing what they think are valid notation, can mangle the results. (say, on line 1,230 of a 5,000 line program segment….)
?Mordor/1 = ?
One does not simply go into Mordor
Does this equation behave badly with the transitive property? As we learned in high school, a(b+c) = ab + ac. So:
8 ÷ 2(2+2) becomes
8 ÷ 4 + 4 becomes
2 + 4 becomes
6!
It is simply not a valid expression. Anything that treats it as valid is wrong.
It can be interpreted as:
(8/2)*(2+2)
or
8/(2*(2+2))
Since both are valid, with different results, the expression as written is incomplete.
It shows how poor education is that there is even any discussion.
Phillip:
If I gave you the written expression ʃ x2 / x(x + 1) dx exactly how would you evaluate it manually?
Do you have a rule that says implies exponentiation? Why not just write x^2/x(x+1) and really confuse people?
Or, just keep it simple and consider a/bc or (since you’re enamored with spaces) a / b c?
Ric:
I wrote that expression in Lyx. I don’t know why the exponation symbol did not come through. I didn’t edit the final paste into the reply box closely enough.
Here it is rewritten: If I gave you the expression ʃ x∧2 / x(x + 1) dx exactly how would you evaluate it manually?
You forgot to state exactly how you would evaluate it, you just skipped over doing that.
Thank You Kip an interesting point.
Perhaps this small point of mathematical convention and computer programming could explain the anger from some of the Climate Modelling Teams when others question their results.
Your point with respect to the order of calculation specified within the model is an important observation.
I too remember programmes that gave very different results,depending on the hardware they were run on.
Politically nothing makes an argument more pointless than when both sides Know they are right.
The comments highlight this certainty of many,that the answer is obviously just so.1 or 16.
Maybe Garbage In Gospel Out is truly a matter of perspective.
For when we used slide rules,we had to have a good idea of what the answer should be,which is very different to todays culture where the phone tells us an result to 3 decimal places.
Faith in the machine?
Steve McIntyre called this over 10 years ago,we need an Engineering Grade review of all the processes of Climate Scrying.
Funny how that never happened,as we approach the 10 year anniversary of the CRU Emails.
John ==> I admit that I have been disappointed with the comments, as so few actually take any interest at all in the greater question of the order of calculation in complex computer models such as GCMs.
Despite Strogatz’ detailed analysis and discussion of the issues in the simple problem, we see so many insisting that what they learned in school is the ONLY right way.
I think that climate programmers have just been copying&pasting earlier code and making adjustments to it, without any real analysis of the order of calculation problems. Correcting or changing the order would be a massive new undertaking.
One reader offered a paper on this topic, which I have yet to read.
Fer chrissake, the rule, the rock-solid convention ,that I and millions of other learned regarding the order of operations was “My Dear Aunt Sal.”
Expressions within parens were to be dealt with FIRST, in the same order.
Ergo the answer is 1.
No airplane, no spacecraft will ever again be safe until these RULES are followed by everyone.
No WHAT-EV.
It is not ambiguous.
Using what you were taught in math.
8/2(2+2)
Handle inside the () and get 8/2(4)
Now handle the () and get 8/8
Now handle the “/” and get 1
Now if you want your calculator or spreadsheet to get the same answer, program it correctly.
One way is 8/(2*(2+2))
1
Also, GIGO and Learn to Code.
The answer is 1. Math expressions are spacially contextual. For example, how do we know a number is an exponent?…only by the way the number is spacially placed and sized.
What a calculator does, when the original expression is naively and poorly translated into text, is irrelevant.For example. For example, exponents can’t even be “properly” put into a calculator (the ‘^’ symbol is commonly used to symbolize superscript). Saying that the original expression is the same as “5/2(2+2)” is like saying that five squared is the same as “52”. Spacing and context matters and this is lost in translation when put into a monospaced, super/subscriptless textual calculator.
So, while the intent of the author of the original expression could be misinterpreted, given the spacing and the rest of the contextual clues, the most likely intended answer is “1”. More parentheses could have been used to “nail it down”, but more parentheses can be visual clutter and the spacing communicates the intent sufficiently.
In The Netherlands we use the following sentence to determin what comes first:
Men Vaart De Waal Op en Af.
(People sail the Waal up and down)
M: machtsverheffen = exponentiation
V: vermenigvuldigen = multiplying
D: delen = division
W: worteltrekken = rooting
O&A: optellen& aftrekken = add & deduct
So the answer is 1.
Multiplying goes before division
—–
Also used is: Meneer Van Dale Wacht Op Antwoord
(Mister Van Dale Waits For Answers)
This is considered less accurate because now adding seems to come before deduction
John Dilks,
You are assuming that the / in 8/2(4) applies to everything to the right of it.
Not so.
The slash / applies to the next value, 2 in this case, and the value is terminated by the (
If the writer of the original equation wanted to tell us the it was 8 divided by the remainder of the equation then they would have written it as 8/(2(2+2)) which is what you assume but the original was written as 8/2(2+2).
In every case written and explained fully in Bird: https://jpmccarthymaths.files.wordpress.com/2012/09/john_bird_engineering_mathematics_0750685557.pdf
there are plenty of examples with a / in the middle of a complex equation and it always, always applies to the following term only. If you want the / to apply to the remainder of the written equation, then you must put () around everything you wish to include.
How else could it be?
“How else could it be?”
You have the “computer programming” virus and are ignoring the spaces given in the expression.
If you include the spaces shown in the original expression then the / is no longer next to the 2, a space is. The 2(2+2) becomes a standalone grouping.
And you still haven’t provided any references that says spaces matter.
Ric: “And you still haven’t provided any references that says spaces matter.”
Jimminy Crickets! Look in any math book to see how expressions are written.
from wikipedia: “In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and GROUPING (capitalization is mine, tim) to help determine order of operations, and other aspects of logical syntax.”
Please note carefully that grouping is specified separately from brackets, punctuation, and operations. Grouping by using spacing is perfectly legitimate as a perusal of any math book will show.
And you still haven’t provided any references that says spaces matter.
Look at my references, I used some from the early 20th century. One I didn’t copy (from the FB dicussion where I was more involved) is from Wikipedia. Since you tolerate Wikipedia, visit https://en.wikipedia.org/wiki/Order_of_operations where it says in part:
Again, for the umpteenth time, the original question is a play on ambiguity. In my book there is no excuse for writing ambiguous algebraic expressions, or for defending them, and I still haven’t found any references that refer to spaces as operators or whatever you wish to call them.
BTW, that page continues with more notes on ambiguity:
“And you still haven’t provided any references that says spaces matter.”
Spaces provide for grouping. Why is that so hard to understand?
And you have still not answered how you would evaluate the expression I gave you. Is it embarrassing for you?
Steve,
Please state the page number showing an equation using the / symbol.
From High precision calculator – keisan – Casio :
https://keisan.casio.com/calculator
The result is 1
Equation solving in the context of climate models are an iterative process and I don’t mean the time step, that is a different matter. This iterative process converges to the exact answer which generally is not reached unless one is lucky (the equation happens to be serendipitous …) A margin of error or precision is the target and this is so because of the limitations of the computing environment. There are limited resources, computing time and memory. The iterative process has to stop at one point and the remaining error in the calculation has to be accepted, but it can also be chosen, with trade offs. You can throw more iterations into the equations you don’t feel provide enough precision but you will have to give up something else in return, maybe reduce the precision of some other equation or wait a lot longer for your calculations to complete.
There are different mathematical methods of solving the same problem with variance on how fast convergence to the exact answer takes place, how many terms are used in the equation and how amenable the equation is to a computer implementation.
It is alright to have different calculated values depending on the order one processes the operations and this is something to be expected, because you are managing the margin of error. You get a different answer but it falls into the expected error range. Mixing the order of operations is a good way to check your algorithms. You don’t expect the exact same results, you check that they fall within an accepted range. Margin of error will always increase the further your time steps take you though.
It would be best to frame the problem as a real-world question and then write the correct algebraic formula, as the expression must have originated from something. I see 2 problems that could have led to the poorly written expression:
1. I have 8 apple seeds to put into pots, and there are 2 boys and 2 girls with 2 pots each. How many seeds per pot? 8 seeds divided by 8 pots = 1
2. I have 2 boys and 2 girls each bringing 2 seeds for my 8 pots which require 2 seeds each. How many seeds are the children bringing? 8 pots divided 2 seeds per pot times (2 seeds per boy plus 2 seeds per girl) = 16 seeds. (Sort of an over-specified problem).
Gee, that is an algebra test question… we were told to replace the order of operations word that had ‘and’ with then: addition then subraction then multiplication then division. work it from the inside out using polynomial fractions, if there is missing parenthesis or brackets. In this case, 8 over the rest, just like a good algebra calculator (some are wrong) answer is 1.
The correct answer is 8.5
We have two mathematical models, one gives 1 as the answer and the other 16.
If we do that climate scientists do, which is to take the average of the two models, the answer is 8.5
Kip,
The (2+2) is obviously 4 so what we have after the first step is … 8 / 2 * 4, where the * is implied.
BODMAS processes 8 / 2 * 4 as … (8/2) * 4
PEMDAS processes 8 /2 * 4 as … 8 / (2 * 4)
BOMDAS processes 8 / 2 * 4 as … 8 / (2 * 4)
Unlike additions, subtractions and multiple multiplications, division is not commutative – the order of operations is important.
Any programmer who knows his stuff knows to avoid mathematical ambiguities and will insert parentheses. The question is always whether a given programmer knows his stuff.
Kip…
Imagine you gave a class in university an exam. And at the end there was an extra paper returned with no name. As strange as it might sound, the climate model solution is to take unidentified paper and give everyone else in the class an equal share of the right and wrong answers. Otherwise, the climate models would create or destroy energy – something that is physically impossible.
From my own testing, a very simple problem is to try and build a computer model of a few hundred molecules bouncing around in a closed container under the influence of gravity. Then to realize just how difficult the problem is.
For each molecule you must calculate their trajectory, then calculate the time of the next collision, either with another molecule or the container, which is in itself a huge, a non trivial problem. Then bring the clock forward to that exact point in time of this next collision and move all the molecules to their new positions. Calculate the new paths created by the collision, then solve for the next collision.
You might imagine when molecules get compressed at the bottom of the container by gravity, the collisions can become extremely rapid, and very quickly you end up with round off and loss of precision errors in the calculations, and this shows up as a gain or loss of total energy in the system. The system begins to drift. even though you have not added or removed any energy – something that is not supposed to happen in the real world.
And this error occurs very quickly even using floating point processors with 80 bit precision and a few hundred molecules. It also happens in climate models. They drift – creating or destroying energy along the way. And the solution? The climate model builders calculate the drift at each iteration, and divide this delta up and add or subtract it from the individual elements of their simulation.
Thus, the result is very quickly a “smeared” version of reality, where the unidentified results are divided up among the other students in the class.
ferd ==> see this link:
http://www.miroc-gcm.jp/cfmip2017/presentation.html
Kip,
Here is an article I recently read about the use of mathematical modelling using computers in the world of economics, the same issues apply to climate change science.
T
he evolution of thinking went like this,
1.Holy shit! Do you see what you can do with a computer’s help.
2.Learning computer modeling puts you in a small class where only other members of the caste can truly understand you. This opens up huge avenues for fraud:
3.The main reason to learn stats is to prevent someone else from committing fraud against you.
4.More and more people will gain access to the power of statistical analysis. When that happens, the stratification of importance within the profession should be a matter of who asks the best questions.
https://rwer.wordpress.com/2019/08/09/econometric-illusions/
Vincent ==> One of the huge problems in many fields of science today is that researchers use Statistical Software Packages to handle the statistical analysis of their findings. Most researchers are not trained very well in statistics, so use just what seems right, or use what they have read that other researchers have used — without really understanding what the particular approaches do or what the results of the analyses can actually be used for….thus they get results that are not scientifically sound, thus “Most research findings are false.” John P. A. Ioannidis
Until this thread, I had not heard of different rules of precedence dependent upon whether an operator is written or implied.
I can see where it has come from, and to my mind, is a result of sloppy working.
From Texas Instruments help web page: https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110
” Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?
Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.
This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication is given the same priority. “
So the same manufacture (TI) has a number of calculators on the market that give different answers to the same input!! Looks like more recent calcs do it properly now.
This is why it is important that every formula, equation, question etc is written in an unambiguous manner.
By all means write down your equation how you want, enter it into any calculator you want, but if anyone else is to see your equation, you must remove ambiguity so they understand what you mean and it will not matter which calculator THEY use, they will get the same answer as you.
The answer remains 16.
Not to confound things but how do you know the proper way to pronounce “read”?
Is it “red” or is it “reed”?
Did you read the book?
Have you read the book?
Understanding how words go together to modify other words is a skill to be learned rather than just learning the word all by itself.
The same with this math question. You cannot take each operation solely by itself in a strict order. You need to understand the meaning of the symbols used. Because there is no additional operation indicated between the 2 and the (2+2), the 2 belongs to the (2+2) as an alternative way to express 8. Consequently the 2 does not belong to the 8 or to the division symbol.
Tom: “the 2 belongs to the (2+2) as an alternative way to express 8. Consequently the 2 does not belong to the 8 or to the division symbol.”
Not just this but there is a space between the division symbol and the 2 thus disassociating the 2 even further from the division symbol.