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Post by reTrEaD on Sept 19, 2023 12:38:35 GMT -5
8 ÷ 2(2+2) = ?Everybody and their brother on social media has been exploiting an ambiguity regarding "multiplication denoted by juxtaposition" aka "implied multiplication" and its application within PEDMAS aka BODMAS aka BOMDAS aka BEDMAS. Let's review PEDMAS, then unravel the controversy. PEMDAS (order of operations)
Mathematical operations should proceed from left to right, with the following priorities observed, listed in order.
- Parentheses have the highest priority. Any and all operations within parentheses (or brackets or braces) must occur before operations outside this grouping.
- Exponents have the next priority.
- Multiplication and Division are the next priority and neither one has priority over the other, so we proceed from left to right.
- Addition and Subtraction are the last priority and neither one has priority over the other, so we proceed from left to right.
Now let's apply this to our contentious equation: 8 ÷ 2(2+2) = ? First, do the operation within the parentheses: 8 ÷ 2(4) = ? The 2 juxtaposed with the 4 contained in parentheses implies multiplication so lets indicate that explicitly. 8 ÷ 2 · 4 = ? All that remains is Multiplication and Division which have equal priority, so we proceed from left to right. Division is the leftmost, so let's do that first. 4 · 4 = ? Finally the Multiplication. 16 = ? 16 is the answer, plain and simple. Case closed.
Uhhh, not so fast! Some (myself included) would argue that implied multiplication takes priority over division. Let's examine that claim. In algebra, implicit multiplication has priority over division. Thus, 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n Let's compare explicit and implicit multiplication in an expression with a variable ... EXPLICIT MULTIPLICATION8 ÷ 2 · y = ? Using PEDMAS, we simply process from left to right and division is first. (8 ÷ 2) · y = ?
4 · y = ? In case you didn't notice I had substituted y for (2+2) in the original equation, so let's plug that back in. 4 · (2+2) = ?
4 · 4 = ?
16 = ? (this agrees with our original solution) Implicit MultiplicationWhat happens when the multiplication is implicit? The implied multiplication must take priority over division. 8 ÷ 2y = ? We'll use brackets to indicate the priority of implicit multiplication. 8 ÷ [2y] = ? Now let's substitute. 8 ÷ [2(2+2)] = ? Process the inner parentheses first. 8 ÷ [2(4)] = ? Now the multiplication within the brackets 8 ÷ [8] = ? 8 ÷ 8 = ? 1 = ? The answer is 1 (this does not agree with our original solution) In my opinion, the rule regarding implied multiplication should filter down from algebra to simple arithmetic, and should be specified in PEMDAS. Unfortunately, most explanations of PEMDAS don't differentiate between explicit and implicit multiplication. And that's unfortunate. For a set of rules that was intended to minimize clutter while avoiding ambiguity, it fails miserably.
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Post by JohnH on Sept 19, 2023 14:54:12 GMT -5
Id agree with you, the answer should be 1 (but the PEMDAS version =16?)
But I suppose in real life, there are very few cases where this comes up with simple numbers. The implicit multiplication isn't a feature of normal life using only numerals and can get confusing. While 2(2+2) has a clear meaning, and 2(4) does also though its quirky looking, its not the same as saying 2(4) =24 (!). Or if we start using the '.' symbol as a multiplication, we could get 2 x 4 = 2.4 (!). Europeans get around that to some extent, by using a comma ',' instead of a decimal point.
So I think in practice, we have to drop out of using the implicit notation once its down to a specific numerical calculation rather than a line of algebra.
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Post by unreg on Sept 19, 2023 16:39:54 GMT -5
4 · 4 = ? Finally the Multiplication. 8 = ? 8 is the answer, plain and simple. Case closed. 4 * 4 = 16 (i.e. 4 + 4 + 4 + 4 = 16) But, I got 1 bc I felt the implied multiplication beats division sign… so that goes first. 🙂 Like you pointed out.
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Post by reTrEaD on Sept 19, 2023 18:31:59 GMT -5
Id agree with you, the answer should be 1 (but the PEMDAS version =16?) Yes. Thanks for noticing. I corrected the error in my OP. But I suppose in real life, there are very few cases where this comes up with simple numbers. The implicit multiplication isn't a feature of normal life using only numerals and can get confusing. While 2(2+2) has a clear meaning, and 2(4) does also though its quirky looking, its not the same as saying 2(4) =24 (!). Or if we start using the '.' symbol as a multiplication, we could get 2 x 4 = 2.4 (!). Europeans get around that to some extent, by using a comma ',' instead of a decimal point. Rather than a period, one alternative for explicit multiplication is the "multiplication dot" which is centered vertically in the cap height of the font an example is shown here after the period. · However, I think I read that in Europe this is sometimes used as a decimal point? So I think in practice, we have to drop out of using the implicit notation once its down to a specific numerical calculation rather than a line of algebra. The big problem here is that PEDMAS isn't explicit about the implicit. It lacks any rules to deal with juxtaposition and the absence of an operator. Perhaps PEDJMAS should be the new standard? idk. Until standardization is achieved, the safe way of dealing with this is by increasing clutter to remove any possible ambiguity. NEVER write an expression such as 8 ÷ 2(2+2), instead use the verbose version which says exactly what you mean. 8 ÷ [2 · (2+2)] = 1 [8 ÷ 2] · (2+2) = 16 I'd like to hear from some of the other mathemagicians such as Yogi B and ms for their input on the topic.
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kitwn
Meter Reader 1st Class
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Post by kitwn on Sept 20, 2023 17:00:12 GMT -5
I'm with you that the implied multiplication takes precedence. 2n (two lots of 'n') looks too much like a single entity, a multiplication that was already carried out before everything else started, to then be ripped apart so that we can start all over again.
And in any real world calculation there should be some context to guide you. For example, if two families each consisting of 2 parents and two children are coming to tea and you only have 8 biscuits. The answer to how many biscuits will each guest be able to eat is obviously 1 rather than 16.
Kit
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Post by Yogi B on Sept 20, 2023 17:54:52 GMT -5
Until standardization is achieved, the safe way of dealing with this is by increasing clutter to remove any possible ambiguity. NEVER write an expression such as 8 ÷ 2(2+2), instead use the verbose version which says exactly what you mean. 8 ÷ [2 ⋅ (2+2)] = 1 [8 ÷ 2] ⋅ (2+2) = 16 I'd like to hear from some of the other mathemagicians such as Yogi B and ms for their input on the topic. From a maths perspective, aside from gimmicks, by the time people are using implicit multiplication the majority have likely moved on from using the division symbol and are writing the expression as a fraction instead: \frac{8}{2 (2 + 2)} = 1 ,\hspace{2em} \frac{8}{2} (2 + 2) = 16
But if disambiguating the expression whilst restricted to a single line, I'd be unlikely to add more parentheses for want of limiting extra line noise. For the equals-one case, I'd rely on the normal left-associativity of division and do: 8 ÷ 2 ÷ (2 + 2) = 1 For the equals-sixteen case, inserting the explicit multiplication is enough, extra brackets would be redundant: 8 ÷ 2 × (2 + 2) = 16 Alternatively, as long as we're using a type of division that is equivalent to a multiplication by the multiplicative inverse, I'd do that substitution: 8 × 0.5 × (2 + 2) = 16 But then, as long as we're dealing with scalar multiplication (which, along with being associative, is also commutative), we can reorder the factors in any order of our choosing and so move the 'multiplication by a half' to the end — such that then it no long matters whether the (implicit) multiplication or the division is performed first: 8(2 + 2) ÷ 2 = 16
With actual mathematics we have the benefit of infinite precision, but the most likely scenario where we might need to write such an expression using on a single line is when writing code. In this case we likely have fewer options, for instance IEEE-754 multiplication is not associative; a few examples using standard 64-bit floating point numbers (in Python): >>> # Obvious cases are overflow... >>> a = (1e200 * 1e200) * 1e-200 >>> b = 1e200 * (1e200 * 1e-200) >>> assert a == b, f'a != b\n {a = }\n {b = }\n' Traceback (most recent call last): File "<stdin>", line 1, in <module> AssertionError: a != b a = inf b = 1e+200
>>> # ... or (near) underflow >>> a = (3.141592653589793e-294 * 1e-30) * 1e30 >>> b = 3.141592653589793e-294 * (1e-30 * 1e30) >>> assert a == b, f'a != b\n {a = }\n {b = }\n' Traceback (most recent call last): File "<stdin>", line 1, in <module> AssertionError: a != b a = 4.9406564584124655e-294 b = 3.141592653589793e-294
>>> # but even 'normal' operations lose associativity, >>> # because of rounding to the available precision >>> a = (3.3 * 2.2) * 0.1 >>> b = 3.3 * (2.2 * 0.1) >>> assert a == b, f'a != b\n {a = }\n {b = }\n' Traceback (most recent call last): File "<stdin>", line 1, in <module> AssertionError: a != b a = 0.726 b = 0.7260000000000001
However that's veering off topic. More relevant to this discussion is, in languages which do allow kind of implicit multiplication, how have they chosen the precedence of that operation relative to others. I'm only familiar with two such languages and *SPOILER* they have conflicting opinions. Frink allows for implicit multiplication by both immediate juxtaposition and by separation with whitespace characters, but both have the same precedence as explicit multiplication: C:\>frink -e "10volts / 10kohms" 1000 m^4 s^-6 kg^2 A^-3 (unknown unit type)
C:\>frink -e "10 volts / (10 kohms)" 1/1000 (exactly 0.001) A (current)
C:\>frink -e "8 / 2(2 + 2)" 16
Performing the first of the above calculations on the Frink web interface, returns the same result, but also includes a link to their FAQ on the topic. On the other hand the Julia programming language supports numeric literal coefficients, which have higher precedence than division (and explicit multiplication) but are only syntactically valid in much more limited circumstances than Frink's implicit multiplication. Here's a macro, and its application to the titular problem showing how Julia parses the expression: julia> macro dbg(arg) return quote temp = $(esc(arg)) println($(string(arg)), " = ", temp) temp end end @dbg (macro with 1 method)
julia> @dbg(8 ÷ 2(2+2)); 8 ÷ (2 * (2 + 2)) = 1
If you want modify/run this example you can AttempThisOnline. But note that although Julia has a "÷" operator it represents (integer) truncated division — so, with different numeric values, rational "//" or floating point "/" division would be more appropriate. Another thing to note (and something which catches me out on occasion) is that this implicit multiplication has differing precedence on each side of exponentiation: julia> a, b, c = 3, 2, 1 (3, 2, 1)
julia> @dbg(2a^2(b + c)); 2 * a ^ (2 * (b + c)) = 1458
i.e. at first glance, I sometimes think the above should be equivalent to: 2 a2 (b + c) = 2 (9) (3) = 54 rather than what Julia actually does: 2 a2(b + c) = 2 (36) = 1458
I've seen people argue that P(arentheses) includes the implicit multiplication by a value juxtaposed with them. However, that cannot be correct as doing so would give this greater precedence than exponentiation, i.e. if that understanding were true then: a(b)c = (a b)c
Finally, there is also a related question about parenthesis-less function application: cos π / 3 = cos(π / 3) = 1/2 or cos π / 3 = (cos π) / 3 = −1/3
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Post by thetragichero on Sept 20, 2023 18:43:51 GMT -5
former mathematical sciences major. it's 1, although there are too many numbers for what we mostly did (lots of lowercase deltas and whatnot)
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Post by reTrEaD on Sept 20, 2023 21:51:45 GMT -5
Thanks Yogi. I appreciate the time you invested. I won't have an opportunity to carefully digest the majority of what you posted until tomorrow or the next but I'm likely to glean a bit of insight from that. The last two items caught my eye at a glance so I'll respond to those right now. I've seen people argue that P(arentheses) includes the implicit multiplication by a value juxtaposed with them. However, that cannot be correct as doing so would give this greater precedence than exponentiation, i.e. if that understanding were true then: a(b)c = (a b)c Yes. I've seen that attempt to expand the scope of parentheses and knew intuitively that was a bad path. That explanation and example are an excellent way of showing why. Finally, there is also a related question about parenthesis-less function application: cos π / 3 = cos(π / 3) = 1/2 or cos π / 3 = (cos π) / 3 = −1/3 I only took one semester of trig and that was a few decades ago. I don't remember how to deal with that. Just for S&G I googled cos π / 3 = cos(π / 3) and it recognizes that as an identity. It saw cos π / 3 = (cos π) / 3 as a contradiction.
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Post by unreg on Sept 21, 2023 0:35:22 GMT -5
Finally, there is also a related question about parenthesis-less function application: cos π / 3 = cos(π / 3) = 1/2 or cos π / 3 = (cos π) / 3 = −1/3 I only took one semester of trig and that was a few decades ago. I don't remember how to deal with that. Just for S&G I googled cos π / 3 = cos(π / 3) and it recognizes that as an identity. It saw cos π / 3 = (cos π) / 3 as a contradiction. IIRC, cos doesn’t require parentheses as long as it’s referring to the entire equation. That makes it easier to write equations using sin, cos, tan, arcsin, arccos, and cotan. It’s nice that search engine agrees. I forgot google existed.
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Post by ashcatlt on Sept 21, 2023 21:54:56 GMT -5
Nobody use the inline division symbol after grade school. Nobody in their right mind uses slash notation unless they absolutely have to and when they do, they either use parentheses carefully to avoid ambiguity or they provide some statement about the convention they intend to follow. People who actually intend for their expressions to be calculated correctly don’t write them in such a way that “80% will get it wrong”. That is, these things are deliberately ambiguous mostly with the intent of driving engagement with their posts.
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Post by newey on Sept 22, 2023 6:36:33 GMT -5
Nobody in their right mind uses slash notation unless they absolutely have to and when they do, they either use parentheses carefully to avoid ambiguity And YogiB said: As YogiB has shown above, with a PC and appropriate software, one can make nice legible, understandable, and unambiguous fractional expressions. But it wasn't always that way. On a typewriter keyboard, the slash was usually the only option available. Some fancy typewriters like IBM Selectrics had keys for fractions of one-quarter and one-half, but that was it. (Query: How many remember using one of those? Or knows to what the term "IBM Golfball" refers?) So it seems to me that some of the ambiguity discussed here was, in the past, a consequence of the technology of the time.
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Post by sumgai on Sept 22, 2023 10:56:25 GMT -5
I learned touch-typing (with the good ol' home row) in 1961. I didn't see my first Selectric until I entered the Army in 1966. I didn't actually use one until two years later, whereupon I had access to a pair of golfballs, Elite and Pica. Good times, good times.
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Post by sumgai on Sept 22, 2023 12:15:46 GMT -5
One of my rare moments of lucidity..... PEMDAS is the order of the day. It has one written rule, and one unwritten rule. Respectively, they are: a) work from left to right; b) scan the whole equation first to ensure that any parens and/or exponents are worked out first, then go back to the left-most term. One could suppose that another unwritten rule would be that there is no implication in showing multiplication of a parenthetical term - it's explicit, and that's all she wrote. There has never been an (intentional) case where someone wrote an expression and intended for one term to not be a multiplier of a parenthetical term. Accidental, yeah, sure, but intentional? I think not. And do remember that many, many expressions have two parenthetical terms placed right next to each other, without any other method of showing multiplication is intended - that's just a given. Err, that is, lacking any other operator immediately preceding the term. (Which, going back to our typewriter, that particular device had neither a centered (up and down) "x" nor "." to show intended multiplication. But still good times. ) So, the original equation shows that we scan first, and find that (2+2) must be determined before all else. Then we multiply that by 2 (PE MDAS, multiply before division) and finally, although we seemingly worked backwards (right to left), we derive the answer of X = 1. And yes, multiplication does come before division, as well as addition comes before subtraction. - we didn't make up 'PEMDAS' just because it easily rolls off the tongue. As in our original example, too many cases can be shown where reversing that order will render an incorrect answer. HTH BTW, nice presentation, reTrEaD. Good discussion, everyone. sumgai
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Post by reTrEaD on Sept 22, 2023 13:47:38 GMT -5
So, the original equation shows that we scan first, and find that (2+2) must be determined before all else. Then we multiply that by 2 (PE MDAS, multiply before division) and finally, although we seemingly worked backwards (right to left), we derive the answer of X = 1. That's not PEMDAS as it's currently taught. And therein lies the problem. Order of operations: 1 - Parentheses. 2 - Exponents 3 - Multiplication or Division (both have equal precedence, work from left to right as either of these appear) 4 - Addition or Subtraction (both have equal precedence, work from left to right as either of these appear) If we observe these rules as they are stated ... 1 ÷ A × B = 1/A × B (this legit) 1 ÷ AB = 1/A × B (this is where PEMDAS forces an incorrect answer)
PEMDAS drools and PEJMDAS rules. (Or PEIMDAS if you prefer.) Juxataposition aka " Implied multiplication" due to the absence of an operator should have its own priority, below Exponents but above (explicit)Multiplication or Division.
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kitwn
Meter Reader 1st Class
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Post by kitwn on Sept 23, 2023 0:13:24 GMT -5
Many of these problems arise from trying to write equations on a single line using a machine instead of a pencil. When I was at school I don't think I ever saw the / symbol used for division, my text books (I've just blown the dust of one of them to check) invariably write division within a formula in fractional form as shown by YogiB. In this modern, digital age I think we need a new rule:
Rule 1) Apply PEDMAS
Rule 0) Write every equation in such a way that PEDMAS can be applied unambiguously.
Kit
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Post by sumgai on Sept 23, 2023 0:50:41 GMT -5
1 ÷ A × B = 1/A × B (this legit) 1 ÷ AB = 1/A × B (this is where PEMDAS forces an incorrect answer) Err, I think your second example should be 1 ÷ AB = 1/A x 1/B. At least that's how I remember the distributive property. Or did I miss something in the translation? Because otherwise, 1/A x B being the same re-write for both equations should yield the same answer, no? sumgai
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Post by unreg on Sept 23, 2023 2:12:29 GMT -5
1 ÷ A × B = 1/A × B (this legit) 1 ÷ AB = 1/A × B (this is where PEMDAS forces an incorrect answer) Err, I think your second example should be 1 ÷ AB = 1/A x 1/B. At least that's how I remember the distributive property. Yes sumgai, you remember the distributive property well. —- Ummm… why is there a mass fixation on a math acronym? At least here, lots of posters look up to “PEMDAS”. I blocked that acronym out of my head, when I was supposed to be learning it, bc acronyms are English or maybe just text; while, math is numbers and symbols. At that time, I couldn’t possibly use both math AND English at the same time (learning trig “sine” would have been impossible for me, then ). I remember a scan tron question, “what does PEMDAS mean?” That was scary until I read the four choices. Is it insanely hard to just memorize “(), ^, */, +-“? And then just add in the implied multiplication after the carrot (or exponent sign)?
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Post by reTrEaD on Sept 23, 2023 9:03:43 GMT -5
In this modern, digital age I think we need a new rule: Rule 1) Apply PEDMAS Rule 0) Write every equation in such a way that PEDMAS can be applied unambiguously. Why adhere to a system (PEMDAS) that promotes ambiguities and force ourselves to avoid them? Why not adopt a system (PEJMDAS) that eliminates these ambiguities in the first place? The idiom: "If it ain't broke, don't fix it!" ... does not apply here. PEMDAS is broken. Let's fix it!
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Post by reTrEaD on Sept 23, 2023 9:04:12 GMT -5
Err, I think your second example should be 1 ÷ AB = 1/A x 1/B. At least that's how I remember the distributive property. Or did I miss something in the translation? Because otherwise, 1/A x B being the same re-write for both equations should yield the same answer, no? Yes, that's the problem with PEMDAS. It doesn't differentiate between explicit multiplication and implicit multiplication. And since the division is the leftmost operation, PEMDAS says it should be done first. Thus you get a different answer if you adhere to PEMDAS than you do if you apply the distributive property. Ambiguity is.
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Post by reTrEaD on Sept 23, 2023 9:04:55 GMT -5
Ummm… why is there a mass fixation on a math acronym? Because it's being drilled into the heads of our students. I place much of the blame on the NEA. Pompous Educators Misteaching Distressed Algebra Students
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Post by reTrEaD on Sept 23, 2023 11:12:31 GMT -5
Just for S&G, let's try some online resources to evaluate these expressions: If we Bing: solve: 1 ÷ A × B The first result is Microsoft Math Solver, so let's use that resource. Microsoft Math Solver for: 1 ÷ a × b The division is performed first and ultimately we get: b/a
And now Microsoft Math Solver for: 1 ÷ ab The division is performed first and ultimately we get: b/aImplicit multiplication is treated exactly the same as explicit multiplication. This is not good! This is where the application of PEMDAS fails us.
Now let's Google: solve: 1 ÷ a × b Google solves this directly. The division is performed first and ultimately we get: b/a
Finally, let's Google: solve: 1 ÷ ab Google does not attempt to solve this expression. Instead, it just provides search results to sites where that expression appears. Why does Google automatically provide a solution for an expression where PEMDAS correctly interprets explicit multiplication, but not for an expression where PEMDAS would incorrectly interpret implicit multiplication?Is Google incapable of recognizing implicit multiplication? Or is Google intelligent enough to recognize the PEMDAS problem and it chooses to avoid the controversy?
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Post by unreg on Sept 23, 2023 15:57:40 GMT -5
Is Google incapable of recognizing implicit multiplication? Or is Google intelligent enough to recognize the PEMDAS problem and it chooses to avoid the controversy? That search engine has many problems. - After checking two other sites it seems some may be doing ok: www.mathway.com/AlgebraAfter entering 1/xy and then choosing “simplify” it returns: The 1/xy is written correctly there; though I bet the copy&paste will show it incorrectly written here. And, I bet 1/X * 1/Y is not considered a simplification, there, bc that requires a multiplication sign in the middle. www.mathpapa.com/equation-solver/After entering 1/xy that site rewrites my entry as 1/X*Y before I even attempt solving that… so that site is sort of kind since it declares its incompetence before being used.
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Post by JohnH on Sept 23, 2023 16:07:11 GMT -5
We seem to be getting rather distracted by the small issue of solving the fundamental basis of human mathematics, which is merely the foundation of all science and engineering, when we should really be getting on with the more important work of rewiring the worlds guitars?
Thinking about just one day as an engineer, Im likely to be interpreting, writing and communicating maths in multiple different ways eg:
1. Quick numbers in a calculator or calculator app, which if its of good quality post about 2000AD, will display a full line and then follow PEMDAS to evaluate it, but if its older, or simpler (including the latest basic calc app built into Windows), will do each operation as you enter it, with a different result
2. I might revert to my favourite HP calculator, with RPN notation, which sorts out priorities in a much more compact and logical way. Great if you learnt to be used to it years ago, plus the BUTTONS feel REALLY nice!
3. Checking or setting up a spreadsheet, in which we use ^, * and / etc, plus more ( ) brackets than you can poke a stick at, just to be sure
4. Handwriting a calculation on a pad. And this leads to another source of conundrum: If you represent division using a / symbol, then PEMDAS tells us how to apply it. But if you instead morph the division into a horizontal line with calc above and calc below, does the transition from '/' symbol to horizontal line then give the division a higher priority? I would say yes-maybe! A similar issue occurs in writing slides for an engineering lecture, except its easier to read than my 2023 handwriting.
5. Working out numbers in my head, which is probably a place that such numbers should stay
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Post by reTrEaD on Sept 23, 2023 18:10:41 GMT -5
4. Handwriting a calculation on a pad. And this leads to another source of conundrum: If you represent division using a / symbol, then PEMDAS tells us how to apply it. But if you instead morph the division into a horizontal line with calc above and calc below, does the transition from '/' symbol to horizontal line then give the division a higher priority? I would say yes-maybe! A similar issue occurs in writing slides for an engineering lecture, except its easier to read than my 2023 handwriting. I would say yes-definitely! ... whenever the numerator and/or denominator has more than one term, or the denominator has more than one variable or a variable preceded by a coefficient, and PEMDAS is invoked. When it comes to division represented by a horizontal bar, the division is treated as though there are parentheses around the entire numerator and parentheses around the entire denominator, even though those parentheses are not actually there. This type of division has a low priority in the order of operations. If you morph one of those divisions with a horizontal bar, to a slash or a ÷, and neglect to place those 'invisible' parentheses, you're gonna have a problem! 2. I might revert to my favourite HP calculator, with RPN notation, which sorts out priorities in a much more compact and logical way. Great if you learnt to be used to it years ago, plus the BUTTONS feel REALLY nice! I'm glad you mentioned calculators. Some HP calculators employ PEMDAS some employ PEDJMAS. But according to the author of the following video, there is no chronological pattern. They're just all over the map. In 1996, TI regressed from employing PEJMDAS to now using PEMDAS on the TI-83 most most of the later families. In 2006, Casio also regressed to PEMDAS. However, they have since returned to PEMDAS. Apparently Sharp remained committed to PEJMDAS and never reverted. YAY! Around the 8:58 mark in the video, she discusses the communications she had with some of the calculator companies and why they regressed. At the 12:17 mark, she shows how Casio is doing something rather interesting with their fx-100AU. She's my second most favorite Australian.
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Post by JohnH on Sept 23, 2023 21:07:24 GMT -5
For a real Hewlett Packard, of the type whose grandfather helped land the Space Shuttle, and which every engineering student of the 1970's and 1980's lusted after and is still using today (unless they couldn't afford one and bought a Casio instead) it HAS to be RPN (Reverse Polish Notation) or else it denies their HP heritage. In RPN mode, the input for our puzzle would be, if working left to right with the intent to respect PEJMDAS:
8 Enter 2 Enter 2 Enter + x ÷
answer is 1
Or if one wants strict PEMDAS:
8 Enter 2 ÷ 2 Enter 2 + x
answer is 16
If anyone wants to remember these magnificent old engineering work-horses, many of them have been lovingly recreated as Android apps, mostly for free or for a few $, on Google Play
And I agree about the lady in the video. As smart and beautiful as she is, our favourite Aussie will always be Gumbo.
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Post by Yogi B on Sept 25, 2023 21:44:49 GMT -5
At the 12:17 mark, she shows how Casio is doing something rather interesting with their fx-100AU. I have two older Casios, a fx-7400G PLUS and a fx-991ES PLUS, circa 2008 & 2010 respectively (IIRC). Both follow PEJMDAS, though neither do the bracketing trick. I do have a vague memory of some 2011/2012 Casio calculators doing some odd new thing, whether it was this bracket insertion I do not know. What the fx-991ES PLUS does have is "natural textbook display", which gives the ability to enter calculations as fractions (amongst other things) — probably why calculators didn't earlier jump out to me as being restricted to a single-line interface. The fx-100AU shown in the video also has this ability: the fraction key is the first in the uppermost complete row of function keys. Even now, when I was checking the behaviour of my calculator, my muscle memory meant I automatically reached for and pressed the fraction key before correcting it to the division key.
Finally, let's Google: solve: 1 ÷ ab ... Is Google incapable of recognizing implicit multiplication? Or is Google intelligent enough to recognize the PEMDAS problem and it chooses to avoid the controversy? There is more than just one ambiguity at play here. Maybe not quite obvious in that specific example, but we're pretty close to an example which does demonstrate the issue (and some further WolframAlpha weirdness). First, another example that shows WolframAlpha recognising multiplication by the juxtaposition of two variables as higher precedence than division: 1 ÷ ab. However, it does not give the same higher precedence to implicit multiplication (which I'm proposing should not necessarily be synonymous with juxtaposition, in this case due to the presence of whitespace between the symbols): 1 ÷ a b. So, using that information, what if we wanted the juxtaposition behaviour but with just one extra symbol, say an 's'? 1 ÷ abs, oops! It's now treating the denominator as something entirely different: a function name. With simple text entry systems there is ambiguity between the juxtaposition of multiple one-character names, and any of the ways in which those characters can be recognised as one or more multi-character names.
P.S. if you think switching between calculators with different opinions on PEMDAS versus PEJMDAS is bad — programming languages, with their expanded number of operators, are far worse. Linked here are the operator precedence tables of JavaScript, Julia & Python. Even for operators with common syntax across the three there is disagreement as to their precedence. For example, compare the relative precedence of the following operator groups: the basic arithmetic operators ('*', '/', '+', '-'), the bitwise shift operators ('<<', '>>'), the bitwise arithmetic operators ('&', '|'), and the comparison operators ('<', '<=', '>=', '>') — though there is some commonality, each of the languages has a distinct ordering.
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Post by reTrEaD on Sept 26, 2023 3:45:31 GMT -5
There is more than just one ambiguity at play here. Maybe not quite obvious in that specific example, but we're pretty close to an example which does demonstrate the issue (and some further WolframAlpha weirdness). First, another example that shows WolframAlpha recognising multiplication by the juxtaposition of two variables as higher precedence than division: 1 ÷ ab. However, it does not give the same higher precedence to implicit multiplication (which I'm proposing should not necessarily be synonymous with juxtaposition, in this case due to the presence of whitespace between the symbols): 1 ÷ a b. I suppose we might be able to consider Implicit Multiplication synonymous with Juxtaposition, so long as we use the term Assumed Multiplication for the latter expression you provided. Or we could get a bit more cryptic and refer to the former as JiM and the latter as WaM (Whitespace assumed Multiplication). One thing I said earlier, is troublesome in this situation: Juxataposition aka " Implied multiplication" due to the absence of an operator should have its own priority, below Exponents but above (explicit)Multiplication or Division. In your first example there is Juxtaposition and an absence of operator. In your second example there is also an absence of an operator without true juxtaposition. The whitespace is ambiguous in WolframAlpha. It seems to default to assuming the whitespace is to be treated as a multiplication operator (explicit multiplication), but also presents the option to treat the whitepace as a comma with trailing space.
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Post by reTrEaD on Sept 26, 2023 13:21:19 GMT -5
Yogi B , I've been playing a bit with WolframAlpha and I see an inconsistency I can't explain. 8 ÷ 2(2+2) = 16 | coefficient juxtaposed with term contained in parentheses | PEMDAS |
| 8 ÷ 2b = 4 b | coefficient juxtaposed with variable | PEMDAS |
| | variable juxtaposed with variable | PEJMDAS | 🡄 Why the shift here? |
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Post by Yogi B on Oct 12, 2023 14:47:02 GMT -5
(which I'm proposing should not necessarily be synonymous with juxtaposition, in this case due to the presence of whitespace between the symbols) I forgot to add: "in typed out maths", but that should be obvious. If you have to resort to grabbing a measuring implement to decide if there's a larger gap between symbols in handwritten (or pretty printed) maths, the author's intent has already been diluted to a point where the reader is basically guessing. But when we can easily determine the exact sequence of characters used, it is pretty clear that "1 / ab" is intended to mean something different to "1/a b" (outside the influence of any specific set of parsing rules). The real problems only occur when the whitespace (or lack of it) is 'symmetrical'. I've been playing a bit with WolframAlpha and I see an inconsistency I can't explain. I think WolframAlpha is trying to be (mostly) compatible with the syntax of Wolfram Mathematica, and likely follows Mathematica's operator precedence. Oddly/interestingly, without even considering implicit multiplication, this splits multiplication to give: PEMDMAS. The easiest way to find the relevant section is probably to Ctrl+F for "Divide[": reading a handful of rows downwards you'll find 'normal' (commutative/element-wise/scalar) multiplication indicating it has lower precedence; whereas looking upwards you'll find non-commutative multiplications (dot & cross product, as well as a generic non‐commutative multiplication) have higher precedence.
Note: while the 'usual' dot product of vectors is commutative, in Mathematica the dot is really matrix multiplication (which is non-commutative) with extra automatic transposition of row/column vectors as appropriate. (Use of the dot allows matrix multiplication to be distinguished from 'normal' multiplication which it performs element-wise.) Though, the main thing to note is that only whitespace implied multiplication is listed in that table, in general Mathematica treats immediate juxtaposition of symbols as a single combined symbol. That is, "ab" is an entirely distinct symbol, having no relation to either "a" or "b". However, also note that (as is common to many programming languages), though a variable name (or in this instance a symbol name) may contain digits, it cannot begin with them — see: Symbol Names and Contexts. Therefore, "2b" is treated as a number followed by a symbol, equivalent to "2 b". Where WolframAlpha differs from Mathematica is that the former takes a fuzzy DWIM ( Do What I Mean) approach, making a best guess for the intended meaning of the given input. Since "ab" doesn't have a common mathematical meaning (as a function name, an operator name, the (Romanized) spelling of a Greek letter, etc.) and variables (in mathematics, as opposed to computer science) generally consist of only a single character, it evidently assumes multiplication of "concatenated variables" as the most likely intent. Plus, since it appears the user has opted for a stronger grouping of these symbols than Mathematica's rules would normally allow, it's not unreasonable to make the assumption that this form of multiplication should have higher precedence than usual. Of course, as with anything which employs DIWM, this results in an inherently chaotic system. For instance the "1 ÷ abs" example I posted earlier, or that (by default) 1/ab is not even interpreted as a mathematical expression.
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Post by bassplayerordinaire on Nov 29, 2023 18:50:36 GMT -5
And in any real world calculation there should be some context to guide you. For example, if two families each consisting of 2 parents and two children are coming to tea and you only have 8 biscuits. The answer to how many biscuits will each guest be able to eat is obviously 1 rather than 16.
I would like to lead with the confession that I am not a fan of Math. Flunked algebra once and passed the second time with a D--. I also can whip up a wicked spreadsheet every now and again. Go figure!
With that said, I truly appreciate this particular point in this discussion about an equation in the title. For example, I could argue that there is no absolute in mathematics as it relates to reading a series of symbols left to right, right to left, up, down, all around. Yes, left to right is the more common practice, but is there truly a different outcome if the equation is worked from right to left? The question mark was used to represent the variable outcome of doing something to the left of the equal sign and its result does not impact the read of the equation. So the parenthesis tell me to do that action before multiplying the result with the 2. Without any more of the string to the left, the question mark would represent 8. And yes, 8 divided by 8 equals 1. My going fancy-schmancy on this musing of mine is to put on display how I was taught to "process" a math equation using numbers and fractions and rules. In other words, I was taught the 2 prior to the parenthesis must be "decoded" into its simplest form before being put into action with the division of the results of the sub-equation. So here's a question. In this world of automated algorithms we all now live in, who knows how many algorithms are doing equations with a left to right read versus any other variation? [and then the bass player walks away from their words and towards their instrument of choice, unaware of all that was transpiring as the stage goes dark]
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