Rene
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Well, I do live further north than most of the folks who live in Ontario.
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8 / 2 (2 + 2) =
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Rene
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So, that's where can I get some of those 160 ounce gallons of MoGas? The ones in Canada are stale. But, I don't have a Monrovia Sectional.
My high school valued quiet and conformity. I learned that just being quiet earned one a 'B' for 50% of the class grade. ('A's were for those the teach liked.) So, I just read and took the required tests. Nobody left behind was interpreted to mean moving at an injured snail's pace. So, "bad" is a matter of opinion. The teachers who lasted more than a semester did as the district administration told them to. And they were generally happy to ignore anyone not bothering them or specifically asking for help (although individuals sometimes conflated the two).
By definition, a school district is a political subdivision, filled with politics, and politicians, so I won't go there. I guess the state college system is too, and each feels that the other is "wrong". That's why I liked math. It was supposed to just have one correct answer that was repeatable and verifiable.
Pinecone
En-Route
I have an HP-41CX app on my iPhone. And have an RPN calc (Excaliber) on all my computers. And a couple of -41s on my desk
Once I learned RPN, I have not owned a "regular" calculator.
I worked for HP with the HP-41C came out. I got one. Later sold to a friend and I got a -41CV. It is still around the house.
Pinecone
En-Route
The great thing about the -41 was the programming modules. I think I still have an Aviation Module laying around.
Not to mention the card reader and printer.
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Snorting his way across the USA
I have a 42S sitting in my desk right now, but if I need a calculator, I have Excalibur on my computers. I got a 28S when I was going through undergraduate ME school. It was a folding thing that had a big LCD screen for graphing. Then the 48 series came out. I borrowed a coworker's 48X with a survey pack to take with me for the survey portion of the California civil PE exam.
I liked the 42S, and used it for much of my initial engineering career, up until handheld calculators became redundant. I see on EBAY that these things are fairly valuable now.
I liked the 42S, and used it for much of my initial engineering career, up until handheld calculators became redundant. I see on EBAY that these things are fairly valuable now.
Jetter2
Filing Flight Plan
PEMDAS.
Multiplication/Division carry the same weight (same with addition/subtraction) - and are read from left to right (like a book)
8/2(2+2) == 8/2*4 = 16.
Matthew
Touchdown! Greaser!
This is how I remember my early math lessons in fractions and general math. I think that still sticks with me (“There are two ways to write “divide by”…”): rewriting the equation with a horizontal line vs slant, “divide by”, can give two different answers. I still have to think hard about looking at an equation like this because my brain wants to visualize it by substituting the horizontal line.
dtuuri
Final Approach
No doubt. If you want the computer to render "16" that's how you must write it. But if you follow math law, isn't the expression 2(2+2) equal to the expression (2+2)2 ? Then, the computer would come up with 4, right? A law-abiding human would still come up with 1 though, no?
jimhorner
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Yes, both expressions are equivalent.
But, neither a computer nor a math savey human would come up with 4 as the answer for either of the two expressions. Expressions in parentheses are evaluated first and then multiplication and division from left to right and then addition and subtraction from left to right.
2(2+2) = 2(4) = 8
(2+2)2 = (4)2 = 8
And in no way would either expression evaluate to 1.
The only difference between the human way and a computer expression is that the multiplier operator is assumed by a human by convention. A computer usually would require a multiplier operator (typically an *) like so:
2*(2+2) or (2+2)*2
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dtuuri
Final Approach
In the context of the OP where the expression is the denominator of 8?
jimhorner
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I didn't realize you were talking in the context of the original expression. In the post I responded to, you were only discussing the commutative property of multiplication. 2(2+2) = (2+2)2.
In the context of the original expression, the order of operations is clear.
Division is not commutative. 8/2 is not equal to 2/8 for example, but 2*8 = 8*2. In a similar manner, addition is commutative, but subtraction is not.
As has been said many times, one must evaluate expressions according to the well defined and accepted conventions of math.
Recursivly one must apply:
1. Evaluate expressions in Parentheses
2. Apply exponentiation
3. From left to right, evaluate multiplication and division
4. From left to right, evaluate addition and subtraction.
8/2(2+2) is not equal to 8/(2+2)2
8/2(2+2) = 8/2(4) = 4(4) = 16
8/(2+2)2 = 8/(4)2 = 2 * 2 = 4
The above two expressions are not equal. One can't arbitrarily rewrite an expression and expect to get the same answer. The conventions for evaluating expressions are straightforward, well-documented, simple to apply, and can be found in most any textbook at the pre-algebra level.
The one and only denominator in the original expression is 2.
Arguing otherwise makes no sense to me. If one wants to make up new rules, one faces an uphill battle to getting the new rules accepted.
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dtuuri
Final Approach
Actually, I had in mind the Distributive Law mentioned in the link I posted earlier. Isn't 2(2+2) or (2+2)2 each equal to (4+4) = 8, hence as the denominator of 8 the result is 1?
dtuuri
Final Approach
Which Law states that? Would Newton or Einstein agree? What piques my interest in this thread is I never heard of PEMDAS, but have heard of the various math laws at least 62 years ago. I had to learn 'em twice. Once, in algebra class, again in summer school after I flunked.
I always thought they were
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AV8R_87
Cleared for Takeoff
Same question can be applied to your way.
It's all about grouping of terms. Using your grouping logic we'd get 8/2+6=1?
dtuuri
Final Approach
Not by MY logic. I'd get 10.
Matthew
Touchdown! Greaser!
I already posted something similar-
In my brain:
8
— is the same as 8/2*4
2*4
But, to exact definition of MDAS, they aren’t equivalent and I have to be very careful about internet deals like this one. I guess all that chalk dust from my math teachers who wrote it the first way must have shorted something out. I was taught left to right, top to bottom. I tend to look at the ”/“ as a shortcut to the “—“, since it fits on one line, and interpret it that way.
But the “-:-“ symbol (I can’t figure out how to get that on my phone keyboard) would be interpreted correctly.
8 -:- 2*4
I guess I lose the internet today.
In my brain:
8
— is the same as 8/2*4
2*4
But, to exact definition of MDAS, they aren’t equivalent and I have to be very careful about internet deals like this one. I guess all that chalk dust from my math teachers who wrote it the first way must have shorted something out. I was taught left to right, top to bottom. I tend to look at the ”/“ as a shortcut to the “—“, since it fits on one line, and interpret it that way.
But the “-:-“ symbol (I can’t figure out how to get that on my phone keyboard) would be interpreted correctly.
8 -:- 2*4
I guess I lose the internet today.
How about a learning academy- people pay money to send their children here:Which Law states that? Would Newton or Einstein agree? What piques my interest in this thread is I never heard of PEMDAS, but have heard of the various math laws at least 62 years ago. I had to learn 'em twice. Once, in algebra class, again in summer school after I flunked.arcaneself-evident rules. After all this time, I may have found a use for them.
Khan Academy
Here's another one (same site- so I count them as one):
Here's another one:
What is the Order of Operations in Math? Understand the Rules
The order of operations in math is a list of the steps to take when doing a math problem. It is important to know the order of operations so you do not make any mistakes.
www.learner.com
Penn State University refers back to the Kahn academy site above:
All from left to right. It's a standard convention.
Newton and Einstein probably would have some differences. Newton was one of the founders of calculus, but we (and Einstein did use) use the notation of his contemporary, Leibnitz, instead for integral calculus.
dtuuri
Final Approach
Too late. The Law of Primacy, ya know...
jimhorner
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The distributive property certainly applies, but it's important to apply it properly and with the conventions appropriately followed. The distributive property doesn't allow one to short circuit the entire expression.
Your statement: Isn't 2(2+2) or (2+2)2 = (4+4) = 8 is certainly true. However, in the original problem:
8/2(2+2), one must first use the correct order of operations applied appropriately. The division must occur first.
I.e. 8/2(2+2) = 4(2+2) (division and multiplication happen left to right)
Now, applying the distributive property gives: (4*2 + 4*2) = (8+8) = 16
Note that all of these operators are binary. Without parentheses grouping a more complex expression into one operator, only the numbers to the immediate left and right are affected by the operator.
multiplication, division, addition, and subtraction all require a left and a right operand. Exponentiation, also, is a binary operator (5^2). There is a unary operator which looks like subtraction (the minus sign) which, when put in front of a single operand with no left operand, has the effect of negating the operand.
The operands in the original problem for the division are: 8 and 2, not 8 and 2(2+2). So the division is 8/2 not 8/(2*(2+2)). I keep seeing the implication that the denominator in the original expression is 2(2+2). That's incorrect. By the mathematical order of operations, the denominator is 2.
Maybe writing it this way will help those who see fraction bars to see the correct problem:
8/2(2+2) can be correctly rewritten as:
8
--- x (2+2) = 4 x (2 + 2) = 4 x 4 = 16
2
Where the x above is the multiplicative operator and not a variable.
It is incorrect to rewrite the original expression as:
8
--------------
2 x (2+2)
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jimhorner
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Generally accepted rules for order of operation conventions accepted and successfully applied by generations of mathematicians, engineers, scientists, and computer programmers. These rules are well documented in multiple references that I and others have linked above. These go back to at least the 16th or 17th century and Descartes and Liebniz would certainly recognize them. Many millions of people have been taught these and apply them successfully. Check any pre-algebra level textbook.
dtuuri
Final Approach
I've been consulting my collection of them. The thing is, they all use the vinculum, which avoids controversy. Even my physics texts. I bet Einstein and Leibniz used 'em too. To me, and a lot of others, 2(2+2) is a single expression of an implied product and as such ought to be simplified first. You would agree, I take it (EDIT: I guess you don't, my mistake), if the expression were explicitly described (*). It's software that can't wrap its mind around implied operations and causes me lots of grief whenever I try to use Excel.
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Here's where I get hung up, and I think maybe dtuuri is as well?
For some reason my mind considers the 2(2+2) as needing to be the first operation. If it had the * in there 2 * (2+2), I would not feel that way. But I have nothing to base that on but my feeling. I feel like the lack of * operator binds the 2 to the parenthesis operation. Again, I have nothing but my feelings to base that on.
I would literally never write it as 2(2+2) unless I wanted that done first, but then again, I'm a programmer by trade and that wouldn't work anyway.
You wouldn’t write 2 2 and assume the * between them. I think the lack of * implies a tight relationship.
Let’s say x = (2+2). If you substitute x into the formula you get 8 / 2x. Would you still divide first? I wouldn’t. And for me it’s due to the lack of the implicit * operator. If you wrote 8 / 2 * x, I would divide first. And then I would wonder if that was what was intended. Lol
JMO
For some reason my mind considers the 2(2+2) as needing to be the first operation. If it had the * in there 2 * (2+2), I would not feel that way. But I have nothing to base that on but my feeling. I feel like the lack of * operator binds the 2 to the parenthesis operation. Again, I have nothing but my feelings to base that on.
I would literally never write it as 2(2+2) unless I wanted that done first, but then again, I'm a programmer by trade and that wouldn't work anyway.
You wouldn’t write 2 2 and assume the * between them. I think the lack of * implies a tight relationship.
Let’s say x = (2+2). If you substitute x into the formula you get 8 / 2x. Would you still divide first? I wouldn’t. And for me it’s due to the lack of the implicit * operator. If you wrote 8 / 2 * x, I would divide first. And then I would wonder if that was what was intended. Lol
JMO
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MauleSkinner
Touchdown! Greaser!
People take shortcuts. People don’t always understand the implications of the shortcuts they take. Or, as I noted earlier, they do understand the implications, and in this case the implication is a controversial result.
The confusion isn't really about the order of operations, it's about the meaning of /, and it is generally accepted that the slash is not the equivalent of the vinculum.
A vinculum is a special symbol that works similarly to parentheses to group numbers and symbols together. Another example of its use would be in a radical expression, such as
Since the vinculum is its own unique symbol, it should be apparent that a slash / cannot be substituted for it. Per Merriam-Webster, a vinculum is ": a straight horizontal mark placed over two or more members of a compound mathematical expression and equivalent to parentheses or brackets about them."
A slash mark / does not have the same meaning and is not equivalent to parentheses.
Exactly. The slant / is not the same as the vinculum. To put the entire expression 2(2+2) in the denominator requires either parentheses or a vinculum.
dtuuri
Final Approach
You mean we can't solve this problem without additional symbols than the original post? "Two people each have 2 red boxes and 2 green boxes and 8 apples between them. If the apples are evenly distributed, how many apples would go in each box?" That's how the OP looks to me, answer is 1ea.
If what you suggest is really the problem statement (pure speculation, btw), you don’t need additional symbols, just the correct ones. Use a vinculum instead of a slash.
Why don't you consult the links I left for you? They explain things quite well.
Software wants things written unambiguously. It's been that way since Charles Babbage and Lady Lovelace, if not prior to them.
Most of us have done so using the rules posted in the thread.
Newton did. (https://archive.org/details/1686-newton-principia-1ed/page/194/mode/2up ). So did Leibniz (https://old.maa.org/press/periodica...nizs-papers-on-calculus-differential-calculus ). Einstein used a mixture. He either implied multiplication or used a dot symbol to represent multiplication. The citation may be confusing to you as he also used partial differentiation operators that appear to be division, but are not. (https://einsteinpapers.press.princeton.edu/vol6-doc/40 ), and he did use a vinculum too.
So do I: https://pubs.acs.org/doi/10.1021/acs.jchemed.0c00437?articleRef=test
I'll note also that all of these are typeset, not written in a blog post, or a thread in an aviation forum.
Try using the Microsoft equation editor for anything more complex than the equations in my paper, or try writing it in LATEX and inserting it here.
We don't use 4 beam beacons anymore- we've moved onto GPS. Likewise, math notation has moved on from sometime before WW2 to allow teletype and computers to show equations.
dtuuri
Final Approach
I don't need it explained, I need validation for how I learned it. It's pretty obvious you need to talk to the computer in computer language. They didn't have them when I first took algebra in 1961, at least in high schools. We were more concerned (well, the teacher was) with historical math principles.
Say you had a fraction in the denominator, like 8/1/2(2+2). I know you invert and multiply, so 8x2(2+2) = 64. If you treat the OP's problem likewise, it looks like this: 8/2(2+2)/1, which resolves to 8x1/2(2+2) = 1. Correct?
Btw, sorry, I can't do vinculums.
flyingcheesehead
Taxi to Parking
For some of us, it wasn't PEMDAS yet, it was "Pretty Please My Dear Aunt Sally". PPMDAS. "Powers" instead of "Exponents" but otherwise the same.
Right answer, but wrong method. There hasn't been much disagreement about the parentheses being first yet in this thread, at least until now. The division does NOT occur first, the parens do:
8/2(2+2)
8/2(4)
4*4
16
(Yeah, it's pretty much the same but if you're gonna do it that way, it's 8/2(2+2) = 4*4 = 16.)
jimhorner
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Agreed about the usual convention of Parentheses first. I've made that point multiple times in this thread with the last being in my post #171. What my example was intended to show was why applying the distributive property first to the 2(2+2) term as [mention]dtuuri [/mention] was trying to do was wrong because that was doing the right multiplication prior to the left division. The only way the distributive property could possibly be used is how I demonstrated. That was the point of my example.
Now, on the other hand, I wouldn't say what I did was wrong either. Math, particularly at the higher levels (Not that I'm saying this simple expression is higher math; it's not) does leave open room for flexibility in correctly applying rules to solve a problem. Integration by parts is one example. ILATE is a good guide to that, but not always.
jimhorner
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When I learned this stuff starting in 7th grade in 1979 or so, there were no mnemonics at all being taught. It was more like this:
The teacher said "learn and memorize this:"
Order of precedence:
1. Parentheses
2. Exponents
3. Multiplication and Division left to right
4. Addition and Subtraction left to right
No mnemonics, just a simple statement of the rules with the instruction to learn it and commit it to memory followed by some in class work at the board with students going up to work problems followed by a large number of homework problems to practice, followed by a test to check. Then we moved on to other things, like solving for x, equations of a line, etc.
Frankly, I think the whole mnemonic thing is unnecessary here. As has been shown abundently in this thread, the mnemonics have added to confusion because some people have taken it to mean that multiplication has precedence over division or addition has precedence over subtraction. Do students really need these mnemonics? Why not just say: "Learn this"?
edit to point out that I'm being a bit hypocritical here in that I referenced a mnemonic (ILATE) in my previous post. In my defence, I haven't integrated by parts since helping my son with his AP calculus ten years ago, and all I remembered at the time from my own calculus taken in 1983 was the mnemonic ILATE. I couldn't tell you what it stands for or how to apply it...
Turns out Electrical Engineers (at least for me as an analog chip designer in the early part of my career) use a lot of Calculus in college to explain concepts, not so much in the real world where numerical integration and differentiation tend to be the rule.
I also remember low-dee high - high dee low all over low squared. That's, what, how to take the derivative of a quotient? Good thing I'm not paid for my calculus skills, they're rusty.
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flyingcheesehead
Taxi to Parking
Because mnemonics help you learn faster, and if you can learn faster and have a better way to commit it to memory, you can learn more.
I remember my dad telling me that stuff I learned in middle school and high school is stuff that he didn't do until college. My second grader is now doing things I don't think I did until fifth grade. As time goes on, better methods of teaching are found and used.
Point taken about this particular mnemonic, though. You have to be paying enough attention to remember that MD and AS are of equal precedence.
I think all of the oxygen I've breathed since calculus has been used to oxidize my calc skills.
Google tells me that your thing is in fact the quotient rule so maybe you're not as rusty as you think.
AV8R_87
Cleared for Takeoff
I'm on your side on this one. Valid for all the stuff we do on airplanes as well. A better understanding of aircraft systems makes you remember which ones are required better than ATOMATOFLAMES.
jimhorner
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Not correct.
Maybe stop thinking of these problems as fractions with denominators and think instead in terms of dividends and divisors. Same thing, I know, but the whole fraction thing is throwing you off. It seems you think the problem you wrote ( 8/1/2(2+2) ) is this:
Is that right? If so, then that is not the correct way to interpret the linear form of the expression. To get that, you would have to write it this way:
8/(1/(2(2+2))).
Try both in excel:
= 8/1/2*(2+2)
=8/(1/(2*(2+2)))
the first will give 16, the 2nd will give 64.
Instead your new expression is this (as fractions):
For some reason it seems you're strongly binding the 2(2+2) together as a single term. That's not correct. 2(2+2) = 2 x (2 + 2) or in most programing languages 2*(2+2). If that needed to be bound together as a term, it would have to have parens around it: (2(2+2)).
8/(2(2+2)) can be thought of as a fraction with the (2(2+2)) being the denominator. 8/2(2+2) isn't that. The only denominator in this, original problem, is the 2 following the /.
Try writing it with the old fashioned multiply and divide signs: (I can't insert an obleus divide sign so I'm using a hyphen colon hyphen instead) I'll use your new expression instead of the one in the original problem.
8/1/2(2+2) = 8 -:- 1 -:- 2 x (2+2)
Applying the rules:
parens first
8 -:- 1 -:- 2 x 4
now multiplication and division left to right:
First dividend is 8, divisor is 1 (8 -:- 1 = 8)
8 -:- 2 x 4
Now dividend is 8, divisor is 2 (8 -:- 2 = 4)
4 x 4
Now do the multiply
16
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I don't understand. That seems like asking for validation that Christopher Columbus discovered the Americas. Never mind that it was already discovered- the people that were living here would be surprised by that statement. The Vikings also beat him to it. Yet how many of us were taught that Columbus discovered America? We even have a day commemorating that "discovery". You'd do better by looking at the links and learning how to do it properly.
The left-to-right notation has been taught since at least 1912:
Earliest Uses of Symbols of Operation
mathshistory.st-andrews.ac.uk
This seems to contradict this statement:
Also, you never did a square (or other) root? The vinculum is that bar above the expression being operated upon. √2+x is different from √(2+x) (I can't do the vinculum here, so I used parentheses. I can do it in word and other software). In another linear notation, I'd do (2+x)^0.5 or (2+x)**0.5.
Or do you mean that you can't do them in this thread (as per my note above)? If so, that's one reason we use the linear notation and the order of operations.
dtuuri
Final Approach
The expression 2(2+2) is welded together in my mind as a single value because of some long-ago math training. I'm not alone in this thread on that point, although feeling a bit abandoned at the moment. I'm looking for the source of it. You all miss the point that we understand what you do these days and why you do it, but it doesn't make it easy for us old-timers who learned these rules before they invented battery calculators.
That's a pretty interesting history, but I suspect the author isn't much fun at parties. Left-to-right is non-controversial, so I don't see the need to discuss it.The left-to-right notation has been taught since at least 1912:
Earliest Uses of Symbols of Operation
That I have textbooks that only use vinculums yet I myself can't do them on a computer or in a forum or in, say, Excel is a contradiction? How so? I use the required syntax in Excel, albeit with much misery. As far as I know, vinculums are not possible.
Can't do them except by hand.
jimhorner
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What your actually doing here is incorrectly evaluating the multiplication and division right to left instead of left to right.
Your new expression:
incorrectly Right to left:
8/1/2(2+2)
8/1/2(4)
8/1/8
8/0.125
64
correctly left to right:
8/1/2(2+2)
8/1/2(4)
8/2(4)
4(4)
16
Original problem:
incorrectly right to left:
8/2(2+2)
8/2(4)
8/8
1
correctly left to right:
8/2(2+2)
8/2(4)
4(4)
16
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dtuuri
Final Approach
I have NO trouble getting 64 the first way — the cleaner, simpler looking way.Not correct.
Maybe stop thinking of these problems as fractions with denominators and think instead in terms of dividends and divisors. Same thing, I know, but the whole fraction thing is throwing you off. It seems you think the problem you wrote ( 8/1/2(2+2) ) is this:
Is that right? If so, then that is not the correct way to interpret the linear form of the exoression. To get that, you would have to write it this way:
8/(1/(2(2+2))).
Try both in excel:
= 8/1/2*(2+2)
=8/(1/(2*(2+2)))
the first will give 16, the 2nd will give 64.
Instead your new expression is this (as fractions):
For some reason it seems you're strongly binding the 2(2+2) together as a single term. That's not correct. 2(2+2) = 2 x (2 + 2) or in most programing languages 2*(2+2). If that needed to be bound together as a term, it would have to have parens around it: (2(2+2)).
<snip>
jimhorner
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Yea, but that equation is not the appropriate way to interpret the linear equation. It's simply wrong.