8 / 2 (2 + 2) =

[Flatiowa]

The fact that this thread has gone on this long makes me understand why I can’t have a plutonium powered thermoelectric generator unit to power my home and a hydroponic greenhouse…, and why fusion power plants are AlWAYS just 50 years away

 

[MauleSkinner]

I was issued my HP-15C in 1984 at USMA.

Two years behind my brother.

Not that that’s worth much.

 

[jimhorner]

Fortunately, a (or 3) isn't zero. If it were, then I'd invoke the 11th Commandment - Thou Shalt Not Divide by Zero

If a were 0, it wouldn't be a 2nd order polynomial, and the quadratic equation wouldn't be needed.

bx + c = 0 => x = -c/b

 

[MauleSkinner]

"where's an English major when you need one?"

In this case, you want a technical writer, not an English major.

 

[Initial Fix]

In this case, you want a technical writer, not an English major.

Hey now ….. I married a tech writer (who bailed on engineering after calc 1). Many have met her at RR

But back to the math problem. It’s just a poorly written equation syntactic wise. As written, and without context, we just don’t know the intent to discern the answer.

 

[murphey]

If a were 0, it wouldn't be a 2nd rder polynomial, and the quadratic equation wouldn't be needed.

bx + c = 0 => x = -c/b

Very true, it’d be the def of a straight line.

 

[MauleSkinner]

Hey now ….. I married a tech writer (who bailed on engineering after calc 1). Many have met her at RR

But back to the math problem. It’s just a poorly written equation syntactic wise. As written, and without context, we just don’t know the intent to discern the answer.

There ya go..ask her.

Problem with English majors is they like to use multiple different terms for the same thing. Probably an English major that wrote the original equation.

(And FWIW, I was stubborn enough to wait until I failed Diff Eq.)

 

[jimhorner]

But back to the math problem. It’s just a poorly written equation syntactic wise. As written, and without context, we just don’t know the intent to discern the answer.

It's not poorly written, though. It's very parsable with standard, commonly understood syntax rules of math that have been valid for hundreds of years. Any engineer, math major, or scientist would agree on the interpretation.

As a data point, I showed the problem to my 91 year old father (Mechanical Engineer), 86 year old mother (8th grade math teacher), and wife (Masters degree in Electrical Engineering) All of them, with no prompting from me ( Also Masters in Electrical Engineering), correctly identified the answer as 16 with only a couple of seconds mental computation each. The rules are clear and unambiguous, as is the problem.

 

[AV8R_87]

Would you agree then that if a horizontal line was used instead of a slant line, the solution would be 1?

Yes

I interpret the slant line to be equivalent to a horizontal line with everything past the slant under it, unless otherwise bracketed.

That's the error. If the slant line applied to the whole expression after, and not just the first term, the whole expression would be in a set of parantheses or square brackets.

This reminds me of those grammar exercises where a comma, or lack of a comma, changes the whole meaning.

Let's eat grandma
vs
Let's eat, grandma

 

[Half Fast]

I interpret the slant line to be equivalent to a horizontal line with everything past the slant under it, unless otherwise bracketed.

That's the error. If the slant line applied to the whole expression after, and not just the first term, the whole expression would be in a set of parantheses or square brackets.

I think this might be clarified a bit by considering why we use the slant / for division.

Back when the earth was young and dinosaurs roamed the planet and I was an engineering student in college, we typed our reports on simple typewriters, not computer word processors. I doubt you young'uns remember those, so you'll just have to trust me a bit. Unless you had an IBM Selectric with interchangeable typeface balls, you had no way to type mathematical symbols. Plus and minus and multiply (+, -, x) were no problem, but for a divide symbol ÷ you had to type a colon : then back up and type a dash - through the middle of it.

That was a PITA and seldom looked good, so using a slash / was an alternative and was treated like a ÷ symbol. If we wanted a long expression in the denominator, we would use two lines, with the numerator underlined (which was also a PITA) and the denominator typed below it. If we needed to fit it onto a single line, we'd use the / but with parentheses to show the denominator clearly.

Since the / was simply an alternative for ÷, the equation in question could be written as

8÷2x(2+2)​

using conventional symbols.

We then evaluate it by solving the parenthetical expression first, then since only division and multiplication remain, we proceed from left to right.

8÷2x(2+2) = 8÷2x(4) = 4X4 = 16.​

​

Multiplication and division are equivalent operations so there's no priority of one over another. We just proceed left to right.

This is exactly how all the computer languages I've used will evaluate it and it's how RPN will handle it. It's even how Excel evaluates it.

​

BUT, this is all rather ambiguous and can be confused easily. In any serious technical paper we would use brackets and parentheses to eliminate the possibility of error.

 

[Half Fast]

Let's eat grandma
vs
Let's eat, grandma

I sorta like, "I shot an elephant in my pajamas." WTH the elephant was doing in my pajamas I haven't a clue.

English can be terribly ambiguous, but in everyday matters we usually have a priori knowledge of the subject matter that makes the correct meaning clear. That's seldom the case in mathematics.

 

[Rene]

. . . . They certainly write/read right to left. Kinda makes my mind explode.

How about intermediate Ancient Greek written in boustrophedon style. One row right to left, the next left to right, then right to left again.

Efficient, I guess.

 

[Rene]

No the correct answer is always 42

So says Deep Thought.

 

[Rene]

This reminds me of those grammar exercises where a comma, or lack of a comma, changes the whole meaning.

Lets eat grandma.
Lets eat, grandma.

Commas save lives.

 

[Rene]

RPN is a much more efficient way of using a calculator, isn't it?

We had an old mechanical adding machine at the hardware store that worked somewhat the same way. If you bought eight $0.59 bulbs, and four $1.29 pots, I'd punch [50][9][8][x][1][20][9][4][x][=], then look up the sales tax on a paper card, say $0.62, and add it [60][2][+][=].

 

[Rene]

I thought it might be useful to discuss this whole topic in a bit more detail to explain the "why" about how the convention is that multiplication and division have equal precedence. Feel free to skip this and move on to something more exciting. I've been a math geek at least as early as when my 7th grade math teacher, Mrs. Sweeny, taught algebra to me in 1979.

It's well known that multiplication and division are inverse operators and are really two sides of the same coin. Dividing by 10 is equivalent to multiplying by 0.1. Multiplying by 1.25 is the same as dividing by 0.8, etc. 1/2 = 0.5, 3/4 = 0.75 etc.


So, for example: 2/2 = 2*0.5 = 1

Since multiplication and division are so closely related, mathematicians over the centuries have determined that the best convention is to treat the two operators as equivallent in precedence.

Take, for example 2 / 2 / 2

The convention is to evaluate from left to right:

( 2 / 2 ) / 2 = 1 / 2 = 0.5

If the convention had been defined to evaluate right to left it would be:

2 / ( 2 / 2) = 2 / 1 = 2
But that's not the agreed upon convention. It's left to right.

Now since dividing by 2 is the same as multplying by 0.5, the above could alternatively be written as

2 / 2 * 0.5 or 2 * 0.5 * 0.5 or 2 * 0.5 / 2

Using the left to right convention with multiplication having equal precedence, all of these evaluate to 0.5

If multiplication is given precedence over division or the evaluation direction is different, then different answers are found.

2 / 2 / 2 = 0.5 (correct according to convention when evaluated left to right)

2 / 2 / 2 = 2 (when evaluated right to left)

2 / 2 * 0.5 = 2 (when multiplication is given precedence)

2 * 0.5 * 0.5 = 0.5 (same answer evaluated either direction)

2 * 0.5 / 2 = 0.5 (same answer regardless of order or precedence)

So, there are different answers possible with basically the same calculation, and that leads to confusion and ambiguity. In order to deal with this, people have developed a set of syntax rules for the language of math that are generally agreed upon. A different set of rules could have been developed, but that's not what was done. What has been agreed upon generally is that for written equations (and most but not all computer languages), infix notation is used primarily, with multiplication and division having equivalent precedence and evaluated left to right. Addition and subtraction also have the equivalent precedence (lower than multiplication and division) and are also evaluated left to right. In the written form, multiplication is understood to be implied when writing dissimilar types adjacent to each other as in 2x, (x - 3)(x + 4), 2(2 + 2), etc. Most programming languages don't parse that correctly requiring: 2*x, (x-3)*(x+4), 2*(2+2).

Note, that there are other conventions used in some places.

The programming language Lisp uses prefix notation. In Lisp, 8 / 2 (2+2) would be written as: (* (/ 8 2)(+ 2 2))

In Reverse Polish Notation (RPN), a postfix way of calculating, 8 / 2(2 +2) would be calculated as

8 2 / 2 2 + *

But those are somewhat special cases. Most programming languages (python for example) write it this way:

8 / 2 * ( 2 + 2 )
Python evaluates the above as 16,

Excel: =8/2*(2+2)
Excel also evaluates that as 16

You quite obviously would have been a far better math teacher than my school has ever seen. Much to your benefit.

 

[Rene]

Please consider losing the parochialism . . .

"The system of measure formerly known as English"?
Perhaps we could license this symbol.

 

[Rene]

My post:

One. And I literally just completed high school.

Was not meant to say that I am an authority. Rather it was a response to:

It’s 1 and I didn’t learn order of operations 100 years ago.

And pointing out that in addition to the article saying that it was taught that way 100 years ago, it has been continuously taught that way up through a year or two ago and probably will be taught to students next year as well.

It shouldn't surprise me that I was taught "wrong math", it goes perfectly with the "wrong science", "wrong history", and "wrong grammar" that I had already discovered.

 

[murphey]

I think this might be clarified a bit by considering why we use the slant / for division.

Back when the earth was young and dinosaurs roamed the planet and I was an engineering student in college, we typed our reports on simple typewriters, not computer word processors. I doubt you young'uns remember those, so you'll just have to trust me a bit. Unless you had an IBM Selectric with interchangeable typeface balls, you had no way to type mathematical symbols. Plus and minus and multiply (+, -, x) were no problem, but for a divide symbol ÷ you had to type a colon : then back up and type a dash - through the middle of it.

That was a PITA and seldom looked good, so using a slash / was an alternative and was treated like a ÷ symbol. If we wanted a long expression in the denominator, we would use two lines, with the numerator underlined (which was also a PITA) and the denominator typed below it. If we needed to fit it onto a single line, we'd use the / but with parentheses to show the denominator clearly.

Since the / was simply an alternative for ÷, the equation in question could be written as

8÷2x(2+2)​

using conventional symbols.

We then evaluate it by solving the parenthetical expression first, then since only division and multiplication remain, we proceed from left to right.

8÷2x(2+2) = 8÷2x(4) = 4X4 = 16.​

​

Multiplication and division are equivalent operations so there's no priority of one over another. We just proceed left to right.

This is exactly how all the computer languages I've used will evaluate it and it's how RPN will handle it. It's even how Excel evaluates it.

View attachment 136577​

BUT, this is all rather ambiguous and can be confused easily. In any serious technical paper we would use brackets and parentheses to eliminate the possibility of error.

Actually, it was due to Western Union teletypes. Hence everything was uppercase.

 

[Half Fast]

Actually, it was due to Western Union teletypes. Hence everything was uppercase.

Not surprising, but I also wouldn’t be surprised to learn it began with printing presses and limited type characters.

The point is that / is merely an alternative to ÷, and neither one means that everything following is part of the divisor.

 

[dtuuri]

The point is that / is merely an alternative to ÷,

You mean, it's not an alternative to the vinculum?

EDIT: Rather than wait for an answer, I went searching and found a nice proof that the answer is NOT 16 but 1. Here's the main point:

"The expression to the right of the division sign must be processed as and simplified to an individual, inseparable term, because it is in the form of a Distributive Property expression. It has parentheses after all, so it must be dealt with before being divided into 8."​

8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough

I thought I’d take a stab at explaining the current viral math problem.

sundaymorninggreekblog.com

Doesn't mention "vinculum", but the parentheses seem to call for the same treatment under the Distributive Property as if the slash were a vinculum.

 

[Half Fast]

You mean, it's not an alternative to the vinculum?

Right. It's not.

 

[Cap'n Jack]

"The system of measure formerly known as English"?
View attachment 136584 Perhaps we could license this symbol.

You're the one that called it "American measurement".

The only people who have been to the moon used "American" measurement.

It was always "English". Also, the post seemed to show a disdain for other ways of doing things (the metric system). I'm "fluent" in both and use either one as needed. Please forgive me if I misunderstood.

It shouldn't surprise me that I was taught "wrong math", it goes perfectly with the "wrong science", "wrong history", and "wrong grammar" that I had already discovered.

I'm curious about what you mean by "wrong science" and "wrong history". Please don't reply about those in this thread because both science and history have become politicized in recent years. You may reply to me via PM if you wish as I don't want to politicize the thread, or close it for politics. That you knew the item quoted below suggests they taught you history better than you may think. Alternatively, you read a bit outside of classes:

How about intermediate Ancient Greek written in boustrophedon style. One row right to left, the next left to right, then right to left again.

 

[Cap'n Jack]

You mean, it's not an alternative to the vinculum?

EDIT: Rather than wait for an answer, I went searching and found a nice proof that the answer is NOT 16 but 1. Here's the main point:

"The expression to the right of the division sign must be processed as and simplified to an individual, inseparable term, because it is in the form of a Distributive Property expression. It has parentheses after all, so it must be dealt with before being divided into 8."​

8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough

I thought I’d take a stab at explaining the current viral math problem.

sundaymorninggreekblog.com

Doesn't mention "vinculum", but the parentheses seem to call for the same treatment under the Distributive Property as if the slash were a vinculum.

I understand where they are coming from in that citation. However, like the rest of language, things have evolved. We have computer languages, listed earlier in the thread, that resolve the ambiguity using a left-to-right order. The citation cites Wolfram where they mention that they will evaluate from left-to-right the equation a/bc to be (a/b)*c rather than a/(b*c), which is a tacit acknowledgement of how things work now. As mentioned by others, over use of parenthesis to ensure clarity is better. Alternatively, although it is a pain to use, I'll use the equation editor in word and use the vinculum and other symbols as needed. Wolfram software can also use the formally written equation and their equation editor is easier to use.

 

[Matthew]

I did a lot of math in my career working on embedded systems.

Ambiguity in an equation like a/bc could be a real headache, especially years later when taking over some legacy project written by someone long gone from the company.

Write it out with parentheses or do something like “x = bc; y = a/x”, or “x = a/b; y = xc”. I always tried to avoid multiple operations on the same line, mainly because some of the equations could get pretty complex. Say what you mean. It isn’t that hard.

 

[TCABM]

…The citation cites Wolfram where they mention that they will evaluate from left-to-right the equation a/bc to be (a/b)*c rather than a/(b*c), which is a tacit acknowledgement of how things work now.…

Ambiguity results in interpretation and, just maybe, consensus among like-minded individuals. That consensus does not remove the ambiguity, nor does it render other interpretations wrong.

am·bi·gu·i·ty
noun
the quality of being open to more than one interpretation; inexactness.

 

[Half Fast]

However, like the rest of language, things have evolved.

True, and it's not even a recent evolution. This goes way back. All the calculators I've tried, both RPN and AOS, give 16 as the result. Programming languages have worked this way for decades. As an undergrad back in the 1980s (and yes, I have a degree in mathematics) I would never have interpreted / as a vinculum and neither would my professors.

At bottom, this isn't really a mathematics question. It's a technical writing style question, and therefore subject to a consensus opinion. The consensus, as evidenced by the great majority of programming languages and calculators and spreadsheets, is that the correct interpretation yields 16 as the correct answer.

As in all writing, though, clarity is important. Proper use of parentheses and brackets can eliminate ambiguity.

 

[dtuuri]

Right. It's not.

After perusing some of the comments in the blog I cited, there were actually 43 mentions of "vinculum". I think the author gives a competent defense of Rene's teacher's teaching, btw. One of the commenters made this point, which is how I see it:

"When any division statement is read aloud, the words, “divided by,” separate the numerator from the denominator, regardless of which division symbol is used (obelus, solidus or vinculum). Therefore, the statement​

​

60 ÷ 5(1+1(1+1))​

​

is properly written as the fraction…​

​

60​

______________​

​

5(1+1(1+1))​

​

…which has a quotient of 4."​

​

Comment by Dee — March 12, 2024 @ 10:22 am​

 

[Cap'n Jack]

Ambiguity results in interpretation and, just maybe, consensus among like-minded individuals. That consensus does not remove the ambiguity, nor does it render other interpretations wrong.

Why did you think we didn’t know what ambiguity means, that you had to post a definition?

Consensus does resolve ambiguity since there is agreement on how to read a mathematical equation. In the example that started this thread, the values of 1 and 16 are very different. Different enough to have consequences in real life.

 

[MauleSkinner]

After perusing some of the comments in the blog I cited, there were actually 43 mentions of "vinculum". I think the author gives a competent defense of Rene's teacher's teaching, btw. One of the commenters made this point, which is how I see it:

"When any division statement is read aloud, the words, “divided by,” separate the numerator from the denominator, regardless of which division symbol is used (obelus, solidus or vinculum). Therefore, the statement​

​

60 ÷ 5(1+1(1+1))​

​

is properly written as the fraction…​

​

60​

______________​

​

5(1+1(1+1))​

​

…which has a quotient of 4."​

​

Comment by Dee — March 12, 2024 @ 10:22 am​

But where does “read aloud” show up in mathematical rules? It’s still got the ambiguity of the original formula unless you verbally define the parentheses.

 

[Cap'n Jack]

After perusing some of the comments in the blog I cited, there were actually 43 mentions of "vinculum". I think the author gives a competent defense of Rene's teacher's teaching, btw. One of the commenters made this point, which is how I see it:

"When any division statement is read aloud, the words, “divided by,” separate the numerator from the denominator, regardless of which division symbol is used (obelus, solidus or vinculum). Therefore, the statement​

​

60 ÷ 5(1+1(1+1))​

​

is properly written as the fraction…​

​

60​

______________​

​

5(1+1(1+1))​

​

…which has a quotient of 4."​

​

Comment by Dee — March 12, 2024 @ 10:22 am​

None of those people have any more expertise in math than the rest of us. I get 36 from 60/5*(1+1*(1+1)), unless the spaces you placed around the division symbol have meaning. If so, please explain that to us.

Saying a math equation and writing it should give the same result, and parenthesis should be used as needed.

 

[TCABM]

Why did you think we didn’t know what ambiguity means, that you had to post a definition?

Consensus does resolve ambiguity since there is agreement on how to read a mathematical equation...

Only among like minded individuals. For the rest of the non-programming language (people not trained to resolve ambiguity in exactly one manner), the equation is still ambiguous, something not everyone appears to accept.

 

[MauleSkinner]

Only among like minded individuals. For the rest of the non-programming language (people not trained to resolve ambiguity in exactly one manner), the equation is still ambiguous, something not everyone appears to accept.

You’re saying that even the ambiguity is ambiguous?

 

[Rene]

You're the one that called it "American measurement".

It was always "English". Also, the post seemed to show a disdain for other ways of doing things (the metric system). I'm "fluent" in both and use either one as needed. Please forgive me if I misunderstood.


I'm curious about what you mean by "wrong science" and "wrong history". Please don't reply about those in this thread because both science and history have become politicized in recent years. You may reply to me via PM if you wish as I don't want to politicize the thread, or close it for politics. That you knew the item quoted below suggests they taught you history better than you may think. Alternatively, you read a bit outside of classes:

To my knowledge, only the USA still uses the system of weights and measures developed by the English. So, by default (abandonment) it is the "American" system. "(F)ormerly known as English" was just my being cheeky. (Which is harder to tell online.)

My public school taught me in the sense that it physically shared a building with the public library (thank you Mr. Carnegie) and I was usually (so long as I was quiet) allowed to sit in the back of class - while the teach repeated him or herself 35 times - and read whatever I wanted to. Many of the boys who had IQs over 100 were diagnosed with ADD because they couldn't stand the boredom without making noise.

 

[Cap'n Jack]

To my knowledge, only the USA still uses the system of weights and measures developed by the English. So, by default (abandonment) it is the "American" system. "(F)ormerly known as English" was just my being cheeky. (Which is harder to tell online.)

Liberia and Myanar also use the English system. Emojis help to tell that you are being cheeky. If you wanted to be correct, it would be the "American customary system" as there are differences from the imperial system, mostly in volume.

My public school taught me in the sense that it physically shared a building with the public library (thank you Mr. Carnegie) and I was usually (so long as I was quiet) allowed to sit in the back of class - while the teach repeated him or herself 35 times - and read whatever I wanted to. Many of the boys who had IQs over 100 were diagnosed with ADD because they couldn't stand the boredom without making noise.

Assuming this is true, they simply didn't teach you anything. They may well have taught you the "right" history, science, and grammar, but by your own admission, you were reading rather than listening to the teacher(s)- I assume you had more than one teacher. They were all bad?

 

[Cap'n Jack]

Only among like minded individuals. For the rest of the non-programming language (people not trained to resolve ambiguity in exactly one manner), the equation is still ambiguous, something not everyone appears to accept.

@Salty, @Half Fast , and I are hardly "like minded". Just look at our conversations here. I doubt we are "trained to resolve ambiguity in exactly one manner". We simply apply conventions that some people were taught incorrectly, simply haven't learned, or forgot. Our spelling is the result of "consensus" as well as how we use word ordering in the USA. Making things more complicated is the need for writing equations in a linear fashion on texts, tweets, or fora such as this where it is difficult to write equations in a non-linear fashion. Non-programmers do use calculators and spreadsheets, don’t they?

 

[Cap'n Jack]

I think the author gives a competent defense of Rene's teacher's teaching, btw.

I don't think so. Please see what she was taught below.

I'm not saying this is empirically correct. But, along with every student in my district, I was taught that each operation has a separate order of importance:
Parenthesis,
Exponents,
Multiplication. . . so it is above division,
Division . . . so it is below multiplication,
Addition,
Subtraction.

As stated by others, multiplication and division are equivalent in precedence. Your citation doesn't apply to these comments.

 

[dtuuri]

As states by others, multiplication and division are equivalent in precedence. Your citation doesn't apply to these comments.

I was referring to her conclusion, as taught, that the answer is "1". Btw, I didn't add any spaces anywhere. I pasted from the blog. In fact, I removed a space from the numerator that made it visually render too far removed from the rest of the fraction.

 

[Mikey52]

And so for the reason of crashing things vs landing things.

 

[Sac Arrow]

We had an old mechanical adding machine at the hardware store that worked somewhat the same way. If you bought eight $0.59 bulbs, and four $1.29 pots, I'd punch [50][9][8][x][1][20][9][4][x][=], then look up the sales tax on a paper card, say $0.62, and add it [60][2][+][=].

That sounds like something you would find in a Canadian farming community.