I want to close this series with a topic that arises constantly, both in classrooms and on social media: How do you evaluate an expression like \(a\div bc\) or \(8\div 4(3-1)\), where the multiplication is indicated without a specific symbol? There are several reasons one might want to interpret this differently than the rule we’ve discussed, that multiplication and division are done from left to right. We’ll look at this first from the perspective of students and teachers, and then (next time) investigate some historical issues to close out the series.
Two ways to evaluate ax÷by
Let’s first look at one of the earlier questions we had about this issue, in 1999, to set the stage:
Order of Operations The problem was presented like this: a = 1.56 b = 1.2 x = 7.2 y = 0.2 ax/by = ? Here are two ways that I solved it: 1) I first rewrote the problem as [1.56(7.2)/ 1.2](0.2). Second, a was multiplied by x. The product was 11.232. Then, since no parentheses were present, I followed the order of operations and divided 11.232 by b, which was 1.2. The quotient was 9.36. Then I multiplied 9.36 by y, which was 0.2. The final answer was 1.872. 2) The other way, the first thing I did was multiply a by x. The product, which was 11.232, was set aside for the time being. Then b was multiplied by y, which gave the product of 0.24. The problem was now solved by dividing 11.232 (or ax) by 0.24 (or by) to reach a final answer of 46.8. Can you please tell us which answer is correct and why?
(Note that at that time, the only way to type division in our email was to use the slash, \(a/b\), which I generally assume represents an expression actually written as \(a\div b\). I will occasionally be inserting an obelus, ÷, where we made rough attempts to simulate it.)
The first way follows PEMDAS literally, as usually taught and as I’ve presented it here, by evaluating from left to right as \(a\cdot x\div b\cdot y = ((a\cdot x)\div b)\cdot y\).
The second sees it as \(ax\div by = (ax)\div (by)\). This isn’t explained as following any taught rule, but just as doing what looks right, either because the division is read as if it were a fraction bar, or just because “by” looks like it belongs together as a unit. We’ll be seeing several reasons students have given for doing this.
Though I had been with Ask Dr. Math less than a year, this was already a familiar question, which I wanted to answer thoroughly for the sake of the archive:
You are not alone in wondering about this. We have had several other questions about expressions similar to yours, from confused teachers and students who have found that different books or teachers have different answers, and even calculators disagree.
Note that it is not only students doing what feels right, but also some textbooks and calculators that follow the second method.
A new rule, or what looks right?
I elaborated on the two methods, taking the PEMDAS version as correct (though I’ll have some second thoughts on that):
As written, your expression
ax/by
should be evaluated left to right: a times x, divided by b, times y. The multiplication is not done before the division, but both are done in the order they appear. Your first solution is right.
Some texts make a rule, as in your second solution, that multiplication without a symbol ("implied multiplication") should be done before any other operations in an expression [except exponents], including "explicit multiplication" using a symbol. Following this rule, you would multiply a by x, then multiply b and y, then divide one by the other. Some (probably most) texts don't mention such a rule - but some of those may use it without saying so, which is far worse.
I think I had made up the term “implied, or implicit, multiplication” when I answered my first question on the topic a few months before, to refer to multiplication indicated by just putting two numbers or variables or parenthesized expressions next to one another – “juxtaposition“, as others call it – like \( ab\) or \( 2b\) or \( a(b+c)\), as opposed to explicitly writing \( a\times b\) or \( a\cdot b\).
We had seen some questions from students whose textbooks taught only the usual PEMDAS, yet evaluated the second way in examples or solutions, without comment. This might have been due to the answers in the back being written by someone other than the author, but it is an inexcusable inconsistency.
Why would an author make this extra rule? I have had different opinions at various times about whether the rule is a good idea, but have always recognized that it is not what is usually taught:
I don't know of a general rule among mathematicians that implied multiplication should be done before explicit multiplication. As far as I'm concerned, all multiplications fit in the same place in the order of operations. It's not an unreasonable rule, though, since it does seem that implied multiplication ties the operands together more tightly, at least visually; but the idea of Order of Operations (or precedence, as it is called in the computer world) is supposed to be to ensure that everyone will interpret an otherwise ambiguous expression the same way - so if some texts change the rules, or if people do what feels natural, the purpose has been lost.
A rule that is not a rule is worthless, no matter how reasonable it is. Yes, the “new rule” is the natural way to read \(ax\div by\) because \(by\) looks like a single entity; but until everyone teaches that, we can’t do it and expect to be understood by all readers.
In particular, many students assume that it represents a horizontal version of \(\displaystyle\frac{ax}{by}\):
The problem here is that the expression looks as if it were meant to be
ax
----
by
In the Dr. Math FAQ about writing math in e-mail, one of our recommendations is to use parentheses wherever possible to avoid ambiguity, even where the rules should make it clear, because it can be easy to forget them in some situations.
So in e-mail we would write it like this:
ax/(by) or (ax/b)*y
depending on what is intended.
By using parentheses, we can avoid writing something that people who were taught different rules, or who ignore the rules they were taught, might take differently than we intend.
Calculator issues
In my research for another Dr. Math "patient," I found that some calculators have experimented with this rule. Calculators have somewhat different needs than mathematicians, since they have to take input linearly, one character after another, so they are forced to make a decision about it. On the TI Web site I learned that they deliberately put this "feature" into the TI 82, and then took it out of the TI 83, probably because they decided it was not a standard rule and would confuse people.
The link there went bad long ago; but when a specific question about a calculator came up in 2008, I quoted from what TI said in their Knowledge Base:
Implied Multiplication and TI Calculators ... Solution 11773: Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators. Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators? Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper. This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Handheld in TI-84 Plus Mode. Implied and explicit multiplication are given the same priority.
This makes it clear that calculator designers have to decide on their own rules, which don’t have to be the same as rules for writing on paper; but educators seem to have convinced them to keep things as much the same as possible for students’ sake.
In conclusion (back to the 1999 answer):
So to answer your question, I think both answers can be considered right - which means, of course, that the question itself is wrong. I prefer the standard way (your first answer) when talking to students, unless their own text gives the "implicit multiplication first" rule; but in practice if I came across that expression, I would probably first check where it came from to see if I could tell what was intended. The main lesson to learn is not which rule to follow, but how to avoid ambiguity in what you write yourself. Don't give other people this kind of trouble.
Subsequently, we had many more questions about this; I’ll just quote a few unique bits from some of those answers.
Old-fashioned math?
Here is a typical example of a school conflict, from 2000:
Order of Operations Dispute The problem reads: N ÷ ml where n=12, m=6, and l=3. I believe the correct answer should be .6666, as 12 divided by 18 equals this. My husband agrees with me. My son came home very upset from school, with a note from his teacher that the answer was wrong. She indicated that I should have divided the 6 (m) into 12 (n) before I divided the 3 (l) into the equation. Her answer was 6. My son is very upset with me; his teacher told him I was doing "old fashioned math." Do I need to go back to school?
The problem is \(N\div ml\), and the parents are doing the multiplication first. I replied, in part:
I can give you some good news and some bad news. First, the bad news: according to the usual order of operations rules now taught, your answer is wrong. ...
I explained the standard rules, and added:
BUT... You are not alone in your opinion. This part of the rule - doing multiplication and division together - is probably the last rule to have stabilized; I know that in the 1920's, at least, there was no agreement. It seems that an agreement developed, but it is unraveling now, as I hear from many students whose texts answer questions like this the way you did. It appears that they are adding an unstated rule, which seems entirely reasonable in this context, that an implied multiplication (indicated by simply placing two variables or expressions together, as in "ml") should be done first. It certainly looks as if it should mean that. The problem is that, although I've heard of this rule being followed frequently, I've hardly ever heard of it being taught, so these texts are not following their own stated rules.
I’ll have more on the history next time.
Since this type of expression is so ambiguous, with people disagreeing on the rules, and the rules being easy to overlook, my own opinion is that neither your answer nor the teacher's is right: the question is wrong. No responsible mathematician would write such an expression; we would just say
n
---
m l
so there would be no question about its meaning. After all, the purpose of rules is to allow us to communicate clearly, not to help us trick students and start fights among families.
So you may in fact be "old-fashioned"; or you may be on the cutting edge. In any case, I'm afraid you'll just have to learn how they are doing it in class, and follow along. There shouldn't be many more issues like this to worry about.
More recently, the fights tend to be on social media!
Misapplying the distributive property
I’ll close with the most recent archived discussion. This question is from 2017:
Even More on Order of Operations I'm curious to know what the answer is for this: 8/4(3 - 1) Following strictly PEMDAS, the answer is 4: 8/4(2) 2*2 4 However, if you follow the distributive property, you get 1: 8/((4*3) - (4*1)) 8/(12 - 4) 8/8 1 Which one would be correct and why? Both are valid, so I'm conflicted as to what would be the correct answer. It should be right or wrong, not two different answers being right.
I answered with a collection of my standard answers to this sort of question; even my first archived answer on the topic in 1999 was largely a standard response I had given to others before. Here, I’ll just look at a few points I made that haven’t been fully covered above.
I first summarized what was going on:
The problem is not a conflict between PEMDAS and distribution; it is that strict interpretation of PEMDAS conflicts with one's natural impression of the meaning of the expression, so that you unknowingly apply an alternative interpretation when you think you are just applying the distributive property.
If you recall earlier statements that PEMDAS is (a) in harmony with the properties of operations, and (b) fitting with the visual impression of our notation, then some alarm bells should already be going off!
When you distributed, you ASSUMED that it was the 4, not 8/4, that was multiplying the (3 - 1). In doing so, you were bypassing the rules and just doing what felt right. If you followed the rules AND distributed, you would get this: (((8/4)*3) - ((8/4)*1)) ((2*3) - (2*1)) 6 - 2 4
It is not really the distributive property that led to the “wrong” result, but the fact that in distributing, the 4 was seen as the multiplier.
Those who say you should distribute first are putting the cart before the horse: you can't apply tricks to evaluate an expression before you first know what it MEANS, but they are thinking that the distributive property affects the meaning. (In fact, the distributive property is a waste of time here, because it makes you do two multiplications where only one is needed!) The meaning is determined by the order of operations. Is the multiplication supposed to be done before or after the division?
I have observed that many students learn to distribute so well that they automatically do it when it is not helpful, as here, and even when it is not applicable, as when there is no addition at all!
Why students do it
In an unarchived answer in 2012, I listed reasons students have given for doing the multiplication first; on this occasion the question was about the expression \(a^2\div 4b + c\):
In fact, there are several different reasons people have given (this is a very popular question), some of which are better than others. As your friend argues, the rules as usually taught tell us to do all multiplications and divisions left to right (within any cluster of them), and make no exceptions that would cause 4b to be evaluated first. Many of us here would agree with that, and be done with it. Some people would evaluate 4b first because of a misunderstanding of PEMDAS, thinking it means multiplication should be done before division. I think you know they are wrong. Another wrong reason, applied to a slightly different sort of expression, is a misunderstanding of parentheses: the rule that parentheses "come before" everything else leads them to believe that in an expression like 12/4(4-1), the multiplication 4(4-1) has to be done first. But the rule about parentheses really only says that what's INSIDE parentheses has to be evaluated first; the result is treated like any other number. (I sometimes call this the "sticky parentheses" view.) Another reason given in relation to this second type of expression is the idea that the distributive property forces you to do the multiplication first, because they first evaluate 4(4-1) = 4*4-4*1 = 12 and then divide; but this begs the question, because the only reason they took the 4, rather than the 12/4, as the multiplier on the left, is that that's the way it looked to them. And, of course, the distributive property is only a way you may, if you wish, rewrite an expression to give the same value; it is outside of the question of what the expression in itself MEANS. Ultimately, most people probably do it just because it feels right: the 4b looks closer together, so we naturally tend to want to do that first. But they can point to no rule that justifies that; and since math is about proof, and about doing what you KNOW is right, not just what feels right, this is not good.
For an example of “sticky parentheses”, see
Is the 2 Related to the Numbers in Parentheses?
For an example of seeing the division sign as a fraction bar (and a long discussion of not being swayed by visual appearance), see
Order of Operations and Fractions
Back to the 2017 answer …
Avoidance is the best policy
In books and handwritten math beyond the elementary level, we hardly ever use the horizontal division symbol, but use fraction bars instead, which leaves no ambiguity. As a result, the math community has never had a need to make a choice on this situation! It's essentially been left undefined, and it is textbook authors who came up with explicit "rules" to describe what is really just a language that developed organically, based not on carefully stated rules but on tacit agreement. So which is the "right" way to read such an expression depends on what rules are in force in a particular community (math class, journal, etc.) -- and what was intended by the writer.
I closed with a plea for peace:
As a result, in problems such as this, the error is being made primarily not by those who give "wrong" answers, but by those who post the problem in the first place (or pass it on). Anyone who really wants to do math correctly will want to communicate clearly about it, and will avoid anything ambiguous or uncertain. They should either fully parenthesize, or use the horizontal fraction bar, which makes the order clear:
6 6
-------- or ---(2 + 1)
2(2 + 1) 2
Arguments on social media about this sort of thing are a waste of time. But thinking about our conventions can be very enlightening. Next time, I’ll close everything out with a look at history, and some solid reasons to think the “new rule” is in fact correct.
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Implicit Multiplication takes higher precedence than division. So for 6 / 2 (1+2) is 6 / [2*(1+2)]. Another example… say we have a sum A / B where A = 6 and B = 2(1+2) if I write it out in full we can see that 6 is on one side and 2(1+2) is on the other. the 2 does not some how magically jump from the B to the A so that A = 6*2 and B = 1+2.. so intuitively , we can see that an implicit multiply is the right approach to this lousy ambiguous equation !! hopefully… !
As I’ve indicated here, and more fully in the next post, Order of Operations: Historical Caveats, I don’t disagree with your conclusion; but I disagree with your argument.
Your first statement is just an assertion without proof. In order to show what actually takes higher precedence, you would need to show evidence of how it is used in mathematical publications, not just state your belief.
Your argument starting from A/B is fallacious, because when you substitute in an expression, you need to use parentheses to ensure that the order is not changed: you should write 6 / (2(1+2)). Otherwise, you could make the same argument to say that if A = 6 and B = 1+2, then A/B means 6/1+2 = 6+2 = 8, which is not true! Rather, A/B would be 6/(1+2) = 6/3 = 2. All you’ve really done is to assume that 6 / 2(1+2) means A/B because it looks that way!
But again, if we didn’t have books around the world that follow both interpretations, and if I could decree a rule for the world to follow, it would probably be to take implicit multiplication first. Even then, however, many people would misinterpret it, so I would still recommend adding parentheses!
Can we change PEMDAS to PEJMDAS?
(J for Juxtaposition)
Does that fix all the ambiguity to please both left-to-right and multiply juxtaposition first?
Not at all!
First, we can’t decree a rule; that requires agreement of all parties.
But even if this were how everyone stated it, it would not help, because PEMDAS is only a reminder of the rules, not a complete statement of them, and it is often misinterpreted. I think you are doing so: MD does not clearly imply either “M before D” (as some students take it) or “M and D, left to right” (as it is intended), so JMD would not clearly communicate whatever you intend it to mean. If you mean “juxtaposition first, then explicit multiplication and division left to right”, you need to say it — and then sit back and wait for people to say you’re wrong!
I learned the Order of Operations as BEDMAS, I’ve also seen it represented as: BODMAS and PEDMAS. So PEMDAS may seem to imply that multiplication takes precedence, but that’s not true for all school districts.
We were also taught that the division and multiplication were evaluated in “the order that they appear” (meaning left to right). I wasn’t explicitly taught that implied multiplication takes precedence, but I came to realize that I would automatically give it precedence anyway.
Like you, I have come to realize that the best practice is to avoid confusion by structuring the expressions so that any vagaries are avoided.
Implicit Multiplication always takes precedence because ‘ab’ etc are single entities in just the exact same way as ½ or any other fraction is considered as a single quantity.
Where in the rules does it say that implied multiplication takes precedence?
Perhaps you have missed the point that different textbooks or teachers teach different rules. You presumably have been taught the form that doesn’t mention implied multiplication. Be sure to read my next post, which shows that the rules as commonly taught in America do not represent actual practice accurately, but are oversimplified.
On the other hand, Steve Smith, to whom you are responding, is using an invalid argument; the correct interpretation of any communication depends on what is agreed upon by the community, not on mere sense impressions or personal assertions. This is a question of grammar, which is essentially arbitrary. So you are right to refer to rules. The only issue is, Whose rules?
(Math written in LaTeX)
To my understanding, the printing press had an impact in how equations were oriented due to it’s in-line requirements. There’s also an argument for cost per character, but I don’t know enough about that currently.
But I believe our confusion with how these symbols operate, stem in part from in-line requirements with printing.
I would construe that divisions and fractions are the same, but are used in different contexts;
$$2 \div 8 = \frac{2}{8}$$
This being the case for me, an equation such as \(6 \div 2 (1 + 2)\) would be equal to \(6 \times \frac{1}{2} (1 + 2)\).
This way, the answer will always be unambiguously \(6\), rather than \(1\); as you can solve from any direction or from any position.
To do this, I use the PEMS method; Parenthesis, Exponents, Multiples, Sums.- – Division being a form of Multiplication and Subtraction a form of Sums.
Thanks for writing.
It is, of course, true that the problem arises because of writing expressions inline; that’s the only situation in which it occurs. It is also true that division and fractions are essentially the same thing.
The problem is that the obelus, \(\div\), does not inherently indicate its operands as the fraction bar does, so we are left to decide whether the denominator is only the 2 (making it \(\frac{6}{2} (1 + 2)\)), or includes all that follows (making it \(\frac{6}{2(1 + 2)}\)). That is, someone, somewhere has (had) to decide on the appropriate order of operations.
On the other hand, what you have done here is not just to treat the division as a fraction, as I just did and as you seemed to be saying at first, but to take the division as multiplication by a fraction, which is a stronger argument. It is essentially the same argument I used in another post for doing division from left to right in the first place.
Unfortunately, one could still use the same argument to say we are dividing by \(2(1 + 2)\), making it \(6 \times \frac{1}{2(1 + 2)}\); this does often make more sense visually. And I think the argument I make for that interpretation in the next post has a lot of merit.
But the only way to ultimately resolve the issue is to make a worldwide decree (and recall many books in circulation!). Neither you nor I have that power. I think this form of expression is just inherently ambiguous, and the only real way to fix it is to avoid using the obelus at all.
Agreed; the obelus should be avoided when presenting equations.
Doing a bit of reading and avoiding a paywall or two, I found that the ISO does not recommend the use of the obelus.
It is true though, I was under the impression that the obelus implied it’s operands the same way multiplication does; there’s is little to no consensus from what I’ve found. I’m a little embarrassed that I hadn’t thought to verify something I assumed to be concrete.
Much appreciated.
I should add that even if we banned the obelus in algebra, the slash would still be used in typing math questions, and we would still get “1/2x” from students, sometimes intending \(\frac{1}{2}x\), and sometimes intending \(\frac{1}{2x}\), no matter which rule we taught, just because it would feel right to them in their immediate context.
“…They should either fully parenthesize, or use the horizontal fraction bar, which makes the order clear.”
With reading and mainly solving formula from left to rights, it may be safe to imply that “1/2x” is in fact \(\frac{1}{2} x\).
However, you’d run into a similar problem as with the obelus. Case in point would be modern Arabic mathematical notation being read right to left; “x2\1”. Better to have a universal system that reduces conflict between different linguistic notations.
I can’t help but to agree with your assessment. The better option would be to avoid in-line equations wherever possible. But for those instances where it’s not avoidable… use parenthesis.
Feel like I went in a logical circle, but learned a bit more along the way.
I’m programming an expression parser (just for fun) and ran into this problem when trying to implement implicit multiplication.
After reading all of the above, I made up my mind.
Math expressions are a language. And ambiguous language is no language as the sole purpose of language is to pass information from one person to another.
Consider the sentence: “There is a bird in a cage that can travel through time.” Rather than have a bar fight over this, it’s better to pick up the phone and call the person that wrote the sentence and ask for clarification.
This suggests that I should not accept implicit multiplication preceded by explicit multiplication or division.
But I also want my parser to handle errors with grace, that is, ignore the error and continue parsing (and resolving the expression if possible).
So, I’m going to opt for higher precedence for implicit multiplication, because it’s the more intuitive interpretation.
But the parser will always throw a warning that it’s going for this interpretation.
The warning is what’s important here: it’s the phone call to the writer of the expression. It also “unambiguates” the situation by clearly stating the assumption made.
About the story of the teacher and the father: the teacher missed an opportunity here to devote a full lesson to ambiguous expressions using the example at hand as a starting point. That would have been a lesson of wisdom.
Agreed! The only thing I’d say differently is that human language is generally ambiguous! We need mathematical language to be better than that.
I often recommend the “call the author” approach to dealing with ambiguities; one example is to state your interpretation of a problem on a test before working on it (assuming you aren’t in a position to raise your hand and ask). I equate this to “asking the boss” about what you are being asked to do, rather than doing what you think it means, and finding out later that it’s wrong. Your solution is a good one.
Of course, makers of calculators and computer languages get to state their rules explicitly, so they can decree an interpretation; you don’t have that freedom.
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The implicit multiplication takes precedence. All algebraic operations confirm this, as we treat an expression such as “3x” as belonging together always, such that when you include “3x” in an equation, it is interpreted to be “(3 times x)”.
That doesn’t constitute proof. In fact, your example is trivial, and does not illustrate the issue at all. You are making an assertion, and nothing more.
Keep in mind that we are talking about the equivalent of grammar, not of mathematics proper. To prove that something must be interpreted in a particular way, you have to show that it is taught this way, worldwide. And it isn’t.
Be sure to look at the next post after this one, Order of Operations: Historical Caveats, which includes some historical evidence. It also includes my view that your claim would be a good idea, if only we could decree a rule on our own.
It is actually taught that way worldwide. Only north American teachers differ but everybody do traditional algebra the same way even the North Americans. That should be proof enough. It only became a problem after computers or calculators couldn’t comprehend this. U don’t change centuries of algebra because a machine can not comprehend it, u improve the machine. Now there are calculators that do understand this.
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One way I was taught to look at it would be to interpret what’s inside the brackets as ‘group of.’ So, when reading
6÷2(1+2) I say to myself six divided by 2 groups of (1+2), which gives an answer of 1. A real life example of this equation could be the following: I have 6 apples and I want to divide it equally amongst two groups of children, each group having 2 boys and one girl. How many apples would each child get? Looking at the equation in this way one automatically uses implied mutliplication by juxtaposition and treats the 6 as the numerator and the 2(1+2) as the denominator whether written with an ÷ or a /. There really shouldn’t be any confusion.
There would be no confusion if everyone agreed! But since that is not true, it’s far better just not to write such expressions.
And, of course, the fact that one application of the expression fits your interpretation of the expression is not sufficient to say that the expression must always mean that!
Multiplication denoted by juxtaposition has always been on higher precedence than explicit multiplication. And this is true in all Algbera and Physics Textbooks. I have never seen any Algbera and/or Physics Textbook that interpreted or treated 3x ÷ 3x as counterintuitive as (((3•x)÷3)•x) but rather they always treat it as (3x)÷(3x) which is of course equal to 1 not x². Remember that 3x in itself is already a product result of the factors 3•x, we put the product result in juxtaposition because the value of “x” is still unknown, so whatever the value of “x” maybe it is meant to be multiplied by the 3.
The most and well respected Physics Textbook, The Feynman Lectures on Physics uses this convention, giving higher priority to implied/implicit multiplication over that of explicit multiplication and division.
The same is true with The American Mathematical Society which says, Multiplication denoted by juxtaposition has higher priority than Division.
The American Institute of Physics is also saying the same thing – Do not write 1/3x unless you mean 1/(3x).
Latest models of EDUCATION APPROVED SCIENTIFIC CALCULATORS Like CASIO, SHARP & CANON used in board exams give higher priority to implicit multiplication or multiplication by juxtaposition. The only Calculator I can find that uses the strict left to right rule putting implicit and explicit multiplication in the same order or rank is the TI CALCS which is only approved and used in the US.
Again, the left to right convention is not an absolute rule, and is only applies to explicit MDAS operations, but when MULTIPLICATION is implied by juxtaposition, the convention of juxtaposition applies. Higher Mathematics and Physics would be a complete mess if we just going to strictly apply and follow a convention and just disregard the very well established convention of implicit multiplication especially in higher Mathematics.
I find this link very informative though
https://youtu.be/4x-BcYCiKCk
I can’t say I disagree with any of this, except for your use of the words “always” and “never” at the top. I wouldn’t be very surprised if this is largely (entirely?) an issue restricted to American schools. When I’ve looked at good textbooks, they’ve most often just avoided writing expressions in which the issue came up; but many of the early questions I answered about the issue involved students who at least thought they were taught that implied multiplication was no different from explicit multiplication, but found that the answers in a separate workbook treated it differently. It would not surprise me terribly if it were the teachers who were interpreting the rules too rigidly, perhaps because they were not well-trained mathematically beyond what their textbooks told them.
The excellent video you refer to, as you probably noticed, uses me as a source (by way of a student paper that largely copies from me and others, with attribution), and covers much of what I’ve said here and in the next post. I’ve never taken the time to look at a large variety of calculators, but (as noted in this post) I was aware that TI took their cue from (American) teachers. Similarly, much of what I’ve had to say has been directed at American students, so that I have had to tell them to do as they were taught.
For anyone who wonders what mathematicians, scientists, and engineers really, truly do in practice about the treatment of implied multiplication in the order of operations, here’s a wild and crazy idea: Instead of just guessing or just believing someone else’s assertions, _look carefully at a sample of relevant literature and see for yourself._ In more detail:
(1) Pick a bunch of sources (books, articles, college class lecture notes, etc.) of the sort likely to contain mathematical formulas. I suggest focusing on material that is _not_ specifically intended for the primary or secondary education market and that is _not_ specifically about the topic of “order of operations” itself.
(2) Look through your sources for expressions that would have different meanings depending on whether or not an implied multiplication gets precedence over a division operator (“/” or “÷”) textually to its left.
(3) For each such expression you find, try to determine the _author’s intended meaning_ based on the context, and not merely based on your view about “correct” order of operations.
(4) See what patterns you notice. Do all, or almost all, authors who write expressions of the form “A/BC” consistently mean (A/B)C? Do (almost) all consistently mean A/(BC)? Do you ever find examples of both kinds of meaning in the very same document—and if so, do the authors seem to follow any consistent rule about when to give precedence to implicit multiplication and when not to? Are there cases where you find it hard to reliably determine the author’s intent? …
I’d enjoy seeing replies from any readers who _actually try_ the experiment I’ve just suggested. I’ve tried it myself, and I could say a bunch of things about what I found, and also about the processes of selecting sources, searching for relevant expressions, and discerning authors’ intent. But rather than make this comment far longer than it already is, I’ll stop for now and wait to see whether anyone else has observations to report.
A good research project! I’d also be very interested in what percentage of such writings use the inline form at all, to see whether I am right that we usually avoid it, and also that we usually parenthesize for clarity. In my experience, it’s very hard to find; but I have never taken the time for an extensive search of a wide variety of materials.
20/5(2*2)= 1 or 16?
multiplication before division
20/5(2*2) = 20/5*4 = 20/20 =1
parentheses before multiplication then division
(20/5) * 2 * 2 = 4 * 2 * 2 = 4 * 4 = 16
parentheses next to each other imply multiplication
(20/5)(2*2) = 4 * 4 = 16
I was taught multiplication before division. Everyone that I ask younger than 40 responds with 16. Was I taught wrong or has math changed?
I think this is answered in the post. What is taught on this point varies between times, between regions, and between individual authors or teachers. (And students hear differently, too!) Students need to do as they are taught, which unfortunately is not consistent. This is why it is best never to write such an expression.
My impression is that, at least in America, textbooks have increasingly taught a strict left-to-right rule, which is not necessarily what mathematicians themselves would follow.
Math Logic dictates that IF the teacher of 6th grade students wanted an unambiguous 16 as an answer to 8/2(2+2) = ?, the problem would have been changed slightly but have same amount of characters to indicate that 8(2+2) is in the numerator and 2 is the sole denominator.
Teacher and or textbook KNOWS the answer they want, Teacher presents the problem. They’d have written it as 8(2+2)/2 = ? to avoid incorrect answers and have all arrive at 16.
Seeing it the other way, it is an indication that the problem creator wants 1 as an answer with 8 being the sole numerator and 2(2+2) as denominator.
Yes, to be explicit, they could have written 8/(2(2+2)) – 10 characters instead of 8 characters. BUT implicitly 8 characters is enough.
Simple logic.
Interpreting the problem does involve a bit of problem solving and logic of the nuances of mathematics when not all the specifics are mentioned. Linear equations do have nuances. Word problems have their own as well.
I think the implicit multiplication shorthand rules were meant to address certain nuances like this but they were not widely practiced/taught.
These simple shorthand rules do make sense by clearing up issues that arise such as these linear equations as well as trying to interpret a physicist’s proof that covers an entire 6’x8′ chalkboard. Intuition sometimes is all you have to go on if you cannot ask questions when analyzing a complex proof and you have to make a judgement call individually.
Implicit multiplication (juxtaposition) should be more widely taught about its place in the order of operations. Factorials are also not in the mnemonics but they too have their place.
I probably agree with your conclusion, but not with your reasoning (and especially not with another comment you made that I discarded, which contained a silly conspiracy theory).
A teacher’s goal is not to get a student to write some particular number, but to follow the methods that have been taught. If the teacher believes that multiplications and divisions are to be carried out strictly left to right, then the teacher in writing this expression would want the student to say 16. And such a teacher would not see any ambiguity.
Once again, this is not primarily a matter of logic, any more than we can logically say whether a language should have the verb before or after the subject. Languages vary. The issue is whether the rules taught by a particular teacher accurately represent the “language” of math as used by its “native speakers”.
The comments section of this page (like many on related topics) is getting long; I consider comments a bad place to have enlightening discussions in the first place. I invite you, and anyone else who has ideas on this topic, to write to us at our Ask a Question page, which is better set up for such discussions. When we have enough interesting content, I will make a new post to show those ideas that are not already covered in the existing posts.
I’m going to block further comments here in order to facilitate that.
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