Reflections on maths, learning and maths learning support, by David K Butler
I don’t like BODMAS/BEDMAS/PEMDAS/GEMS/GEMA and all of the variations on this theme. I much prefer to use something else, which I have this week decided to call “The Operation Tower”.
The titles of the five blog posts are:
I met with some lovely Electrical and Electronic Engineering lecturers yesterday about their various courses and how I can help their students with the maths involved. And of course complex numbers came up, because they do come up in electronics. (I have not the slightest clue how they come up, but I am aware that they do.)
I asked them what notation they used for complex numbers, and they told me about using j instead of i, and then about polar form using either reᶿʲ and r∠θ. I told them that the students will have seen the notation r cis(θ) at school for those and I was met with righteous indignation. What a horrible notation! How could anyone think that was a good idea!
It was rather a surprise considering their wonderful openness and friendliness up to this point. But this is not the first time I have been met with this sort of response to cis from more-or-less nice people. Many a mathematician has expressed to me their disdain for this little notation, and I have to say I am bamboozled. Why do people hate it so very much?
I actually rather like cis. I like how cuddly-looking it is with all those curvy letters. I like how wonderfully easy it is to write simply with no special symbols and with no powers that have to be typeset at a higher level than the text. I like how it sounds like you’re singling out one of your many sisters when you say it aloud.
I think cis(θ) is friendlier than eⁱᶿ because it doesn’t require you to suddenly believe that imaginary powers of real numbers are a thing and believe they ought to be this unusual combination of cos and sin. I think it’s also friendlier than ∠θ because we’re not co-opting a symbol we saw in geometry for a new and strange usage. (Not to mention the fierce pointiness of ∠ compared to the cuddly roundness of cis.)
Finally, the thing I like the most is how cis(θ) is a function that takes real numbers and produces complex numbers, and those numbers are on the unit circle. For me It’s such a nice thing right from the start to recognise that there are such things as functions that produce complex-number outputs, and that they can have rules to manipulate them just like ordinary functions. Indeed, writing it with a multiple-letter name like all their other familiar functions forges this lovely connection. I also love that multiplying a complex number by cis(θ) rotates it around the origin θ anticlockwise, because of course it does since cis(θ) is a position on a circle. (If anything, I think it wouldn’t hurt it if we renamed it “cir” to highlight its connection to a circle rather than highlighting its connection to cos and i and sin, even if that means losing its familial pronunciation and a little roundness.)
Here’s a little geogebra applet to play with cis as a function from the real numbers to the complex numbers: https://ggbm.at/MWnm5TJA
I know it hides the thing about complex powers of real numbers, but I think it’s important to hide that for a bit, lest the eⁱᶿ feels too much like a fancy trick. Plus I think it’s good to be familiar with this friendly little function for a while before discovering that it also solves a problem of complex powers. Indeed, I would love it if students were familiar with this friendly little function even before introducing the concept of polar form.
So there are the reasons why I actually rather like cis. But there is one more reason I think mathematicians and professional maths users shouldn’t hate on cis: because it means hating on the students themselves. Our students put a lot of effort into understanding cis, and to loudly announce how much you hate it is telling them that they wasted all that effort and that their way of doing things is less sophisticated and babyish. Not the best way to start your relationship with them as their university teacher. I’d prefer it if people started off with acknowledging the coolness cis and then pointed out that the other notations are more convenient for doing what they want to do, or allow them to write formulas in more magical-looking ways, or allow them to do some cool stuff with powers. Then maybe we might not get them offside early in the piece.
So please, stop hating on cis(θ)!
My first post of 2018 is a record of some rambling thoughts about remainders. I may or may not come to a final moral here, so consider yourself warned.
What has prompted these ramblings today was reading this excellent post by Kristin Gray about her own thoughts on division and remainders. In that post, I saw the following:
7÷2 = 3R1
For some reason, this bothered me. For some reason it’s always bothered me. Today I think I realised what the problem was: In my head “7÷2” is a number, and “=” indicates that two things are equal, but 7÷2 can’t be equal to 3R1 because 3R1 is not a number. It is only today that I realised that 3R1 isn’t a number.
How do I know 3R1 isn’t a number? Well firstly it’s two numbers. One is a number of groups and the other is a number of objects. I don’t even know how big the groups are – it could be 3 groups of 2 and one left over, it could be 3 groups of 7 and one left over, or 3 groups of 200 and one left over. I can hear people saying that actually all plain numbers could mean any number of different units, and a 7 could be 7 cm or 7 ducks or 7 groups of 200. But the 1 here is definitely 1 of something, while the 3 is some unknown size of groups of that same something. That seems like a totally different kind of unit issue than with a plain ordinary number.
Also, if it really is a number, then I should be able to place it on a number line, but where does it go? The 3 I certainly know where it goes, but what about the 1. Where does that live? It lives in a completely different land to where the 3 lives, and I can’t really put it on the number line until I know how big the groups are that the 3 represents.
It occurs to me that this is a good way of transitioning to a fraction sort of idea. The fact that the 1 is small relative to groups of size 200 and large relative to groups of size 2, and needing to encode this relative size would lead nicely to a need to write this as 7÷2=3+1/2. What an interesting idea.
My other really big issue is that the “=” sign in this context doesn’t work the way an “=” sign works. If 601÷200 = 3R1 and 7÷2=3R1 then usually the properties of “=” would mean that 601÷200 = 7÷2. But they aren’t equal. I suppose they both produce 3 groups with 1 left over, but that 1 is very different in size relative to the group in each situation! So they’re not really equal are they? Actually, this idea is going back to the same idea I had with the number line, where you need to encode the relative sizes.
My final problem is that if it really is a number, then surely you’d be able to do operations on it. But I don’t really know how you’d do that. You’d expect that if the 3R1 came from 7÷2, then 2×(3R1) would produce 7, but if it came from 601÷200, then what would 2×(3R1) even mean? I’ve been trying to figure it out, but to no avail.
It might be possible to do addition and subtraction, if you knew the groups were the same size. In that case 3R1 + 5R3 would be 3 groups and 1 plus 5 groups and 3, so it should be 8 groups and 4. So 3R1+5R3 should be 8R4. It seems you add the two numbers separately, which is actually super interesting. I’m still a bit worried about what would happen if the groups were size 4, say, because then 8R4 is actually the same as just 9. So now it seems like they are a lot more like numbers than I originally thought. This seems like a very interesting thing to investigate.
As you can see, I’m rather puzzled by remainders and where they stand numerically. I get that the idea of dividing a collection into groups requires us to have a concept of remainder. I just feel weird writing it in this way because these symbolic representations feel like they ought to make numerical and algebraic sense, but here they don’t.
In Kristin’s post, she floated the idea of the equation being not 7÷2=3R1 but instead 7=3×2+1. This second equation I feel completely comfortable with. It is 100% clear what the numbers are doing and the “=” really is acting as an equality here. I still wonder if there’s a more helpful way of representing the division-producing-a-remainder thing though.
And maybe that’s another issue I have with it, that this statement “7÷2=3R1” is about doing an operation and producing a result, as opposed to declaring a relationship, which is what I have come to believe the “=” sign is for. By using something that is not like a normal number and just encodes a description of an answer, are we reinforcing that “=” means “here is the answer”? I don’t know what to do with this question yet, or if it even really matters.
So there you go. I’ve rambled through a whole lot of thoughts and worries about remainders. I don’t have any conclusions or morals or recommendations. But it’s certainly helped me to try to write it all down. I hope it helped you to read it. I’d love to hear your own thoughts on it.
This comment was left on the original blog post:
Deborah Peart 4 January 2018:
You make a great point! Students have enough confusion around equality and the equal sign. Truthfully it should be expressed
7/2=(3×2)+1
Interesting thought!
There is a procedure that people use and teach students to use for finding the inverse of a function. My problem with it is that it doesn’t make any sense, in two ways.
Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff. One of the more important problems was about how the calculator interprets what they type into it, which is really in essence about the order of operations.
The titles of the five blog posts are:
Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, I think.
But for a while I have wanted to add just this one thought to the conversation and now is as good a time as any.
I had a meeting with an international student in the MLC on Friday who has having a whole lot of language issues in her maths class.
She was from the USA.
Yes, the USA. Her problem wasn’t the everyday English; it was with the different terminologies for mathematical things here compared to her experience where she comes from. Her only experience with vectors was in physics where a vector is a quantity with magnitude and direction, whereas in maths class here, a vector is usually just a list of coordinates. She knew how to find a derivative, but had never heard the word “differentiate” used for that action. She had only ever used the word “anti-derivative” for what we call an “integral” most of the time.
I was talking through the transpose of a matrix (something else new to her) and how it interacts with other operations on matrices, and how since it’s normally written as a power it takes precedence over nearby multiplications like powers do. She asked me if, here in Australia, we still use the same order for what is supposed to be done before other things. I said, yes, we do, and told her that most local high-school students use the acronym BEDMAS to describe that order.
She wrote underneath the acronym familiar to her: PEMDAS. First she focussed on the fact that one had DM and the other MD, but reconciled that quickly saying, “Well I suppose they go in the order they come and so it doesn’t matter which way around they are.” But she had no idea what to do with the B.
I told her the B stood for brackets, and I drew what brackets look like: ( ), [ ], { }. And then she freaked out. To her, those things are called parentheses ( ), brackets [ ] and braces { }. I said, yes, those are the official titles of those things, but here in South Australia they’re thought of as different kinds of brackets. If we want to distinguish between them we’ll call them round brackets ( ), square brackets [ ] and curly brackets { }.
And suddenly a light came on for her and a whole lot of stuff people had said this week made sense. She also suddenly understood the very odd look her class tutor gave her when she mentioned the word parentheses. “Yes,” I responded. “Most maths students and tutors here would never have heard the word parentheses.”
And what happened next? Well we’ll have to wait and see. I think we made some excellent progress, and we agreed to keep meeting across the semester to help deal with anything more that might come up.
For me, I’m so glad I knew a little about the differences between Australian and American mathematical-English. (Thanks MTBoS!) And perhaps if anyone is reading this, then you will know too.
| Australia | USA | |
|---|---|---|
| ( ) | round brackets | parentheses |
| [ ] | square brackets | brackets |
| { } | curly brackets | braces |
PS: I find it interesting how the Australian acronym BEDMAS references a general term “brackets” which covers all shapes of bracket, whereas the American acronym PEMDAS references a specific term “parentheses” which only covers one shape of bracket.
These comments were left on the original blog post:
Claire 31 July 2016
This is very interesting. I teach high school in Southern California. When I teach the Order of Operations, my students have seen it before from middle school. They were either taught PEMDAS or Please Excuse My Dear Aunt Sally to learn the order. I always use these acronyms as a way to highlight the limitations of some of the “tricks” they use to memorize math concepts.
It’s interesting that the parenthesis, brackets, and braces are all called “brackets” where you are. Having 1 word for that is super helpful. I tend to say that they are all “grouping symbols” and if it were up to me, the acronym would be GEMDAS. Because then, for example, absolute value symbols are grouping symbols.
We also discuss why MD doesn’t matter the order… That they are inverse operations and part of a mathematical family. Same with AS.
This family idea is helpful to explain where roots and logs might go in the order of operations.
This post gave me a lot to think about.
Geoff Coates 1 August 2016
I helped an African student once who learned his high school maths in French. They used commas where we use decimal points and full stops where we use commas in large numbers. He was very confused for a while …
David Butler 2 August 2016
This came up in the Chemistry labs last semester, when all the pipettes had their volume listed as “0,25mL”.
If you google “fundamental trig identity” you will get many many images and handouts which all list the fundamental trig identity as:
sin2 t + cos2 t = 1
This is, of course in the wrong order and it should really have cos first then sin, like this:
(cos t)2 + (sin t)2 = 1
“But David,” you say, “it’s addition, so it doesn’t really matter what order it’s in does it?” Of course it does! Mathematically it’s the same, but psychologically it’s different. If it really wasn’t different then you would sometimes write cos first and sometimes write sin first, but I can bet you always write it in a particular order. And if you write it with sin first, then you’re making it harder for yourself.
Let me explain.
The reason we have the fundamental trig identity is because the angle t there is a piece of the circumference of a unit circle, and cos t and sin t are the coordinates of the points on that unit circle. If I asked you to write down an x-y equation for the unit circle, you would naturally write x2 + y2 = 1 with the x first. But the x-coordinate of a point on the unit circle is cos t, and the y-coordinate is sin t, so of course that means it’s (cos t)2 + (sin t)2 = 1. Writing your trig identity with the cos first makes it easier to make the connection with the equation of the unit circle. If you write it with sin first you’ll have to continually switch it round!
Also, the order does matter if you’re using hyperbolic trigonometry. Then the formula is (cosh t)2 – (sinh t)2 = 1 and having sin first would be definitely mathematically wrong. For years, I had great trouble remembering which way around this was supposed to go until I realised that the cos and sin were in alphabetical order. From that point forward I always wrote my ordinary trig identity in the same order as the hyperbolic trig identity (in alphabetical order) so that through force of habit I would never get the hyperbolic one wrong.
So, I recommend you start writing your fundamental trig identity in the right order. It might help you remember and make connections to other things!
PS: You may be wondering about the way I put brackets in my trig identities. Don’t even get me started.
In our bridging course (and indeed in Maths 1M and Maths 1A and several other courses) there is a section on differentiating logarithmic functions. One of the classic questions that we ask in such a section is to differentiate the log of some horrifying function, with the intention that the students use the log laws to simplify the original function first and then differentiate. There is something about this particular type of question has long bothered me and I only just figured out how to resolve my issue with it. I’m so excited I need to share it somewhere!
I had a long chat with one of the students the other day about rotation matrices. They had come up in the Engineering Physics course called Dynamics as a way of finding the components of vectors relative to rotated axes. He had some notes scrawled on a piece of paper from one of my MLC tutors, which regrettably were not actually correct for his situation. I know precisely why this happened: rotation matrices are used in both Dynamics and Maths 1B, but they are used in different ways (in fact, there are two different uses just within Maths 1B!). It’s high time I made an attempt to clear up this confusion, especially since three more students have asked me about this very issue in the last week!
In Maths 1B, you learn about Linear Transformations, which are a special kind of function that you enact upon vectors in some dimension to produce vectors in some dimension. It turns out that all linear transformations can be described by representing your vector as a column of coordinates and multiplying it by a matrix. Each linear transformation has its own matrix that works for all the vectors it acts upon. Rotations happen to be a type of linear transformation and in two dimensions there is a formula based on the angle you rotate that tells you what the matrix is. I’ve included just such a matrix in the picture here.
One reason this works is because multiplying a matrix by your standard basis vectors of (1,0)T and (0,1)T gives you the first and second columns of your matrix respectively. But multiplying by the matrix has the same effect as the rotation transformation, so to figure out what these columns actually are, all we have to do is rotate the points (1,0) and (0,1). If you do this, then because of trigonometry, you get the two points (cos θ, sin θ) and (-sin θ, cos θ), which are indeed the columns of the matrix.
Let’s just make sure we know what’s going on here before we move on: You have a point in the 2D plane, you take its coordinates as a column, you multiply this column by the matrix, and you produce a new set of coordinates, which is a new point. So your matrix in effect moves your point from one place to another. The point with coordinates (1,0) moves to the point with coordinates (cos θ, sin θ); the point with coordinates (0,1) moves to the point with coordinates (-sin θ, cos θ).
So now we have that a rotation matrix has cos θ on the main diagonal, sin θ in the bottom left corner and -sin θ in the top right corner. And it tells you where a point moves to under a rotation of θ anticlockwise. (It’s worth noting that it also works perfectly well on the components of vectors imagined as arrows.)
The problem is that over in Dynamics, a rotation matrix does not look quite like this! In particular, the minus sign is in the opposite corner. Why?
The answer is that in Dynamics the rotation matrix is not a description of a transformation of the points or arrows themselves, but a description of how their coordinates change when you transform the coordinate axes. The points themselves don’t move at all, it’s the coordinate axes that move and we just relabel the points with new coordinates.
The reason this works is again because of the standard basis vectors. The point (1,0) has its coordinates recalculated according to the new axes, and its coordinates turn out to be (cos θ, -sin θ); while the point (0,1) also has its coordinates recalculated and its coordinates turn out to be (sin θ, cos θ).
You may notice that this is precisely what the coordinates would have been if you did rotate the points themselves, but in the opposite direction to the original rotation matrix. This makes sense. If you turn your head to match the new coordinate axes, then this is precisely what has happened. Basically, if you rotate the coordinate axes one way, the points “move” the other way relative to the axes.
And this would be the end of the story, except that in Maths 1B you also rotate coordinate axes, and yet the rotation matrix is somehow still not the same as the one in Dynamics! Why?
The reason is that in Maths 1B we rotate axes in the context of equations of curves, and this is quite a different situation from when you rotate axes in the context of the points themselves.
Imagine I have an equation which describes a curve. A point is part of the curve if its coordinates satisfy the equation, and it’s not part of the curve if its coordinates don’t satisfy the equation. But what if I relabel all the points with new coordinates according to a new set of axes? I want an equation for my curve so that a point is on the curve if its new coordinates satisfy the new equation. How do I achieve that? Well I do already have an equation, it’s just in terms of the old coordinates. So if I have a point in the new coordinates, to tell if it’s in the curve, I just need to figure out what the old coordinates are and sub them into the old equation. It ought to be possible to make one equation that encompasses both of these actions – the transferring to the old coordinate system and the subbing into the old equation.
Did you notice what happened there? In order to create an equation that described the same curve relative to the new axes, I had to begin with the new coordinates and transform them into the old coordinates. Let me repeat: I had to go from new to old. The coordinate transformation matrix in Dynamics goes from old to new. To go in the opposite direction I have to have the minus in the opposite corner.
So that’s why the matrices are different. In Dynamics you are moving the axes but not the points, and finding new coordinates for the points. In Maths 1B you are moving the points, not the axes, so the rotation appears to be in the other direction. Or alternatively in Maths 1B you are moving the axes, but you already know the new coordinates and you want the old ones, so you actually are doing the calculation in the opposite direction.
I’m glad we cleared that up!