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 re^{θj} 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 e^{iθ} 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 e^{iθ} 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)!

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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, or 3 groups 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.

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This post is about a game I invented called Home in One Piece. I invented it in 2014 specifically to play outside at student barbecues, after years of trying to think of an effective game using play dough. I’ve taken the physical version with me to various places to play it, including to Twitter Math Camp in the USA. It’s time to let it out into the world and allow people to play it for themselves.

Home in One Piece is a game for two players that is played with play-dough. Each player has eight blobs of play-dough and the goal is to join the play-dough together and move it across the board, so that finally you win by having all your play-dough together in one blob in the home zone (hence the name Home in One Piece). The original version was a pure strategy game, but then I remembered that I prefer games with a chance element, so I created three special dice to control how many of your blobs you are allowed to manipulate during your turn.

Picture of Home in One Piece board in action

To make the game you need the following components:

Here is a YouTube video of a game in action, so you can have a better idea about how these rules work in action.

There are so many things I love about this game, if I do say so myself. I love the physical and three-dimensional nature of it, with the manipulation of the play-dough in space. I love the contradictory feeling of freedom and constraint of being able to make the play-dough whatever shape I want, yet having to hold it down in one spot. I love the fact that the movement condition is continuous, whereas in almost all other games it is discrete. I love sitting on the same side of the board as my opponent, so that we are both looking at the same game and reaching past each other to play. I love the fact that I’ve arranged the dice so it’s possible to roll 0. (If you’ve played my other games you might notice that particular fascination of mine.)

Mathematically I love thinking about what it means to be “one blob” and about the interesting topological shapes that happen when people join and rejoin blobs. I am particularly in love with my dice, which have been arranged carefully to give the right probabilities for the different numbers with only one dot per face. (That was a whole wonderful investigation, I can tell you!) I love secret problem zero of dividing your blob into eight equal pieces.

But most of all I love watching people play it. I love the shapes they make, and the realisations they have about what is possible and not possible — I love the walls and lakes and bridges people build, especially bridges over bridges over bridges. I love the cries of despair when the play-dough breaks and when someone rolls zero. I love how lax people are about their play-dough touching at the beginning and how unforgiving they are about it at the end. I love watching people wander over, fascinated at what the hell is going on with the play-dough here.

I hope you enjoy the game as much as I do. Please let me know how it goes!

To finish off, some tweets of the game in action (note this is the first version of the game, which had the rules printed on the game-board.)

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Last week, I had one of those days in the MLC Drop-In Centre where I was hyper-aware of what I was doing as I was talking with students and by the end I was overwhelmed by the sheer volume of things I had thought about. I decided that today I might attempt to process (or […]

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There is a Twitter account that tweets the prime numbers once an hour in sequence. (The handle is @_primes_.) Since before I joined Twitter, it’s been working its way through the six-digit primes and some of them are very nice. A lot of other people think they’re nice too, based on the fact that they […]

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This post is about my reaction to the book “Choice Words: How Our Language Affects Children’s Learning” by Peter H. Jonston. I was lent the book by Amie and I am very grateful to her because it really is a good book (though it was tough to read with the forest of sticky notes marking […]

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Well I did it. I went to Twitter Math Camp 2017 (TMC17) in Atlanta, Georgia, USA.
I found out about TMC last year, when Tracy Zager mentioned me in her keynote at TMC16, effectively yanking me right into the thick of it. I could see that this was one of the things that cemented together the […]

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I have a whole suite of maths t-shirts that I made myself. One of them simply has the number 65536 on it. It’s been getting a bit of attention over the past couple of weeks, so I thought I might write about it.
65536 is my favourite power of 2. More specifically, it’s 216, which means […]

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The game of Prime Climb
Prime Climb is a wonderful game by Dan Finkel (aka @MathforLove), which you can find out more about here. The board is a path made of the numbers from 0 to 101, coloured by an ingenous and beautiful system. Each player has two pawns which they move around the board by […]

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Yesterday I talked about one of the common responses to people finding out I am a mathematician/maths teacher, that of saying, “I’m not a maths person.” The other common response I get is, “I don’t have a maths brain.” (John Rowe mentioned this in his comment on the previous post.)
This is how I reacted last […]

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