Converses

In maths, or at least university maths, there are a lot of statements that go like this: “If …., then …” or “Every …, has ….” or “Every …, is …”. For example, “Every rectangle has opposite sides parallel”, “If two numbers are even, then their sum is even”, “Every subspace contains the zero vector”, “If a matrix has all distinct eigenvalues, then it is diagonalisable”.

It’s possible to flip these around so that the implication goes the other way: “Every shape with opposite sides parallel is a rectangle”, “If the sum of two numbers is even, then they are both even”, “Every set containing the zero vector is a subspace”, “If a matrix is diagonalisable, then it has all distinct eigenvalues”. This is called the converse of the original statement. The big problem is that while it’s very easy to form the converse, it’s not usually true! In every one of my examples, the original statement is true but the new converse statement is false. It is possible for the converse to also be true, but it’s not the most common thing to happen!

Many students when faced with statements like those in the first paragraph automatically and unconsciously assume that it works both ways, especially when the subject matter is new to them. The number of students who I have seen show that the zero vector is in a set and declare that therefore it’s a subspace doesn’t bear thinking about. I have some theories about why this happens: one is that they assume the statement is the definition of the concept rather than something that happens to be true about it, the other is that they see us use this one rule to show that something isn’t a subspace and assume you can use this one rule to show that something is. But this post is not about the why. It’s about a way of helping students see the problem.

This is a Ball

There’s this awesome picture book called “This is a Ball” by Beck and Matt Stanton. In it, the adult is supposed to read the words, but the words don’t match the pictures that are shown. The idea is to let kids have a chance at being right for a change. This book is so very awesome that when we saw it in Aldi recently as a big book, we just had to buy it, even though there was nowhere in our house to store it!

this-is-a-ball-book

Anyway, there’s a page in the book which describes a monster scaring a princess and her pet dog, only the picture shows a princess scaring a monster and her pet elephant.

this-is-a-ball-page

When the child tells you that you’re wrong, you’re supposed to read the next part of the text, which says, “We talked about this before. Don’t you remember? Look at its tail! All dogs have tails.”

When I read this for the first time I said to myself, “Oh my goodness it’s a false converse!” I realised what power this would have for helping students understand the problem with converses not always being true. It’s such a simple example, and is pitched in such a way that I can use it as an experience rather than just an explanation. It only took a few months for me to get an opportunity to use it…

All dogs have tails

I was with a student last semester who was doing a problem about eigenvalues of matrices. He looked up a theorem in his notes and its corollary which said “If a matrix has all distinct eigenvalues, then it is diagonalisable”. Then he said to me, “So if this matrix is diagonalisable, it must be that all the eigenvalues are distinct.”

“This is it!” I said to myself and quietly drew a picture something like this:

elephant-1

I said, “Look. It’s a dog!”. He looked at me funny, and I said, “You don’t believe me? But look at the tail – all dogs have tails!” I let it sink in for a few seconds. “Oh,” he said. “I see what you did there.”

We were then able to talk about how things don’t always work both ways and to be careful to read things closely to make sure. I am so so very happy I now have a way to bring up this conversation. Thanks Beck and Matt Stanton!

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This is the last (for now) in a series of posts about Where the Complex Points Are. To catch you up, I discovered a way of visualising where the complex points are in relation to the points of the real plane. All the complex points (p+ri,q+si) are in a plane attached to the real plane at the point (p,q). I call these planes the “iplanes”, and I imagine it as a transparent sheet that can be unfolded to lie flat on the real plane in order to see its complex points. Using this model, I’ve investigated where the complex points are on a line, on a parabola, and on the graph of a complex function.

Here are links to all the other posts in the series:

There is one more thing I want to talk about and that’s where the idea came from, because I think it’s an interesting story and I want it recorded somewhere.

Between not beyond

Ever since I first learned about complex solutions to quadratic equations, I have wondered where the complex points are. I never really thought about it deeply, but the general picture in my head was that the complex points were somehow “beyond” the real plane. I imagined them as over there beyond the edge of the real plane, surrounding it so that the real plane was a small plane inside some much bigger plane.

But something doesn’t quite work about this picture. When I studied conics in depth much much later I learned that a “degenerate circle” with equation x2 + y2 = 0 wasn’t just the origin (0,0) as I had originally thought. In fact it was a pair of complex lines that each happen to have one real point. (These lines have equations x+iy=0 and x-iy=0.)

Only recently did this start to bother me. Suddenly I asked myself, if the complex points are beyond the real plane, how can a line made almost entirely of complex points meet the real plane in the origin, right there in the centre?

It occurred to me that the line must somehow travel over or between the real points to arrive at the origin in the middle.

And the seed of an idea began to grow. If the lines have to travel between the other points to reach the origin, then the complex points must somehow be bewteen the real points, not beyond them.

Finite planes – the smallest plane

Once I realised that the complex points had to be between the other points, I started thinking about the problem in terms of finite planes. In a finite geometry, the points are discrete points, and there is plenty of space between them to put extra points, if indeed that’s where they are.

I started out with the smallest possible affine plane (our ordinary geometry is a kind of affine plane). It has four points with coordinates taken from the set {0,1} (also known as Z2) and six lines. The lines have equations x=0, x=1, y=0, y=1, x+y=0, x+y=1. One name for this plane is AG(2,2) – the Affine Geometry of dimension 2 and order 2.

What I needed was to find where the “complex points” were in relation to this plane. Unfortunately, if you’re using Z2, 1+1=0 so -1 is the same as 1. This means the equation x2 = -1 already has a solution, so I can’t add the number i. You actually have to add the solution to x2 + x = 1. I called this number α so that I ended up with a set of four numbers to use for coordinates: {0,1,α, 1+α}.

From this I can build AG(2,4), which has 16 total points, and 20 total lines. I wrote down the equations of all the new lines and tried to figure out how to add the points to my diagram so that the new lines looked like lines.

I knew the lines in my new plane would have to have four points — one for each “number” in the coordinates — and this meant each line already existing needed two extra points. It seemed reasonable to exend them outwards. It also seemed reasonable to try and make the new lines x=α and x=1+α vertical and the new lines y=α and y=1+α horizontal.

Fiddling around with this gave me a nice square arrangement for the points. Then I investigated what all the new lines looked like and I got something like this. (Note I was drawing in crayon on the paper tablecloths at my sister’s restaurant at the time. I didn’t take a photo on the day, but this is a reasonable facsimile.) You can see the lines a little more clearly in this GeoGebra applet.

Unfortunately the extra points were outside the original ones, and not between them as I had hoped. I did notice that each new line that wasn’t part of the original plane met the original plane in exactly one point. I knew this would have to happen because it’s a theorem from finite geometry, but I was glad that everything was working the way it should.

Finite planes — noughts and crosses saves the day!

After the half-success of the smallest affine plane, it was natural to move on to the second-smallest affine plane AG(2,3). This is the plane you get when you add in all the wraparound diagonals to a Noughts and Crosses board. When drawing it on a noughts-and-crosses board, the twelve lines look like this:

If you draw the points as dots and the lines as (curved) lines, you can make it look like this. You can see the individual lines more clearly in this GeoGebra applet.

As you can see, it’s already quite invovled, and I didn’t really like to imagine what would happen when I went up to the “complex” plane — that would mean going from 9 points to 81 points and from 12 lines to 90 lines. The 20 lines of the affine plane of order 4 were enough of a mess!

But I did start to investigate. AG(2,3) has its coordinates taken from Z3, which is {-1,0,1} with the rule that 1+1=-1. This field doesn’t have any solutions to x2=-1, so it was safe to attach the number i this time. I drew my extra points clustered around the existing points and started to investigate the lines I would need. In order to get the lines with equations like x=i to be vertical, I needed to make sure that the points with coordinates (i,something) were lined up above each other. This required me to make sure that within each cluster of points the same imaginary parts were added to the coordinates.

Here’s a picture of my ideas book around this time:

After drawing it on paper, it was time to investigate it more fully, so I worked on creating a GeoGebra applet that would show the points on a specific complex line. It took some fancy footwork with the spreadsheet to make it do what I wanted.

I noticed now that the extra points on every real line lined up neatly with the real line. This was totally awesome, and I was inspired to move on to the real plane to see what was happening there.

Finally iplanes are found

I was all satisfied with how my complex points worked in finite planes. The final step was to think about how this applied to the infinite real plane.

There was a big problem, of course. In the finite planes, there was plenty of space between the points to fit the extra points. In the contiuous real plane we’re already full. Where do I fit those extra points?

My first idea was that they were all squished into each point of the plane — “fat” points, as it were. But this was somehow unsatisfying because I couldn’t actually see where the points were. Then it occurred to me that if I couldn’t squish them into each point, they’d have to be above the plane but attached at the real point. And the iplane was born.

At the start, I was sketching and calculating on scrap paper, drawing a little square around the real point to visualise the iplane. This was giving me enough insight to get a feel for what was going on with lines. The GeoGebra applets in the previous posts are based on this visualisation. It wasn’t until I got out the cellophane and the tap-tap board that I really felt like I had a proper feel for what was going on. This pinkness has transferred to all of my diagrams and all of my thinking about iplanes, but you’ve seen the results of all that thinking already…

iplane-parabola-unfolding

Conclusion

So what is the point of this story? I don’t know. I just thought you (and future me) might like to see my train of thought in creating this idea.

I suppose for me the main moral is that finite geometry helps give me insights into real geometry, and that my habit of making physical models of things is still a worthwhile thing to foster.

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This is the fourth in a series on Where the Complex Points Are.

This one wasn’t part of the original plan, but some interesting thoughts have come up while discussing the previous posts on Twitter, and I needed to process them. So here we are doing an extra blog post before my final epilogue.

A problem of viewpoints

In the introduction, I talked about how I’ve always wondered where the complex points are when you talk about a parabola meeting the x-axis in complex points. I introduced the concept of an iplane: at each point (a,b) in the real plane, there is an iplane attached, which contains all the points (a+ci,b+di). These points are arranged with the x-axis showing the imaginary part added to the x-coordinate and the y-axis showing the imaginary part added to the y-coordinate. I imagine them as being a transparent sheet attached to the point itself and being able to be flattened out to sit on top of the real plane.

 iplane-at-3-2iplane-parabola-unfolding

This representation of the complex points allowed me to investigate the complex points that lie on lines (both complex lines and real lines) and to investigate the complex points that lie on a vertical parabola, and also how lines meet such a parabola. I shared these thoughts on Twitter, and waxed lyrical about them in person to basically every person I met. People were impressed, but there was something in the way they spoke that niggled at me and I couldn’t figure out what it was.

Yesterday I figured out the problem: I was thinking about the lines an parabolas in totally different ways to other people! In my head, the lines and parabolas were sets of points whose coordinates satisfy a condition. To decide if a point is part of the set, you sub its coordinates into the equation and see if it’s true: if it’s true, then the point is on the object, if it’s not true then the point isn’t on the object. I was looking to see which points were on the parabola by simply subbing the coordinates in to see which ones worked. I did this one iplane at a time because it seemed the most reasonable way of organising my search.

But what was reasonable to me was not reasonable to others! Taking each iplane one at a time and looking to see which points in the iplane satisfy the equation was totally alien to quite a few people I talked to. These people totally understood the iplane concept and they thought it was a great idea, but they had no idea what I was getting at by looking in each iplane separately. Yesterday I realised why: they were thinking of the lines and parabolas as an organisation of the input/output relationship for a function. They were looking for an x-axis and a y-axis to “line up” the input and the output of the function formula.

This is of course a perfectly valid way to think of many graphs, especially for the parabolas I was looking at, whose equations all look like “y = something in terms of x”. It’s just that I wasn’t thinking about them that way so my iplanes weren’t designed to match with this viewpoint. So I started thinking about how to make it match.

The complex line

There is one key issue with making the graph relate more closely to the idea of lining up the input on the x-axis with the output on the y-axis: the input is a complex number and it doesn’t really live on the x-axis. But, I dealt with that problem already in the plane by adding an iplane to each point of the plane. Surely I could do the same thing to the line, right? Why can’t I add an iline to each point on the real line? The iline attached at the number a would contain all the numbers of the form a+bi.

With the iplanes, I flattened them out to sit on top of the real plane so I could see the complex points, but I only flattened out enough to see the point I was looking for. I wanted to do the same thing with the ilines, but it’s not really the same feel. Somehow I can’t tell so easily what’s going on with the complex number. Then I realised what I could do is just draw an arrow from the point representing the real part to the complex number in its iline. That is, I could draw the number a+bi as the point a on the number line and an arrow of length and direction b starting from that point. So the following representation was born:

You can investigate what complex-number addition looks like in this representation with this GeoGebra applet, and what complex-number multiplication looks like with this GeoGebra applet. (This representation is not so good for complex multiplication, but that’s ok — different representations are useful for different things.)

A complex x-axis and y-axis

So I was now able to locate a point on the x-axis as an input for a function, and locate its output on the y-axis. Then something rather wonderful happened: I realised that I could combine this point-and-arrow x-coordinate and this point-and-arrow y-coordinate into one point-and-arrow in the plane. The point (p+ri,q+si) became the point (p,q) and the arrow (r,s) starting at that point.

But the point (p,q) is the centre of the iplane in which the complex point (p+ri,q+si) lives! And the arrow (r,s) is the vector-version of the coordinates (r,s) within that iplane! So it turns out that my point-and-arrow x-coordinates and y-coordinates matched up perfectly with my iplanes! I couldn’t be happier!

 

The complex graph of a real function

So now I was able to investigate graphs of complex functions from the perspective of input-output relationships. It’s a whole different way of thinking about things, and I haven’t investigated it to any great depth yet, but so far it is most interesting. One particular thing I’ve noticed is that quadratic functions are rather special when it comes to how the real images of the complex points are arranged — the tangents seem to play a much bigger role than they do for higher polynomials.

I’ve made a GeoGebra applet that will allow you to enter a real function and move the input around on the x-axis to see what happens to the output and the matching complex point on the graph. You can turn on the trace to see what shape is made when you do move the input around. I now have a host of new questions rising in my mind which I hadn’t even considered before, and it will probably take years to sort them out.

But for now I’ll just be content that I have a picture that works for both views of a graph: as a collection of points that satisfy a condition and as an encoding of the input/output relationship of a function.

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This is the third in what has turned out to be a series on Where the Complex Points Are.

Where the complex points are: introduction to the iplane concept
Where the complex points are: on a line
Where the complex points are: on a parabola (YOU ARE HERE)
Where the complex points are: the graph of a function
Where the […]

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This is the second in what has turned out to be a series on Where the Complex Points Are.

Where the complex points are: introduction to the iplane concept
Where the complex points are: on a line (YOU ARE HERE)
Where the complex points are: on a parabola
Where the complex points are: the graph of a function
Where the […]

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When you first learn complex numbers, you find out that they give you ways to solve equations that were previously unsolvable. The classic example is the equation equation x^2 + 1 = 0, which if you’re only using real numbers has no solutions, but with complex numbers has the solutions x=i and x=-i.
As someone who […]

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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 […]

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This post is about my response to TMC16.
For the uninitiated, TMC is short for Twitter Math Camp. This is a conference designed by teachers for teachers with teacher speakers, organised through the collective efforts of the Math Twitter Blog-o-Sphere (MTBoS) — a group of people who blog and tweet about their experiences teaching math(s). That […]

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Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) – ovals, ovoids, conics, quadrics and their cones – and the lines associated with them – tangents, secants, external lines, generator lines. During the first two years, I did talks about my […]

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A few months ago, I learned a new word: “mansplaining”. You may have heard it before, but I never had until this year.
The general idea is that very often, a man will explain something to a woman in a way that seems to be based on the assumption that the woman is incapable of understanding […]

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