Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: mathematics

  • Three types of infinity

    Numbers have multiple purposes, some of which are

    • Locating things in a list, such as the first person to walk on the moon, the second star to the right, the seventh film in the Fast and Furious franchise. These are all answers to the question, “Which one?”
    • Counting things, such as 1 green sheep, 7 dwarfs, 101 dalmations. These are all answers to the question, “How many?”
    • Locating things in time or space, such as 6.5 km away, 500 miles away, 5 years ago. These are all answers to the questions, “Where?” or “When?”

    There are other purposes, such as measuring or comparing or having something to think about, but they aren’t what I feel like highlighting right now.

    The purposes of number give you various models to help think about what numbers are doing when you operate on them, and even if a number began life as a location, you can think about it as a size if that helps. So, even though you know that starting at the 10th person in the list then moving forward three people and starting at the 3rd person in the list and moving forward 10 people are very different, if you think of them as combining collections of people passed so far, you can tell they’re the same.

    But infinity is different. The three purposes of listing, counting, and locating in space don’t line up so neatly with each other when you’ve stopped using ordinary numbers you can reach in time. That causes the infinities that go with different purposes to be different to each other.

    [Disclaimer: I am not going to go into all the mathematical or philosophical detail possible here, and I’m not going to use all the formal terminology. That’s not the point of this post. So if you’re a mathematician or philosopher who knows about this stuff, think carefully about what I’m trying to achieve here before getting upset at me. And if you want to find out more, I’m sure a quick internet search will provide all the details you need.]

    Infinity of location

    First, the infinity represented by the symbol ∞. That infinity is a location. When you draw a line and mark it out with numbers, you’re using numbers as locations. You can go forwards using positive numbers and backwards using negative numbers. And what if you go forwards forever, following all the way to the end of the line? That location is called “infinity” and we label it with the symbol ∞. If you go backwards forever you get to the location “minus infinity” or -∞. So with a number line going both ways, there are actually two infinite locations of ∞ and -∞. These infinities aren’t numbers in quite the same way as other numbers, because really they are only locations and can’t also represent counts of how many things there are

    So, thinking of numbers as locations in space produces ∞ and -∞. What if you think of numbers as serving the purposes of counting or listing? What infinites do you get then?

    Infinity of counting

    When you count literal things like sheep, dwarfs, and dogs, there is always a perfectly good number to count them with. Those numbers are called the natural numbers. But some abstract things don’t have a nice natural number for how many there are, most notably, the very numbers you use to count with. How many natural numbers are there? This is another type of infinity. And its name is… however many natural numbers there are. There isn’t a name, really, though when someone says “infinitely many”, the size of the collection natural numbers usually what they mean. Many mathematicians use the Hebrew letter aleph with a zero as subscript – ℵ0 – to represent this quantity.

    This isn’t the only quantity that is infinite. Think about all the sets of natural numbers such as the set with just 1 in it, the set of just the digits 1 to 9, the set of even numbers, or the set of prime numbers. How many possible sets of natural numbers are there? That’s another infinity that’s even bigger than ℵ0.

    These infinities are called the transfinite cardinals. This is in reference to the word cardinality, which is the size of a set.

    Infinity of listing

    What about the first purpose of numbers I mentioned earlier – listing things and knowing where in the list you are? Well there’s infinities for that too.

    Imagine listing things in order: first, second, third, fourth, … and your list was somehow longer than all the natural numbers you had available, what number would you say after you’d gotten through all the natural numbers and still needed another one? Well some mathematicians call that ω, a lower case omega. What about the next thing in your list? Well that would be ω+1. It doesn’t have it’s own name, it’s just the number after ω. The number after that is ω+2.

    That’s the fundamental thing about numbers used for listing: every number has a number right after it, and that’s exactly what “+1” means. In that sense, the two numbers in an addition mean different things. When I say “10+3”, that means, start at the 10th object in the list, and then go 3 positions after that, which arrives at the 13th object in the list. The first number is where to start, and the second number is how many positions forward in the list to go, and the final answer is where you end up. With this interpretation, it’s actually surprising that 10+3 is the same as 3+10, though if your numbers are sizes of collections of things, it’s not surprising.

    With infinite listing numbers, the connecton to sizes is broken, and so it’s not actually true that you can do addition in any order you want. For example, ω+1 is the number after ω. But 1+ω is the number ω steps beyond 1, which is in fact just ω. It doesn’t matter where you start in the natural numbers, you still have infinitely many of them to go to get past them all, so going that far just gets you to the end, not any further. (Yes I know how confusing that sentence is. Sorry. Infinities are confusing.) So 1+ω=ω ≠ ω+1.

    These numbers with their weird order-matters addition are called the transfinite ordinals. That goes with the fact that they started life as ordinal numbers: first, second, third etc.

    It’s worth noting that this weirdness with addition having different meanings in different orders doesn’t happen with the transfinite cardinals, because combining sets definitely doesn’t have an order. (Though there is other weirdness: ℵ0+1 and 1+ℵ0 are the same, yes, but they’re both the same as ℵ0. This is why you can still fit an extra person in Hotel Infinity even though it’s full.) And with ∞, well you can’t add anything to that at all. It’s a place that isn’t really a number.

    Conclusion

    So there you go. There’s at least three different infinities to describe the number just beyond the ones we’re familiar with: ω, ℵ0, and ∞. They are the next number in the list, the size of the counting numbers, and the location at the end of the number line. We need them because each purpose of number has a different way to imagine what the numbers are and so a different way to imagine what is beyond beyond the numbers. And that different purpose dictates the sorts of operations that are possible on these infinities too.

    This deep connection to purpose and metaphor in maths is something that really appeals to me.

  • When perimeter is equal to area

    Some triangles have the same perimeter (in cm) as area (in cm²). For example, the 6-8-10 right-angled triangle:

    A diagram of a right-angled triangle with the two short sides labelled 6 cm and 8 cm and the hypotenuse labelled 10 cm. Below it are two calculations. Perimeter = 6 + 8 + 10 = 24 cm Area = half times 8 times 6 = 24 cm squared

    What properties of the triangle might signal that this sort of alignment of perimeter and area is possible?

    One answer I discovered to this question is completely surprising and delightful to me:

    If any of the side lengths is 4 cm or less, then the perimiter (in cm) can’t be equal to the area (in cm²).

    It seems remarkable that knowing about just one side of the triangle could rule out a property that ostensibly requires all three sides to confirm, but it’s definitely true, because I proved it. Indeed, the proof itself is one of the reasons I love this fact so much.

    You may want to attempt to prove it yourself before I show you my proof…

    Ok, here’s my proof.

    Consider a triangle with side lengths aa, bb and cc in cm. It’s possible to arrange the sides in order of size, so choose the labels so that a≤b≤ca \leq b \leq c.

    First we work with the perimeter.

    Perimeter=a+b+c>b+c≥b+b=2b\begin{aligned} \text{Perimeter} &= a + b + c \\ &\gt b + c \\ &\geq b + b \\ &= 2b\end{aligned}

    That is, the perimieter is strictly greater than 2b2b.

    Now we work with the area. The area of a triangle is half the base times the height. Choose the side with length bb as the base and let the height be hh.

    The side with length aa goes from the end of the side with length bb to the apex of the triangle. If it goes straight upwards, then the height is equal to aa, but otherwise, the height will necessarily be less than aa. Either way h≤ah \leq a.

    Three triangles, all with the same length base, all with base labelled b. The side to the left of the base is labelled a and is oriented differently in each triangle. In the one triangle, it heads diagonally up and to the left. In another, it goes vertically. In the third, it goes diagonally up and to the right. The vertical height is drawn with a dotted line in the two side triangles and labelled with h in all three triangles.

    So, when working with the area,

    Area=12bh≤12ba\begin{aligned} \text{Area} &= \frac12 bh \\ &\leq \frac12 b a \\ \end{aligned}

    Suppose that at least one of the sides is 4 units or less. Then the shortest side must be 4 cm or less. That is a≤4a \leq 4. And so

    Area=12bh≤12ba≤12b×4≤2b\begin{aligned} \text{Area} &= \frac12 bh \\ &\leq \frac12 b a \\ &\leq \frac12 b \times 4\\ &\leq 2b \end{aligned}

    That is, the area is less than or equal to 2b2b.

    Combining these two facts gives

    Area≤2b<Perimeter\text{Area} \leq 2b < \text{Perimeter}

    And so the perimeter is strictly greater than the area. In particular, they can’t be equal.

    End of proof! Yay!

    There are so many things about this proof that I love. I love that it uses so much inequality reasoning, which I have a particular fondness for. I love that you decide whether the area and perimeter are equal based on comparing them both to a third thing rather than to each other directly. This feels like such a ninja move. I love that the thing you compare them to is just the middle-length side. Indeed, the longest side doesn’t really feature much in the argument, which is surprising. I also love that the 4 appears more-or-less to cancel out in just the right way with the half in the area formula. A different number wouldn’t have worked, so 4 is the perfect one.

    I hope you like my little fact and its proof as much as I do.

  • Complex lines with i-arrows again

    Once upon a time (in 2016), I created a way to visualise where the complex points are in relation to the real plane, and then more recently (in 2022), I modified it to become the concept of i-arrows. I reread those blog posts recently while updating the blog to the new website, and I got all interested in them again. Here is what I’ve been working on over the last few weeks.

    You can read the rest of this blog post in PDF form here. 

  • Gerry-mean-dering

    A recent video from Howie Hua showed how if you split a collection of numbers into equal-sized groups, then find the mean of each group, then find the mean of those means, it turns out this final answer is the same as the mean of the original collection. He was careful to say it usually does not work if the groups were different sizes. Which got me to wondering: just how much of an effect on the final mean-of-means can you have by splitting a collection of numbers into different-sized groups?

    You can read the rest of this blog post in PDF form here. 

  • Why mathematical induction is hard

    Students find mathematical induction hard, and there is a complex interplay of reasons why. Some years ago I wrote an answer on the Maths Education Stack Exchange describing these and it’s still something I come back to regularly. I’ve decided to post it here too.

    You can read the rest of this blog post in PDF form here. 

  • Where the complex points are: i-arrows

    Once upon a time in 2016, I created the idea of iplanes, which I consider to be one of my biggest maths ideas of all time. It was a way of me visualising where the complex points are on the graph of a real function while still being able to see the original graph. But there was a problem with it: the thing I want, which is to see where the complex points are (or at least look like they are) is several steps away from locating them.

    However, in my original series of blog posts, I actually already created a solution to this problem! I can draw a complex number as an arrow on the real line, which starts at the real part and extends in the length and direction of the imaginary part. Anyway, combining this arrow model of a complex number from an x-coordinate and a y-coordinate produces an arrow in the plane. The point (p+si,q+ti) is an arrow based at the point (p,q) and extending along the journey (s,t) from there. 

    This is the representation I need. I have decided to call them i-arrows.

    You can read the rest of this blog post, and all eight blog posts in the i-arrows series, in PDF form here. 

    The titles of the eight posts in the series are:

    1. Where the complex points are: i-arrows
    2. The complex points on a line using i-arrows
    3. Further updates on the complex points on an unreal line using i-arrows
    4. The complex points on a line in finite geometry using i-arrows
    5. The complex points on a parabola using i-arrows
    6. The complex points on real circles using i-arrows
    7. The complex points on unreal circles using i-arrows
    8. The line joining two complex points using i-arrows

    UPDATE: There was a later blog post in 2024 further investigating the line joining two complex points.

  • Where the complex points are: on a real circle

    In 2016 I created the iplane idea, which allows you to locate the complex points on a real graph. Ever since I had this idea, I have wondered on and off about the complex points on a circle. It’s time to write about what I’ve found.

    You can read the rest of this blog post in PDF form, along with the other six posts in the series, here. 

    The titles of the seven posts in the series are:

    • Where the complex points are
    • Where the complex points are on a line
    • Where the complex points are on a parabola
    • Where the complex points are on the graph of a function
    • Where the idea came from for where the complex points are
    • Where the complex points are on a complex line (again)
    • Where the complex points are on a real circle

  • Where the complex points are: on a complex line (again)

    It’s been four years since I came up with the idea of iplanes as a way to organise the complex points on a graph, and in the intervening time I have thought about them on and off. For some reason right now I am thinking about them a lot, and I thought I would write down some of what I am thinking.

    You can read the rest of this blog post in PDF form, along with the previous six posts in the series, here. 

    The titles of the seven posts in the series are:

    • Where the complex points are
    • Where the complex points are on a line
    • Where the complex points are on a parabola
    • Where the complex points are on the graph of a function
    • Where the idea came from for where the complex points are
    • Where the complex points are on a complex line (again)
    • Where the complex points are on a real circle

  • The Seven Sticks and what mathematics is

    This week I provided games and puzzles at a welcome lunch for new students in the Mathematical Sciences degree programs. I had big logic puzzles and maths toys and also a list of some of my eight most favourite puzzles on tables with paper tablecloths to write on.

    One of the puzzles is the Seven Sticks puzzle, which I invented:

    Seven Sticks
    I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them.
    Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them.
    How long is the shortest stick?

    I sat down at one of the tables and I could see the students there were working on the Seven Sticks, so I asked them to explain what they had done. (WARNING! I will need to talk about their approach to the problem, so SPOILER ALERT!)

    They told me about their reasoning concerning lengths that don’t form triangles, and what that will mean if you put the sticks in a list in order of size. They had used this reasoning to write out a couple of lists of sticks on the tablecloth which showed that a certain length of shortest stick was possible but that a longer shorter stick wasn’t possible. And so they knew how long the shortest stick actually was.

    Only they said to me they hadn’t done it right.

    I was surprised. “What?” I said. “All of your reasoning was correct and completely logical and you explained it all very clearly. Why don’t you think you’ve done it right?”

    Their response was that what they had done wasn’t maths. They pointed across the table to what some other students were doing, which had all sorts of scribbling with calculations and algebra, and said, “See? That’s maths there and we didn’t do any of that.”

    I told them that actually what they did was exactly what maths is – reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.

    Only later did I realise the implication of what had happened: these students are coming into a maths degree, which means they have done the highest level of maths at school, and they perceived that something was only maths if it had symbols in it. A plain ordinary logical argument wasn’t maths to them. And a scribbling of symbols was better than a logical argument, even if it didn’t actually produce a solution.

    This made me really sad.

    I wonder what sort of experiences they had that led them to this belief. I wonder what sort of experiences they missed out on that led them to this belief. I wonder how many other students have the same belief and I will never know. I wonder how I can help all the new students to see more in maths than calculations and symbol manipulation, and allow themselves to be proud of their work when they do it another way.

  • Zooming in to see the slope

    A lot of people introduce the derivative at a point as the slope of the tangent at that point, which to me is quite confusing. It seems to me that the reason we want the derivative is that it is a measure of the slope of our actual function at that point, not the slope of a completely different thing. To me, the thing is that the function itself is pretty much straight if we are close enough to it, so when we’re looking really close, saying it has a slope at this point is a meaningful thing to say.

    For years I got at this idea by drawing a little magnifying glass on my function, and a big version nearby to show what you could see through the magnifying glass. But last year I decided to get technology to help me do this dynamically and I made a Desmos graph with a zooming view window that will show you the view on a tiny circle around a point on a curve. (Some of the bells and whistles here were provided with a little help from Andrew Knauft .)

    Here’s what the graph looks like, and a link to the graph so you can play with it yourself: https://www.desmos.com/calculator/pa1cudpc07 

    A screenshot of a Desmos graph. There is a red curve which is the graph of a function, and a small circle centred at a point on the function. A dotted line joins this to a bigger circle showing a zoomed in version of what's in the small circle. There is a zoom factor slider.

    I’ve used this to help students see that the curve itself really does look straight when you’re very close, and so treating it like it is straight there and saying it has a slope is a reasonable thing to do. Students are usually most impressed and love to play with it. It’s also interesting to put in a function with a kink in it like  |x2-1| to show that the kink never goes away no matter how closely you view the function, so having a single slope there isn’t really reasonable.

    Every time I’ve had to search my own twitter account to find the tweet where I shared it , and I couldn’t keep doing that, so here it is in a quick blog post for posterity.