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An irrational problem

Can you find an $a$ and $b$ that are both irrational such that $a^b$ is a rational number? It’s a fairly easy question to understand, if not to answer. I was posed this question in middle school before I had even heard of logarithms. The only tool you will need to solve this is the following exponent rule: $(a^b)^c$ is equal to $a^{bc}$. If you want to answer this on your own, which I highly recommend, don’t read on yet. Continue reading

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Solutions to recent puzzles

Many a Chalkdust article started life as a puzzle. What’s the intersection of these two lines? How many of these cups would it take to make a sphere? How can I cut my birthday cake? …How else can I cut my birthday cake? Writing and doing puzzles is great but, well, sometimes you don’t feel satisfied unless you’ve got the solution. Never fear, we’re here to give you the answers to three puzzles that were set in recent articles! Continue reading

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Christmas competition #3: ‘An eggnog mystery’

This coffee is exactly the kind of strong brew that a mathematician of any kind could turn into a world-class theorem if she drank it at the right time of her career. But that’s not even the most exciting thing out on the banquet floor tonight.

Tonight, the small chalet playing host to half-a-hundred mathematicians is lit up like a lopsided Christmas tree: fitting enough for the season of the conference. Cushioned on all sides from breathtaking views of the slopes by healthy dollops of fresh snow, it would look fitting in a snowglobe or a painting much more so than overhanging a muddy cracked road ribboning the Rockies. But that’s life at the Banff International Research Station for you. Continue reading

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Christmas lights, trees and maths

Christmas is coming! For many it is the most wonderful time of the year, while for others it is simply an excuse to consume mince pies and mulled wine. This month we have been embracing the Christmas spirit with our competitions, don’t forget to take part! In the (very unlikely) event that you are already missing your maths lectures or research, today we bring you a special Christmas blog, in which we talk about some of the maths and science you can find during this holiday season.  Continue reading

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Christmas competition #2: ‘More geometry snacks’

It’s the second Thursday in December, and so time for the second Chalkdust Christmas competition. Throughout this month, we are running a series of competitions, with a fantastic prize up for grabs in each one. This week’s prizes are copies of More geometry snacks by Ed Southall and Vincent Pantaloni.

More geometry snacks is a book stuffed full of really good geometry puzzles, and would make an excellent Christmas present for your maths-loving friends and family. If you’re not lucky enough to win a copy, you can order them here.
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The first Black Mathematician Month outreach event

Earlier this year we hosted the second edition of Black Mathematician Month, with articles including a biography of David Blackwell and an explanation of how Black Panther’s suit could be modelled mathematically. But we were also busy organising an exciting event: a full day of workshops, talks and activities targeted specifically at black students. Read on to find out how it went!
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Christmas competition #1: ‘You can’t polish a nerd’

December has begun, and with it comes our first Christmas competition of 2018! Throughout the month, we will be running a series of puzzles, with a fantastic prize up for grabs in each one. This week, we’ve been lucky enough to get some DVD box sets from science comedy trio Festival of the spoken nerd. Read on to discover what we thought of their latest show, You can’t polish a nerd, and to find out what you need to do to enter the competition.

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Deal or no deal?

Deal or no deal is a television game show first developed in Holland, but now shown in more than eighty countries. It is loved by many, but considered by others to be no more exciting than watching paint dry. The format of the game, and the best strategy to win, have prompted much discussion, with articles on the programme appearing in some very serious economics journals! Continue reading

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Digging for roots in the complex plane

As winter approaches and fall splatters the treetops with 600nm-wavelength light, it is wise, albeit cold, to get out with a rake or a spade to catch the best of the last of warm sun rays. If, however, we are determined to shirk the changeable weather and stay inside with a computer in tow, the complex plane like the outdoor garden will offer plenty by way of uncovered roots, measured out squares, and eddies of discovery.

Let’s set for ourselves, before we begin, some fences we will attempt not to scale for the time being. To start, let us agree that the coefficients, at least, of our chosen polynomials shall be in the reals: a polynomial in $\mathbb{C}[z]$ will be mentioned but not studied in detail. Furthermore, let’s agree to focus our attention on the cubics: they will illustrate our point well enough, and will be simple enough to generalise: we have heard too much in various classes of the doings and dealings of the quadratics, but the cubics — the cubics retain some air of mystery, and will offer, I promise, a sufficient degree of fascination to keep us engaged with the task ahead. So, onwards! Armed with a graphing software, enthusiasm, and knowledge, let’s dig in!

Dramatis polynomiae

$x^{3}-5x^{2}+17x-13$: a real family man.
$x^{3}-5x^{2}+17x-\lambda$: his entire family.
$a_n z^n + a_{n-1} z^{n-1} + … + a_2 z^2 + a_1 z + a_0 $: ever a lady unwilling to commit to the numbers, scary in the full brunt of her generality.

Dress as the cubic formula for Halloween & scare all your learned friends!

A cubic would, via the Fundamental theorem of algebra, have three roots, and always three roots. One could look in ancient textbooks to find the thorny cubic analogue of a quadratic formula, but with modern calculators in hand, the monster of a formula is hardly useful for anything but making your stomach churn. Say we have found the roots by turning to a higher computational power. With these three roots in hand, and desperate for more, we can acquire more by taking the derivative of our polynomial, and finding its roots in turn. Here, even with cubics, we hit upon a marvellous pattern.

The first observation to make is: the roots of the polynomial’s derivative always seem to lie in the convex hull of the polynomial’s roots. This is the Gauss-Lucas Theorem, first implied in a quick note of Gauss’ in 1836 and proved some years later by Édouard Lucas.

We can get more by noticing that the centroid of our polynomial’s roots is the centroid of its derivative’s roots, and the centroid of its double derivative’s roots, and so on and so forth until you zero in on a point. For cubics, the roots form a triangle with a vertex on the real axis, and there is much more mathematics hiding in each triangle, waiting to be uncovered.

We jump first to some geometry. The inellipse of a triangle lies within and touches the triangle’s sides at the midpoints. (The prefix “in-” here is more akin to that of an incircle than to that of inexcusable.) As a mater of fact, it’s the maximal (in area) ellipse that can be inscribed in a given triangle. In 1881, Steiner showed that the foci of the inellipse inside the triangle of a cubic’s roots are exactly the roots of the cubic’s derivative! This inellipse now bears his name. So if we take a family of cubics with the same derivative, the foci of their Steiner inellipses will coincide.

Roots of the $x^{3}-5x^{2}+17x+\lambda$ family.

Let’s invite our cubic for a demonstration. We consider the whole family, $x^{3}-5x^{2}+17x+\lambda$, $\lambda\in\mathbb{R}$, of the relatives and relations of our friend $x^{3}-5x^{2}+17x-13$. The derivative of each such polynomial is the same. So the foci of the Steiner ellipse must be the same, and moreover, the centroid and the foci of the Steiner inellipses are invariant under $\lambda$. What’s more, we can construct a line of best fit of the three roots in the plane, and that’ll remain unchanged!

I’ve included some of the family in the drawings to the right and below, with and without their triangles. You can see them dancing around a single centroid and line of fit, with polynomial $x^3 -5x^2 +17x -19$ coming close, with roots roughly: $1.65, 1.67\pm 2.94$.

The roots moving in their hyperbolic tracks, the foci of the Steiner ellipse stand firm with the changing constant.

More roots, more glory

As you can imagine, the higher the degree of polynomial we start with, the more impressive our laser-focus on the spot that will be its last derivative’s final roots becomes. So, perhaps, if we are to impress our mates over a pint of beer a greater degree, both of the alcohol and the polynomial may well be required. And, after all, the thing looks pretty. Above is our friend $x^3 -5x^2 +17x -13$ along with some of his integrals, all having the same-root centroid in the complex plane.

Probably time to stop

For the green thumbs in the audience, there are always greater flowerbed arrangements to be built and more elegant ideas to be planted. I advise those interested to lasso in some probability theory and answer the query:

suppose we have an $n^{th}$-degree polynomial, $a_n z^n + a_{n-1} z^{n-1} + … + a_2 z^2 + a_1 z + a_0 $, where $n \rightarrow \infty$, and say that the $a_i$ are each randomly chosen: where do we expect to see the roots of this polynomial? We can grow for ourselves a formula for the expectation, a probability cloud valid for $n=3$, and it is interesting what it would condense into as $n$ grows… but this is really taking a leaf blower to the yard and ditching the spade, so we will talk no more of it here. For the answer, see reference 3.

As you can see, there are many avenues left unexplored. Thus you can spend all your time generating spades of polynomials, testing their properties and cackling in your infinite marvel, but perhaps let’s take a break and breathe some of that autumn air before it turns into the crisp of winter.

References

  1. D. Minda, S. Phelps, Triangles, ellipses, and cubic polynomials, The American mathematical monthly, 115, October 2008.
  2. M. Roth, Permutations given by polynomials, a talk given at Queen’s math club (January 26, 2017).
  3. L. Shepp, R. Vanderbei, The complex zeros of random polynomials, Transactions of the American mathematical society, vol. 347, No. 11, November 1995.
  4. J. Walsh, On the location of the roots of the derivative of a polynomial, Annals of mathematics, vol. 22, no. 2, December 1920.
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Talkdust, episode 1

You may have noticed that in the editorial of issue 08, we mentioned that we had begun work on a Chalkdust podcast. Today, we are very excited to announce that the first episode of Talkdust, a podcast for the mathematically curious, is now available.

You can listen to the podcast using the player below, or download an mp3, or search for us in your in favourite podcast app (or add us to your app manually using the RSS feed).

The presenters of this episode of Talkdust are Sean Jamshidi and TD Dang. Music and editing by Matthew Scroggs, and announcements by Tom Rivlin. In this episode:

  • TD and Sean talk about being scooped and taking exams.
  • TD and Sean talk to Tanmay Kulkarni, the author of Somewhere over the critical line from issue 08 of Chalkdust.
  • TD explains a colouring problem she’s been thinking about recently, and describes this picture to Sean:
  • TD thinks of a number and Sean tries to guess it.