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The limit does not exist!

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What’s more fetch than toaster strüdel, foot cream in a jar and a fertility vase from the Ndebele tribe? Arguably maths. More specifically, the mathematics of Mean Girls. If you haven’t seen Mean Girls, it’s a not-too-bad movie about teenage girls, teenage boys, teenage drama, and (a little bit of) maths.

Stop trying to make fetch happen. It’s not going to happen.

It’s not exactly October 3rd*, but its been 13 years since Mean Girls came out, and

  • $13 \times 2=26$
  • $26+2\times2017=4060$
  • $4060-3010=1050$, and
  • $1050\log(2)+\frac{\pi}{1423211345}$ is… just a number.

What we’ll be looking at in this article is the last 10 minutes of the movie, where Cady and the Mathletes from North Shore compete in a maths state championship. After 87 minutes of play it’s a tie, and so the two teams enter the sudden death round, where the teams choose a member of the opposition (happening to both be female) to battle for the trophy.

T. Pak as in ‘Trang Pak the grotsky little byotch’? No, just her brother.

In one corner, Cady Heron, AKA ‘Africa girl’, AKA the ‘used-to-be maths geek but not so much anymore after she met the Plastics but is kind of a geek again because her attempted sabotage of Regina George backfired and descended into chaos’. She is also the protagonist of the movie.

In the other, is Caroline Krafft, the female member of Marymount Prep, who “seriously needed to pluck her eyebrows, whose skirt looks like it was picked out by a blind Sunday school teacher, and had some 99-cent lipgloss on her snaggletooth”. This was when I realised, making fun of Caroline Krafft wasn’t going make me any better at writing this article (or maths for that matter).

The two were asked to solve the following;

Caroline Krafft hastily states the incorrect answer, $-1$. If Cady manages to state the correct one then the North Shore mathletes ultimately win.

In the midst of all this Cady then has a miraculous flashback to the week that Aaron got his hair cut, and as she takes a moment to see straight pass Aaron’s face and onto the board behind, then has an epiphany (the kind we would like to have in our exams sometimes) and quickly realises the function diverges.

Aaron Samuels is now a spin class instructor, so he won’t be expected to know what a factorial is anymore. Thank God.

So we don’t exactly know how Cady did this in under 10 seconds, but we can try to figure out the question for ourselves.

So the question reads$$ \lim_{x\to 0} \frac{\ln(1-x)-\sin x} {1-\cos^2 x}.$$

We’ll consider L’Hôpital’s Rule, seeing as its the most straightforward way of going about it.

If we substitute $x$ with 0 we will find$$\frac{\ln(1)-\sin(0)} {1-\cos^2(0)} = \frac{0}{0}.$$
This satisfies the criteria for being able to apply L’Hôpital’s Rule, which is as follows,$$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.$$

Let’s apply it to our function, starting by considering the limit from the right;

\begin{align}
\lim_{x\to 0+} \frac{\frac{-1}{1-x}-\cos x} {\sin 2x} &= \lim_{x\to 0+} \frac{-x \cos x + \cos x + 1}{ (x-1) \sin 2x} \\
&= \lim_{x\to 0+} [-x \cos x + \cos x + 1] \lim_{x\to 0+}\left[\frac{1}{x-1}\right] \lim_{x\to 0+}\left[\frac{1}{\sin 2x}\right].
\end{align}
Now,\begin{align}
\lim_{x\to 0+} \left[-x\cos x +\cos x +1 \right] &=2, \\
\lim_{x\to 0+} \left[\frac{1}{\sin 2x } \right] &= \infty, \\
\lim_{x\to 0+} \left[\frac{1}{x-1} \right] &= -1,
\end{align}
so by properties of infinity we get
$$\lim_{x\to 0+} [-x \cos x + \cos x + 1] \lim_{x\to 0+}\left[\frac{1}{x-1}\right] \lim_{x\to 0+}\left[\frac{1}{\sin 2x}\right] = 2 \cdot \infty \cdot -1 = -\infty.$$

Repeating this method for the limit from the left, we will obtain the following,
$$ \lim_{x\to 0-} \frac{-x \cos x + \cos x + 1}{ (x-1) \sin 2x} = +\infty.$$

Just checking.

And so since the limit from the right and the left differ, the whole thing diverges. Yay.

Another approach would be to look at the Taylor series approximations for the separate terms at $x$ near 0. We won’t do that though since it’s not as mathsy, but it might have been the quicker way to go, on the assumption that Cady really knows her Taylor series expansions. But

“Cause the next time you see her she’ll be like, Ohh Kevin G!”

*The date where Aaron Samuels asked Cady what the date was.

CC-BY-SA: Mathletes, Matty McRib. Mean Girls, Flickr. Screenshots copyright of Paramount Pictures.

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The Indisputable Existence of Santa Claus – A Review

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Welcome to the 22nd day of the 2016 Chalkdust Advent Calendar. Today we have for you, a book review!

Christmas, a time of celebration, joy and meeting family you never knew you had. Regardless of how joyous Christmas can be it also is undoubtedly stressful for some. What present could I get person $x$? Why does my Christmas tree look so ugly even though I spent 2 weeks decorating it? The simple answer is because you probably drenched it in tinsel and epileptic seizure inducing lights. The mathematical answer?

The Indisputable Existence of Santa Claus” is a recent release by Dr Hannah Fry and Dr Thomas Oléron Evans, Dr Hannah Fry who has previously written “The Mathematics of Love”. This book arranges itself as a step-by-step guide on how to prepare for the (mathematically) perfect Christmas, covering every detail from how to wrap presents according to their surface area to volume ratio, to using Markov chains as a means of perfecting the Queen’s speech. 

We shall start with the big question of whether Father Christmas himself actually exists. The first chapter gives a ‘seemingly’ valid proof of Santa’s existence, from another ‘seemingly’ valid proof of how 1+1=0. You must be thinking, what? Obviously there must be a flaw in the progression of the proof somewhere, which there is, but you can discover it for yourself by reading the book. 

Moving on, let’s have a look at the inconspicuous game of Secret Santa. For those unfamiliar with the concept, the classic approach to this game is simply, you write your name on a small piece of paper, fold it up and throw it into a hat. After a bit of shaking, someone picks out a piece and takes on the enormous responsibility of finding their victim, *coughs* I mean colleague, a present usually around the price range of £5. So why is this a rather inadequate way of organising this game? 

Well first, you risk the chance of picking out your own name. You might think that’s easy enough a problem to solve, just put it back in the hat…But what if you were the last person? Or even the second last person? Everybody knows your name is back in the hat, and they have a greater chance of picking your name, and in some case, 2 people will have each other’s name, Secret Santa is ruined. Goodnight. The book proposes another method of making sure that no one has their own name, and no one else knows who has their name. A clever yet simple solution involving derangements, alas, Christmas is saved, now lets hope your secret Santa isn’t a Scrooge

My favourite chapter in this book has to be the one on the Queen’s Speech. The beginning of the chapter is an analysis of the Queen’s vocabulary score (based solely on the number of unique words in the first 35K words of her speech. Surprisingly, poor old Lizzy scored lower than her counterparts Jay-Z and Shakespeare. Well, it seems as though maths might be able to give her a hand with that, with something special called a Markov chain, that determines the next word to place in a sentence given the word before. I won’t go into too much detail as to give it away; it just so happens that earlier this week I was reading about how Markov chains are used in determining the probability of flipping a coin and getting a certain outcome which is equivalent to another outcome, so this chapter peaked my interest even further. 

What is particularly good about this book is how accessible it is; at the end of every chapter there are endnotes that explain some of the maths mentioned and includes some extra reading material. The only questionable thing in this book might be the chapter about cooking turkey. They used a chicken instead. Enough said.

Overall, a great read, not too technical but with just enough maths to get you thinking. So even if your grumpy aunt Hilda despises anything to do with maths, there’s now a very slight chance she might enjoy it. 

Also the amount of cracker pulling rules I’ve never heard of is remarkable. I am now going to stop pulling both ends of my own cracker, despite my competitive nature.

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07 December

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Welcome to the seventh day of the 2016 Chalkdust Advent Calendar. Today we bring you a concise list of maths-related gift ideas, made to entertain the mathematician within you, and disappoint your not-so-mathsy family and friends. Let’s begin!

Things nobody wants for Christmas

fx-991de_plus-1A classic, the Casio FX. A great present for anyone who is doing GCSE maths or equivalent. A terrible one otherwise. Most of the general public seems to be under the impression that all mathematicians do is solve large sums whilst punching away on their calculator. We’d be happy to remind them that this is the job of an accountant. “No, I do not want to calculate how much each person is paying when we split the bill. No.”

21057977802_038086490c_oScented candles. Uncreative, almost as festive as Febreze Fresh Cut Pine. Would avoid in all cases unless incredibly desperate and in dire need of a Secret Santa present.

Things some people want for Christmas

Fibonacci Clock

Why? Because its almost as hard to read as this animal clock.

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The clock is made up of 5 squares, each with side lengths representing the first 5 terms of the Fibonacci sequence (1,1,2,3,5) arranged in a spiral form. Red indicates the hour, green, the minutes and blue is for when both hour and minute are the same.

To tell the time, simple add up the green and blue squares and multiply by 5 for the minutes (be aware this clock is only accurate to 5 minutes) and the hour is sum of the red and blue squares.

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A terrible clock to have in an exam, a great one otherwise.

“So what’s the time?”
“2×2 white 3×3 white 1×1 green 1×1 red and 5×5 blue.”
“What?”
“It’s 6:30.”

Games

A safe, usually successful and usually amazing choice for cold winter days. Here are my picks for games to play between $n$ or fewer person(s).

Antivirus

Rush Hour’s less appreciated second cousin twice removed. It comes with a booklet of starting positions with difficulty ranging from really easy to really not easy. Simple looking, though terribly addicting.61pjoxc2eil-_sl1012_

Hive

A personal favourite boardless strategy game made for 2 players. The objective is to completely surround the ‘Queen Bee’ with pieces of any colour (white or black). Pill Bugs prove to be great for strategic game play, and Spiders not so much. Grasshoppers are just as under appreciated as Spiders.

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SET Mini round

The same sets, just tinnier. set_mr_layout_2

Because regular SET is so 1974.

Symmetry Groups Wrapping Paper

To top it off, some lovely Symmetry Groups wrapping paper, of which there are 17 different designs, enough for you, your friends and extended family.

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Image credits: CC-BY-SA: Calculator, Jörg Reddmann. Candle, smittenkittenorig.

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04 December

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Welcome to the 4th day of the 2016 Chalkdust Advent Calendar.

Carols! (Nearly) Everybody loves carols, but admittedly we do get tired of hearing the same lyrics over and over again each year. So on the fourth day, we bring you not two, not three but zero calling birds! Instead here is the first of a few Chistmaths carols we’ll be featuring in this year’s calendar that would definitely spice up any party.

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