Skip to content

Dead Mathematicians Society

Watching Robin Williams talk about poetry in Dead Poets Society reminded me of A Mathematician’s Lament by Paul Lockhart — yes, I have already linked to this once before. I want to make all pure maths students watch it, and replace “poetry” with “mathematics” throughout.

Rigour and structure are extremely important in mathematics, particularly so in pure mathematics. This however, does not mean that one should learn these things first; you need to have an appreciation for it. Schools all over the world teach students trigonometry; the sine of any angle in a right-angled triangle is equal to the length of the opposite side divided by the length of the hypotenuse. Are the people in charge of the curriculum under some deluded idea that this is what mathematics is all about? Or do they think that this is somehow “useful math” for the real world?

Instead of teaching students how to compute angles and lengths by plugging it into the same formula over and over again, we only need to teach them that it’s possible. We should start students with the old “compass and straight edge” problems; something akin to a puzzle. These kinds of problems are more in line with what mathematics is really all about, and then learning basic trigonometry afterwards wouldn’t just feel like rote. Or am I being too optimistic here?

The Greeks long suspected this, but it wasn't until April 12th of 1882 that Ferdinand von Lindemann conclusively proved it when he constructed himself the most awesome birthday party possible and nobody showed up.

From XKCD (Obviously)

Last week, I went to the State Library of Victoria for the first time. There I found a ~500 year old copy of a book, which is second only to the Bible in the number of editions published. Euclid’s Elements is almost certainly the most influential mathematical text ever, so it’s only fitting that Euclid be our first dead mathematician. Thankfully copyright ran out 2 millennia ago — because copyright was a thing then, right? You can read it for free from Google here. Go forth and learn the basic building blocks of geometry as it should be taught!

Update: There is a great online version of the Elements here, with some awesome interactive java applets. Thanks to Mat for linking it in the comments!

The Feynman Sprinkler

It’s amazing how many of us understand the basics of quantum mechanics and general relativity, but still get tripped up by seemingly simple things. I’ve mentioned in the past that we’re still not entirely certain why a bicycle stays upright, which I think it pretty spectacular; we can put a man on the moon, but a bicycle thwarts us.

Recently I posted about a particularly nifty way to cool beer and I couldn’t be certain of the exact mechanism. Recently I had a good opportunity to use this method when I was having some knock-off beers with a bunch of physics PhD students, so I asked them to speculate on the mechanism too. It wasn’t immediately obvious to anybody there and even after the ensuing discussion, there still seemed to be some uncertainty. I think I’m pretty convinced now that it’s mainly evaporative cooling now though. (Although I’m yet to demonstrate this by experiment.)

So anyway, the point of this post wasn’t to talk about all that again. As a PhD student, Feynman and his fellow students found a problem in a hydrodynamics textbook that caused a bit of a stir. In fact after the students got nowhere with it, they took it to Feynman’s doctoral advisor, John Wheeler — who is a pretty damn big-shot himself — and even he didn’t have an answer for them. So what is it that thwarted some of the greatest minds of modern times? A sprinkler. A god-damn sprinkler.

"Physics is my bitch"

“Physics is my bitch”
Image credit: A. Jude

Imagine a simple sprinkler; so simple that it’s easier to show you a picture than to actually describe it.

The curly bits are rigid.

The curly bits are rigid.

Hopefully it is obvious that this will start spinning clockwise (as viewed from above) when we turn it on. What if some idiot neighbour kid decides to kick the sprinkler in the pool? Well obviously we go outside and beat that kid up for intervening with science. But what about the sprinkler? Will the sprinkler still keep rotating if it’s under water? Hopefully it is still obvious that the sprinkler will still keep rotating, just as before. This isn’t any different to having the sprinkler still above water but blowing air out instead; it will still propel itself around.

But now for the tricky part; the part that stumped the big-wigs. What if the direction of the water flow is reversed? Imagine that instead of blowing water out of the sprinkler, it sucked it in. If it helps, imagine that you stole the neighbour’s vacuum cleaner — apparently they had it coming — and you connected the vacuum to the sprinkler, where the hose goes in. We use the neighbour’s vacuum because the vacuum isn’t recovering from this. But the point is, the water is now flowing into the sprinkler instead of gushing out of it.

So which way does the reverse sprinkler rotate? Does it keep rotating the same way, or does it change direction? When the physicists discussed this, they all thought the answer was immediately obvious; half of them said it obviously goes the same way, while the other half said it changes direction.

One half: The ingoing water pushes the sprinkler arms in the direction that the water is flowing in, causing it to continue to rotate clockwise.

Other half: The entering water causes a suction effect where they water enters the sprinkler arms, causing it to rotate anticlockwise.

Since there was so much confusion, Feynman and his fellow students did the only reasonable thing they could do; they built a reverse sprinkler using equipment in a lab at Princeton. And like all good scientists, they conclusively established which way it would rotate and patted each other on the back for a job well done… Oh wait. No that’s not what happened at all! Excessive water pressure resulted in an explosion of water and shattered glassware.

It actually turns out that in most cases it doesn’t move at all. But in an ideal scenario (minimal friction) the sprinkler does start rotating in the opposite direction.

This reverse sprinkler is almost always attributed to Feynman now, despite his best efforts to tell people that he just found it in Ernst Mach’s textbook.

Graphiti – Oh yes, I punned.

I don’t have a well-formed opinion on graffiti in general; I can see the art in some of it, while appreciating that it’s still vandalism. But there’s a particular piece of graffiti that’s shown up in Boscombe, England that’s made a few headlines.

Pinched from The Daily Mail.

Pinched from The Daily Mail.

Before discussing what this actually is, let’s take a look at what other people have said about it.

The formula even had Bournemouth maths graduate Holly Crosby in a spin.
She said: ‘This is clearly a very advanced equation and I would like to have a go at cracking it.’
Daily Mail.

BAFFLED Boscombe residents are puzzled by a piece of graffiti in Boscombe Crescent which shows an advanced maths equation
Bournemouth Echo

And well, if you do some Googling then you’ll likely find a few more stupid things that have been said.

So why is this stupid? Well, imagine that you are a pianist – say it out loud, you know you want to – and somebody who knows a few chords mashes them together on a Casio while intermittently bashing their head on the keyboard, and the media starts calling it an “advanced piece of music”. This is what has just happened. In fact, this is pretty much standard practice when people want to depict math in movies. I also wonder who this Holly Crosby person is, and why a “maths graduate” would say such a thing. Let me be clearer,

THIS IS NOT A F^&$*#G EQUATION!

No more than what I described above could be called a music… Here, let me sum it up with a poem:

Flowers are bite my shiny daffodil ass in the wintmmer,
Like adverb piece to the night sky tortoises.
My you duck is one with the mountains,
but carry on my wayward son;
there’ll be peace when you are chocolatey.

It’s deep.

An equation is, by definition, something which equates things. An equation has an equals sign, and on either side of that are two things; two things, which are said to be equal according to the equation. And this certainly isn’t mathematics, because mathematics has context, a point and a whole lot more words.

With all that said, the graffiti looks pretty cool still. I just wish people wouldn’t act like it meant anything more than a drawing of mathematical symbols.

mathgraf1

Oh by the way, did you know that I can speak Greek too?

ασψλ ζβμχθ εδ… Or something.

How many points are in a square?

I’ve touched on the different kinds of infinities before, but since infinity is such a counter-intuitive beast, I thought we could have another look at it. First we’ll have a quick recap though. Consider the following questions:

  • Are there as many positive integers as there are negative integers?
  • Are there as many even numbers as odd numbers?
  • Are there as many integer multiples of 5 as there are, integer multiples of 7?
  • Are there as many prime numbers as there are negative integers?
  • Are there as many integer multiples of 48376, with 17 in their prime factorisation, as there are integers?
  • Are there as many molecules on Leonard Nimoy’s butt, as there are stars in the universe?

It might surprise you that the answer to all of these is actually yes. Except, of course, the last one; it’s off by 3. Of course, there’s an obvious first question here; what the hell to we mean by “how many”? I did address the notion of cardinality[See Also] in the earlier post, but I’ll give a quick reminder here. It is clear that there must be as many positive integers as there are negative ones, because we can pair them up; 1 with -1, 2 with -2, 3 with -3, etc. This pairing association is what mathematicians call a bijection. For each positive integer we can find it’s negative counterpart and for each negative one we can get the positive one back. Just remember that a “bi”-jection goes both ways.

If we can come up with a bijection between two sets of things, then we say that there are just “as many” things in each set; mathematically, we say that they have the same cardinality. So for each even number, we can associate an odd number simply by taking the number before it. We can associate the multiples of 5 with the multiples of 7 simple by pairing 5 with 7, 10 with 14, 15 with 21, etc. I’ll let you think about the others though. Basically we say all these things above are “countable” because we can write them out in a big list and “count” them out, one after the other. Since there is a 1st, 2nd and 3rd number then it makes sense to say there are as many positive integers as there are in each of the sets above.

Also in the previous post, I mention uncountable things; things that you couldn’t ever write in a list. The standard example is the set of all numbers. If we start with 1, what would the next number after it be? 1.1? 1.01? 1.0000000001? We can’t actually write down all the numbers in a list, so the set of all positive numbers is uncountable. The same goes for any interval too! How many numbers are between zero and one? It turns out that there are “as many” numbers between zero and one as there are numbers in general. But what’s more impressive (to me) is that there are many points between zero and one as there are points in a one-by-one square!

jkhg

Don’t be a smart-ass; I don’t mean the four pointy bits.

So to prove that there are as many points in the one-by-one square as there are between zero and one, we need to think up a unique way to pair up the points like before. Any point in the square is given by two coordinates; how far along the horizontal you are (from say, the left) and how far up you are from the borrom. And both ordinates have to be between zero and one. So now what we do is we take a number between zero and one and write out its decimal expansion, say for example, 0.123456789. Then we pair that number with a number in the square by putting every second digit to the first ordinate and every alternate digit to the second ordinate. So this point we’ve chosen would be paired with the point in the square whose coordinates are (0.13579, 0.2468). It’s easy to see that you can take the coordinates for a point in the square and then interlace both ordinates to get back the original point, 0.123456789. It’s obvious that you can pick and number between zero and one, then find its corresponding point on the square. And if you think that’s weird, just wait until you read about the Banach-Tarski paradox!

There is a very slight problem with making sure each number and square-point are uniquely paired. However, since I’m desperately craving your comments, I think I’ll leave it to the commenters to point it out (and fix it).

But does it work in theory?

In my previous post, I told you that adding milk to your coffee quicker will make it stay warm longer. To verify this, I did a little bit of experimenting and it was easy to tell which was warmer. Today, we take the mathematician/theorist’s approach. Because we totally didn’t procrastinate in the office today doing the calculation…

The equation we’ll use to describe our cooling coffee cup is Newton’s law of cooling; it says that the coffee will cool at a rate proportional to the temperature difference between itself and the room.

\frac{dT}{dt}=-k(T-T_{\text{room}})

Which probably means very little to half of the people reading this, but let’s push on! The left hand side of the equation is the rate of change of temperature – like how many degrees does it cool by in a minute – and the right hand side is just the difference between the coffee temperature and the room temperature, multiplied by some constant to account for different things cooling at different rates. The T is the temperature, T_\text{room} is the fixed room temperature and t is the time that the coffee has been sitting for. But since you may not know how to solve ODEs, the solution – spoiler alert – is given by

T=T_\text{room}+(T_\text{initial}-T_\text{room})e^{-kt},

where T_\text{initial} is the starting temperature of the coffee. e^{-kt} is the exponential function and will decrease “exponentially” – yes, that’s actually what that word means – as time goes on. And we’d expect that, right?

Now let’s plug some numbers in to make it easier to follow. Say the room is 25 degrees and the coffee starts at 100 degrees, Celsius. The constant, k, depends on various things and generally has to be measured, however the value of the constant won’t affect which cools faster; we’ll take it to be 0.15 here, when time is measured in minutes.

T=25+75e^{-0.15t},

So let’s say we want to drink the coffee in 15 minutes, at which point coffee will have reached T=25+75e^{-0.15\times15}=25+75e^{-2.25}. Now we add milk. If we like our coffee to be 20% milk and we add it now at 5 degrees, then milk and coffee average out their temperatures to reach thermal equilibrium, giving a new temperature

T_a=0.8\times(25+75e^{-2.25})+0.2\times5.

Now we’ll figure out what the temperature would be after 15 minutes if we had added the milk first. So the coffee and milk reach thermal equilibrium first, and then we say the milked coffee cools by Newton’s law of cooling. So immediately after adding the milk we have

T=0.8\times 100+0.2\times 5 =81

So we’ve got to plug this starting temperature into Newton’s law of cooling and then set t=15 again,

T_b=25+(81-25)e^{-2.25}=25+56e^{-2.25}.

Now we simple check which is larger!

Yep. I went there.

In the name of SCIENCE!!

Moving on…

T_a\approx 27
T_b\approx 31

So with our particular value of k, which is quite large, the coffee would be unpleasantly cool after 15 minutes in both cases. Although as expected, adding the milk earlier left the coffee warmer.

There are some interesting things about this calculation though. Firstly, it turns out that the temperature should be much the same in either case; 4 degrees isn’t overly noticeable. Regardless of the value of k or how long we let the coffees sit for, this coffee scenario will have a maximum difference of 4 degrees between the two options. So it probably doesn’t make much of a difference unless you use a whole lot of milk. If the milk is warmer than room temperature then it’s best to add it later, and if it’s sitting at room temperature then it doesn’t matter either way.

Now if you happen upon a dead body, you can use Newton’s law of cooling to estimate the time of death. Or if you’re the cause of the dead body, you can use Newton’s law of cooling to figure out what to set thermostat in order to make the murder appear to have occurred while you were in a crowded place with lots of witnesses.

And some people say math has no use in the real world… Pfft.

To be a scientist / Coffee and beer.

Recently @realscientists tweeted “When does someone become a scientist?”, and immediately I knew there would be two common answers to this; you’re not a scientist until you’re being paid to do it or you’re a scientist as soon as you do science. I responded with “When does someone become an artist?”, which – not to brag – was retweeted to the masses.

I don’t think anybody ever becomes a scientist. I think a scientist is somebody who puts their leftover pizza in the microwave, turns it on, and thinks “How the f$#& does that even work?” – and actually decides to learn how. When I was a kid, I’d take so much stuff apart to see what was inside it and try to figure out how it worked. Admittedly, once I got down to circuit boards I had no idea. Furthermore, I’m not sure my parents should have let me take apart random electronics but I don’t think either us really knew anything about capacitors back then. I remember when I first learnt what made light bulbs glow, I was determined to make one myself. I was probably only around 10 years old but I remember trying make a jam jar light bulb with a 9V battery and a piece of wire that I painstakingly shaved thinner and thinner. In an entirely dark room, I eventually got the tiniest bit of greenish glow from it for a fraction of a second. It wasn’t reproducible and I soon gave up after that.

Anyway, all that rambling wasn’t entirely the point. I wanted to mention a couple of basic science facts that I make use of when consuming my two favourite things: Coffee and beer.

Somebody please bring me some of this...

Someone please bring me some of this…

When you make a coffee (or tea, if you’re that way inclined), do you consider the time between pouring the coffee into the cup and adding the milk? You should. The rate at which the coffee cools depends on how far from room temperature your coffee is; your coffee will cool from 100°C to 80°C notably faster than from 80°C to 60°C. By the same token, adding 4°C milk to the cup will cool the hotter coffee more than the cooler coffee, so which one wins out? Well as a scientist, who thought he knew the answer when he started typing this, I am have boiled the kettle and am now conducting an experiment.
Results:

  1. My lips and several fingers are now scalded.
  2. Initial suspicions have been confirmed – add the milk as soon as possible if you want it to stay warmer longer.
  3. Now I have some coffee to drink.

Now for the next problem; you’ve just gotten home from a long day and realise that you’ve got no cold beer. At this very instant, you hate the world. But thankfully science comes to the rescue once more! Normally a beer takes about an hour in the freezer to get to a suitable drinking temperature, but ain’t nobody got time for that. However, if you wrap the can/bottle in a wet paper towel before putting it in the freezer then it will be at beer-drinking temperature in about 15 minutes. If you ask the internet why this works, people propose two reasons. The first is that the moist paper towel stuck to the outside of the can conducts the heat from the can far better than the aluminium (or glass); the second is that the dryness of the air in the freezer causes the water to evaporate from the paper towel, and the extra cooling is simply evaporative cooling. I’m fairly certain that evaporative cooling wouldn’t contribute that much to it is but since I can’t find a reliable source to confirm this, I guess I’ll have to perform another experiment!

Results:

  1. What was I doing again? The beer is all gone…

Root 2 and Proof by Contradiction

Today we’re going to talk about root 2. No, I don’t mean what you like to think of as your “first time” because the real first time was way too sloppy; I mean the number which is the square root of 2.

For many students at Monash, the first mathematical proof that they see is a standard proof of the irrationality of the square root of two. Today, you too will learn this. To recap, a number is rational if it can be expressed as a ratio of two numbers. For example; 1/2, 3/4, 5, 10, -17/3, 1001/73 are all rational numbers. Any number whose decimal expansion ends is also a rational number because numbers like 13.283947 can be expressed as 13283947/1000000. So how does one prove that \sqrt{2} is not rational?

Any proof I’ve seen of this involves what it called proof by contradiction, which may sound like an oxymoron of sorts but I assure you it’s legit math. Roughly speaking, it works like this; pretend what you’re trying to prove true is actually false and show that this leads to some kind of logical “badness”. More specifically, a proof by contradiction is set out as follows:

  1. Assume what you want to prove true is actually false (and make no other assumptions),
  2. Follow a logical argument until you reach something known to be false; for example 1=0,
  3. Now you can say “If the statement isn’t true then one must equal zero, which is absurd”.

Before looking at the irrationality of the square root of two, lets first consider a simpler – but also interesting – proof by contradiction; we prove that there are infinitely many prime numbers. First recall that a prime number is a positive integer with no divisors other than one and itself; that is, we can’t divide it by another positive integer without getting a fraction. But really if we want to check if a number is prime, we only have to check that it has no divisors that are prime numbers other than one (or itself). This is because if a non-prime is a divisor then this divisor has its own divisors; rinse and repeat until you have a prime divisor. In fact, every number can be uniquely decomposed into a product of prime numbers; this is called prime factorisation. With that aside, let’s prove by contradiction, that there are infinitely many prime numbers.

  1. Assume there are finitely many prime numbers.
  2. This means there is a largest prime number, so we will call the largest prime number P.
    Now if there are only finitely many prime numbers, then we can multiply them all together to get some new number, greater than P – let’s call this new number Q.
    Let’s look at the number R=Q+1 now. Since Q is the product of all the prime numbers then dividing R by any prime number leaves a remainder of 1 – that is, R is not divisible by any prime number. This means R is a prime number.
  3. Since R is a prime number larger than P, the largest prime number, we have reached our contradiction. This tells us that there can’t be finitely many primes (without breaking logic).

Ok, I lied. I’m not sure if that’s an easier proof or not, but I hope it makes sense. Now let’s talk about the square root of two. First we’ll cheat a little and acknowledge the fact that the square root of an even number can never be odd. I’ve phrased it this way because the square root may not be an integer, but if it is an integer then it must be even. This isn’t too difficult to prove directly, but I’ll leave that as an exercise for the commenters. Go on – do it!

Let’s follow the same idea and prove that it is not a rational number using contradiction.

  1. For the sake of contradiction, we assume that \sqrt{2} is a rational number.
  2. We can now write \sqrt{2}=p/q, for two integers p and q. Furthermore, we can assume that either p or q is odd, otherwise if they were both even then we could divide the top and bottom by 2 until eventually one of them was odd.
    Squaring both sides of the relation above gives 2=p^2/q^2, or q^2=2\times p^2. That is, q^2 is even, which from above means q is also even.
    Now if q is even, then it must be 2 times some other integer; let’s call it r.
    Since q^2=2\times p^2, we have 4\times r^2=2\times p^2; that is, p^2 is even. Again, we make use of the above and can say that p is also even.
  3. The contradiction here is that we began with at least one of p or q being odd, and now we’ve said that they’re both even too. Since they can’t be both even and odd, we’ve reached a contradiction; this tells us that the original assumption that \sqrt{2} is rational must be false.

Often students struggle with the idea of proof by contradiction, however this kind of argument is used in every day life too. For example:

Alice: Hey Bob, you left the damn door unlocked again.
Bob: No I didn’t, I swear.
Alice:

  1. Let’s assume that you locked the front door.
  2. We can safely assume that the house is secure and no intruders have entered.
    From this, we can say that the TV is definitely not stolen.
    But look, the TV is stolen.
  3. Since the TV can’t be simultaneously stolen and not stolen, we have our contradiction; the assumption that you locked the door is must be false.

Alice and Bob soon got a divorce.