Sunday, April 25, 2010

On an Official Sort-of Announcement...

... which is merely informing you that these posts will no longer be occurring on a Sunday. They could occur on any day of the week, and may not occur every week. I do not know which day they will appear on, but I can guarantee you now that it will no longer be a Sunday. Become a fan of my Facebook page to be updated about new posts instantly...
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On Division by Zero

Following on from my post on π last week, another thing that is becoming rather popular within the geek culture is this notion that you cannot divide by zero. This annoys me because I can, and it’s really not that difficult. Just because your calculator can't do it doesn't mean it's impossible.

You may be expecting a warning about here about high mathematical content, but you’re not going to get one. The reason I decided to leave it out is because I don’t regard this as high level maths at all. In fact, it’s pretty fundamental. The average child will have zero covered in maths at school at around the age of seven, and should learn the basics of division at the age of nine. Therefore, I expect the average nine year old to be able to divide by zero.

All you need to get your head around is the fact that there are different types of zero. For example, even though 2 times zero is still zero, it’s a different zero. In fact, it’s twice the first zero, even though it is given the same zero, but this only becomes relevant in a couple of special cases...

In general, a positive number divided by zero gives infinity. A negative number can be made positive by taking out the negative. Zero divided by zero depends on the origin of the zero, so if you take the first zero in the previous paragraph and divide it by the second zero, you’d get a half. Since it’s not usually as simple as that, you’d simply apply L’Hopital’s Rule, which does require some elementary differential calculus, but it’s really not that hard.

Hopefully the world's so-called “geeks” and “nerds” will read this and stop making such a big thing out of zero...
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On Sheep-Pigs

Last week, three mangalitsa (commonly known as sheep-pigs) were transferred to a zoo in Essex in the UK.

It's a woolly breed of pig from Eastern Europe that is starting going extinct. They used to be popular because they're apparently easy to raise and they grow quickly, but since it's much easier to transport fresh meat over long distances, there is less demand for locally raised pigs. People prefer the imported meat because it is less fatty, despite the fact that the meat of the mangalitsa is actually juicier and has more flavour. People are strange.
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Sunday, April 18, 2010

On Communicating with Aliens

Popular belief has it that whatever intelligent life from outside of our solar system that we encounter will be so far removed from us that we will be unable to communicate with them. It is often said that we will need to demonstrate our intelligence to them by showing knowledge of mathematics. Most specifically, by using universal constants such as the value of π . This bugs me a lot, firstly because mathematics is not a universal language (It’s principals may be universal, but to assume that the way in which they apply is universal is completely ignorant. If you disagree, go try to find the similarities between Mandarin and Icelandic), and secondly, because we assume that this intelligent life will also make use of 3.1415... as their magical constant to map circular geometries into linear ones.

Speaking of mathematics, I did promise many months ago that I’d have a warning when I went off about that sort of thing, but then I never made use of it. So, as promised:

If you wish to remain socially functional, do not pay
to much attention to the technical details.

More importantly, the number π , the so called “universal constant” is starting to become extremely popular amongst the sort of people that like to think of themselves as part of the “geek” or “nerd” stereotypes. As a person who is often considered to fall into these stereotypes, I am ashamed that this number is held in such high esteem when I see π as an embarrassment.

π is essentially defined as the ratio of a circles diameter to its circumference, which is all very well. But what mathematical significance does the diameter of a circle have anyway? Well, the sad truth is, very little. I mean, a diameter makes perfect sense when you are talking about a circle, but when you get to more complicated, non-symmetric shapes, how would you define the diameter? A meaningful radius can be defined, even if it varies around the shape, but a meaningful diameter does not exist.

A far more meaningful constant to use would be 6.2831853... i.e. the ratio of the radius to the arc length of the circle. I’ll refer to this as Π . Then you’d be able to write a single equation which contains every single mathematical constant:

Which, in all honesty, makes -1 a far cooler number than π could ever be...

But I do admit... I have absolutely no dislike for pie.
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Sunday, April 11, 2010

On Fruits and Vegetables

Consider for a minute or two the consequences of building a house out of carefully treated and prepared fruits and vegetables. How would the house look? How would it smell? How durable would it be? Would it be a feasible idea in the long term? I mean very long term, as in would it be comparable to more conventional materials, such as wood, bricks, steel, concrete and the like... Think about that for a minute or two, and see what you come up with.

Then imagine a pea. A single green roughly spherical seed from a legume, approximately 7 millimetres in diameter, containing carbohydrates held together in a tightly packed unit by glycosidic bonds and held together with a soft protein shell, to feed a single microscopic cluster of rapidly multiplying and mutating cells that serve absolutely no other purpose except to consume their surroundings, eventually spread out into a complex network of intelligently controlled cells, eventually produce more peas before being discarded completely, to rot and decay and disintegrate and eventually dissipate. Only a tiny fraction of the peas will ever serve this purpose. Most will be eaten by some other cluster of rapidly multiplying and mutating cells, some of these clusters being consisting of tens or hundreds of trillions of cells. All these clusters serve that single purpose: consume, spread and multiply. These are all attached to another single roughly spherical object which is two billion times bigger than the original pea, containing liquid metal silicates held together by a mutual gravitational compression, with a brittle magnesium silicate and aluminum silicate shell which serves no purpose, and only exists because it is stuck in a never-ending plummet towards (although eternally missing) yet another roughly spherical object one hundred times bigger than itself. This sphere itself only exists as the result of an uncontrolled self-sustaining nuclear fusion explosion that was detonated by a collapsing gas cloud formed in another unbelievably massive, yet microscopic explosion billions of years previously. The collapse of the gas cloud has been temporarily stopped by the nuclear reaction, but will inevitably resume as soon as the gaseous fuel has been used up, until it is a hundred thousandth of its current size. This is in turn drifting in its own slow never-ending plummet towards a single point of oblivion along with several billion similar collapsing gas clouds, and this point of oblivion is drifting away from several billion other similar points, but gradually slowing down, until the mutual pull of all the billions of points in the universe cause a collapse into a single point, which is only a figment of the mathematics’ imagination, and doesn’t truly exist in the real world, and this temporary quantum fluctuation can no longer be considered to truly exist. Peas really are fascinating.

Did you forget about the house? Think of it now. And now think of it tomorrow, and in the next week, in the next year, in the next decade, in the next century, in the next age... Is it still a house? Did you think of it as a hollowed pile of untreated decaying vegetable matter, or did you consider the carefully treated and prepared fruits and vegetables that formed a single carbon supercrystal that will last undamaged for millions of years.

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Sunday, April 4, 2010

On Some More Indecisive Mathematics

Solution time! I don’t just like to point things out, but I like to answer questions too. Last week’s post posed an interesting paradox. Those of you with knowledge of science, mathematics or logic would know that reality does not allow paradoxes, and they always arise out of some misunderstanding.

It’s all a trick, actually. A magic trick, like where the magician slips a coin out from behind your ear, or has an extra card up his sleeve. It’s only a matter of distracting the audience when a third envelope is added into the problem. It was never mentioned explicitly, but I’m sure you all noticed that there are in fact three possible amounts in the calculation, namely 0.5x, x and 2x, even though in the problem, there are in fact only two envelopes. The correct solution goes as follows. Let the envelops contain amounts x and y. The relation and values between these two is not important at this stage. Assuming equal probability of having either envelope, the expected value is (x+y)/2. Now the relationship y=2x can be introduced, giving an expected value of 1.5x, which is what intuition tells us anyway. Granted, this is still more than x, however, this solution made no assumptions about the contents of the envelope we are holding, so the expected value has a 50/50 chance of being more or less than what we already had originally... Which is what intuition tells us anyway.

Consider now a completely different problem of a similar nature. And this time, there is no paradox, no lies, no hidden tricks. I will be completely honest with you on this one. Consider that there are now three envelopes, and only one contains any money (and I know which one does, but you don’t). I let you choose one envelope at random. I then show you the contents of one of the remaining envelopes, which turns out to be empty. I then give you a choice to either keep the envelop you have, or switch it with the unopened envelope I have.

In this case, the maths behind it all reveals that you are twice as likely to get the money if you change your mind, even though your intuition tells you it shouldn’t make a difference. Think about it for a while, and then read the greatly simplified (rather non-mathematical) explanation in the comment.

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