Math Fact of the Day

[Mashi]

Here, I will be giving you a daily Math fact.  99.9999999% of the time, it will be a frivolous fact.  But it will be a fact!!!


Today's Math fact is that, of 2D shapes with the same perimeter, a circle will have the largest area.  And of 2D shapes with the same area, a circle will have the smallest perimeter.

What this implies (through some higher level Math I won't go into) is that, of 3D shapes with same surface area, the sphere has the largest volume.  Likewise, of 3D shapes with the same volume, a sphere will have the smallest surface area!

I won't go too in depth, but this is part of the reason why things like our planets and the Sun are more or less spherical in shape.  For the Sun (and stars in general), having a high Volume to Surface Area ratio compared to all other shapes allows the star to keep as much energy stored in itself as possible (which is related to its Volume), while releasing as little energy to the environment as possible (which is related to its Surface Area).

[mikey]

Oh come on we totally all knew that.
Unfortunately circles are much too awkward to fit together so we use 2:1 rectangles more often.

[Bloop]

I didn't know? ;_;

[Dude]

Fixed the title.

[Ruto]

@Mashi

Maupertuis principle? That was what we used to explain the frequency of spheres and circles in nature during this calculus course I took years ago with this rather awful calc professor.

[Mashi]

Oh come on we totally all knew that.

i went through the trouble of using LaTeX to show you that you had better like it!!!!!!!!

Also, Ruto, yes!!!

[Mashi]

Math Fact #2:

An event with 0 probability is still possible.  We just define it to be almost surely improbable.

Take this example: I'm thinking of an integer between 0 and ∞.  What's the probability that you'll guess it correctly the first guess?

Let's assume there's an equal chance of you choosing any integer as your guess.
You can't use this equation all the time, but we can use it in simpler cases of probability like this one:

In this case, a success would be guessing the correct number and a trial is considered to be any number that can be guessed.  This means that the number of successes is equal to 1 and the number of trials is equal to ∞.

However, 1/∞ is an algebraic impossibility.  So we instead have to make it a bit more complicated and transform the equation to this form:

Where n is # of trials, we find that the result is equal to 0!

But is it not possible to guess my number despite it being infinitely unlikely?  It is still possible!  Perhaps my number was 2038432 and by the offchance, you guessed it correctly.  Incredibly improbable but possible, no?

So clearly there's something wrong with how we do statistics or how we interpret them.  This is a case of the latter!  An event (in this case, the correct guess) with a probability of zero is one that is statistically insignificant, but not necessarily impossible.
If this probability zero event seems unreasonable to you, think of it from a different perspective.  Let's say that we have a coin that will always have a 50% chance of landing on heads and a 50% chance of landing on tails (so it'll never land on its side or otherwise).  Let's say we want to calculate the number of times we flip heads a Number of trials, n.  In which case, we can use this formula:

So there's a 50% chance of getting heads for one flip, 25% for two consecutive fllips, 12.5% for three, etc.  All the way down to ∞ flips, where the probability is zero.
But it's certainly imaginable to flip every single time if we could go on flipping infinitely many times, no?  Just really really really really really really really really really unlikely!


You might see this probability zero event problem in applied statistics from time to time.  This is in part due to how calculus works and how infinity hates us and makes everything complicated.

This is a common example, but let's say you have a dartboard and a dart.  Let's also say that you hypothetically will always throw the dart onto the board, no matter what.  So the only places you can hit with the dart is on the dartboard.  Let's also say that you throw the dart in an equally random way each throw, so that the dart has an equal chance of landing anywhere on the dartboard.
Contrary to intuition, the probability of hitting a specific point on the dartboard is 0!  The reason for this is because, in calculus, we consider each point on the dartboard to be a differential of essentially negligible area.  If that's rather recondite to go over, think of it like this: the length of a point is 0, the area of a line is 0, the volume of an area is 0, etc.  In this case, we have a point and we're trying to find its area, which turns out to be 0.  And you can't really hit something without an area!
Of course, in real life, we don't hassle ourselves with dealing with differentials.  Doing so allows us to find meaningful results in probability.  So instead of finding the probability of hitting a point (i.e. differential), we find the probability of hitting a very small area instead.  That way, we avoid the dilemma with the probability zero events while still finding statistically significant results.



tl;dr - ∞ h8s us

[Maelstrom]

Makes sense.

[blueflower999]

These are really interesting.   

[Mashi]

ooops i screwed up

it's supposed to be and not

I'll fix it.

[mikey]

Makes sense.

It's cause he used limits.
ANYTHING IS POSSIBLE WITH LIMITS!
Like finding the slope of a line with only one point given lol

[Jamaha]

i went through the trouble of using LaTeX to show you that you had better like it!!!!!!!!

Ooh.

[braix]

my head hurts

[Bloop]

I saw weird shapes and shit
I don't understand weird shapes
I only know the ancient greek pronouncation for them

[Mashi]

sry short update today i was busy

We use slope to determine the rate of change of a certain measurement.  An example would be a graph of displacement versus time, where the slope would yield the velocity of an object.

A similar concept is that of curvature, which in non-technical terms is exactly what it sounds like; how curvy something is.  A line has a curvature of 0.

The mathematics requires to calculate this requires some calculus, so I won't go into it, but the curvature of a circle is 1/r, where r is the radius of the circle.

So what happens as r approaches 0?  The circle becomes smaller and smaller and we begin to measure to curvature of a point, which is ∞!

And what happens as r becomes bigger and bigger?  The circle becomes bigger and the curvature approaches 0!  In fact, as r approaches ∞, we get a curvature of 0.

Do you remember what has a curvature of 0?  A line!  So, in a sense, a line is the same thing as a circle, but with an infinite radius.