# Archive for the ‘mathematics’ Category

## Probability and statistics videos

Posted by Ed on October 15, 2017

Here are some good videos on probability and statistics. I also found them perfect as a quick refresher.

Probability Course by Hossein Pishro-Nik
Video and online book

ActuarialPath
Videos

And here a full university course:

Probabilistic Systems Analysis and Applied Probability
Videos and assignments

## How to fix a wobbly table (with Math)

Posted by Ed on August 27, 2014

Here is a nice Numberphile video on how to fix a wobbly table with math:

## Optical illusions with impossible motions

Posted by Ed on March 3, 2014

The balls on the ramps seem to defy gravity, hmm…

## Toys in Applied Mathematics

Posted by Ed on February 20, 2014

Watch this great talk by Tadashi Tokieda presenting some surprising phenomenona in toys. This is quite fun to watch.

## Bagels and mathematics

Posted by Ed on February 15, 2014

## Free Math textbooks

Posted by Ed on February 14, 2014

Check out the website by the American Institute of Mathematics with a list of approved textbooks:
http://aimath.org/textbooks/approved-textbooks/

Subjects are calculus, differential equations, linear and abstract algebra, real analysis, introduction to proofs, combinatorics, discrete structures and probability.

## How to draw regular polygons

Posted by Ed on November 20, 2012

In this blog post we will derive a formula for the interior angle $\alpha$ of an n-sided regular polygon. We will examine how $\alpha$ depends on n. This allows us to draw the regular polygon.

Regular polygons
A regular polygon is a figure whose sides have the same length (equilateral) and whose interior angles are equal (equiangular). Let’s draw one with 3 sides. We can easily construct it if we know the interior angle $\alpha$. For the triangle it is $\alpha$ = 60°.

Let’s do the same for a square, for which the interior angle is $\alpha$ = 90°.

Now, can we do the same for a pentagon (5 sides)? What about a 13 sided regular polygon?

Obviously, we could if we knew how to choose $\alpha$. Before you continue reading you may want to ponder how to determine the internal angle. So, spoilers ahead.

Derivation of interior angle

We have two equations.
$\alpha + \beta = 180^\circ$ (1)
$n \cdot \beta = 360^\circ$ (2)

We solve (2) for $\beta$: $\beta = \frac{360^\circ}{n}$ (3)

Plug (3) into (1): $\alpha + \frac{360^\circ}{n} = 180^\circ$ (4)

Multiply (4) by n: $n \alpha + 360^\circ = n \cdot 180^\circ$ (5)

Substract 2*180° on both sides: $n \alpha = n \cdot 180^\circ - 2 \cdot 180^\circ = (n-2) \cdot 180^\circ$ (6)

Divide by n: $\alpha = \frac{(n-2)}{n} \cdot 180^\circ$ (7)

This is the interior angle for an n-sided regular polygon.

Problems

1) Show that the formula is correct for the triangle and the square.

2) What is the interior angle for the pentagon?

3) What is the sum of all interior angles in a 7-sided regular polygon?

4) What is the sum of all interior angles in an n-sided regular polygon?

5) What happens with the interior angle if you make n bigger and bigger?

1) Before we derived the formula we already knew that the interior angles are 60° and 90° respectively for the triangle and square. Let’s check if the formula gives us these values.

For the triangle we set n=3 and get $\alpha=60^\circ$.
For the square we set n=4 and get $\alpha=90^\circ$.

2) For the pentagon we set n=4 and get $\alpha=108^\circ$.

3) and 4) The sum of all interior angles in an n-sided regular polygon is $n\alpha = (n-2)\cdot 180^\circ$. For a 7-sided regular polygon we set n=7 and get $7\alpha = (7-2)\cdot 180^\circ = 5 \cdot 180^\circ = 900^\circ$.

5) If we make n bigger and bigger, the value for $\alpha$ approaches 180°. We can express this with the limit:

$\lim_{n\rightarrow \infty} \alpha$
$= \lim_{n\rightarrow \infty} \frac{(n-2)}{n} \cdot 180^\circ$
$= \lim_{n\rightarrow \infty} \left(1-\frac{2}{n}\right) \cdot 180^\circ$
$= 180^\circ$

## The sound of hydrogen

Posted by Ed on January 20, 2012

Henry Reich has a series of videos on physics and mathematics each lasting for a minute on his channel minutephysics. He explains topics by using handdrawn pictures and makes them look like an animation. Here are a couple of his excellent videos (among them one that explains how to create the sound of hydrogen):

## Introduction to Markov Chains

Posted by Ed on January 15, 2012

In the following I will give an easy example on Markov Chains. I will assume that you know how to multiply two matrices.

Example:
Suppose today it’s Monday and you want to celebrate your birthday on Wednesday with an outdoor party. Of course, you are interested in the weather and you find this data:

If sunny today, then tommorow:
80% probability for sunny
20% probability for rainy

If rainy today, then tomorrow:
60% probability for sunny
40% probability for rainy


This data could for example have been collected during summer. Note that the weather tomorrow only depends on the weather from today, so e.g. it does not matter what the weather was a week ago. So today on Monday, you go outside and see that it is a rainy day. Therefore, the probability that it will be sunny tomorrow is 60%. The problem is that your birthday is not tomorrow but in two days on Wednesday. What is the probability then?

A good idea is to visualize the situation as a graph:

These are the ways to Read the rest of this entry »