Tag Archives: teaching

A math operation table for kids

I had such a thing in Greece when I was a kid. By pulling the i-th and j-th band on a plastic board their product was being revealed. That's how I had learned the multiplication table very quickly.

I wanted to buy something like this for my little one, but I couldn't find anything similar. All math tables work now with buttons. So I decided to construct one by myself.

At first I designed it with xfig: the prototype is table and the full elements are board and leisten. Then I printed them on 300gr carton and cut them accordingly; the board:

20160130_212859and the bands:

20160130_212923which were also cut into 20 elements:

20160131_114614Then I put it all toghether:

  1. bottom carton
  2. temlate carton, where I wrote the numbers but after putting it inside
  3. horizontal bands
  4. vertical bands
  5. cover carton with 100 quadratic holes
  6. stickers (s. below)

I fixed the band sliding by attaching small white stickers on the boundary between the bands (width = 1.4mm). On the right hand side, I fixed paper pieces with paper glue on the front side and post-it glue on the back side, so that I can pull them off and change the number board inside the construction (e.g. this one is for addition, I can change the template inside for sutraction or multiplication). Here the board with all squares closed:

20160201_121539Here with all bands open:

20160201_121722

And here some operations:

20160201_12160720160201_121628

 

The Adventure of π to the Normal distribution

Today, after giving the definition of the normal distribution a student asked me a very interesting question: how does the number π emerge in the Gaussian curve? The question is legitimate, because the Gaussian curve does not show any relation to the circle curve. I improvised an brief answer, and I will try to give here the path of this fascinating adventure of π. Firstly, the distribution function of a normal distributed random variable with expectation μ and variance σ2 is

\displaystyle\Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(t-\mu)^2}{2\sigma^2}}dt

and its limit is

\displaystyle\lim_{x\rightarrow\infty}\Phi(x)=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(t-\mu)^{2}}{2\sigma^{2}}}dt=1

This is a normalised form of the Poisson integral:

\displaystyle\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}

Now the question is why this equals  \sqrt{\pi} . For this I would give 2 answers:

1. Path

Substitution x2=t gives

 \displaystyle\int_0^\infty e^{-x^2} dx =\frac1 2\Gamma \left(\frac 1 2\right) = \frac{\sqrt{\pi}}{2}

And how is \Gamma \left(\frac 1 2\right) = \sqrt{\pi}?

This can be explained over the Gauss form of the Gamma function:

 \displaystyle\Gamma (x) = \lim_{n\rightarrow\infty}\frac{n!n^{x-1}}{\prod_{i=0}^{n-1} x+i} = \lim_{n\rightarrow\infty}\frac{n!n^{x}}{\prod_{i=0}^{n} x+i}

These 2 different forms give at ½ when multiplied with each other the double of the Wallis product:

 \displaystyle 2\lim_{n\rightarrow\infty} \prod_{k=1}^n \frac{k^2}{k^2 - \frac 1 4}=\pi

Now, why does W(n):=\frac{k^2}{k^2 - \frac 1 4} \prod_{k=1}^n converge to \frac \pi 2 for n\rightarrow\infty (Wallis product)? This is because of the recursive relation

 \displaystyle \int_0^{\frac \pi 2} \sin^nxdx = \frac {n-1} n \int_0^{\frac \pi 2} \sin^{n-2}xdx

The recursive resolution of the terms of the following fraction yields:

\displaystyle1=\lim_{n\rightarrow\infty}\frac{\int_0^{\frac \pi 2}\sin^{2n+1}xdx}{\int_0^{\frac\pi 2}\sin^{2n}xdx }=\lim_{n\rightarrow\infty}W(n)\frac 2 \pi

and therefore the equation of the Wallis product.

We see here that π comes from the sine function to the approximation of the Wallis product, then as the value of Γ(½) and from there it evaluates the square of the Poisson integral. A long path!

2. Path

The second path that I will present is shorter and more direct. Because

\displaystyle\left(\int_{-\infty}^{\infty} e^{-x^2}dx\right)^2=\int_{R^2} e^{-x^2-y^2}dxdy

calculating the left side should give the expected result (π). This can be done by integrating over a circle with centre in 0 and radius a (B(a)) and then by transformation to polar coordinates:

\displaystyle\lim_{a\rightarrow\infty}\int_{B\left(a\right)}e^{-x^{2}-y^{2}}dxdy=\lim_{r\rightarrow\infty}\int_{0}^{2\pi}\int_{0}^{r}re^{-r^{2}}drd\phi=\pi


References:

Forster Otto: Analysis 1; Differential- und Integralrechnung einer Veränderlichen, 6,. verbesserte Auflage, o.O., vieweg, 2001.

Κουνιας Στρατής, χρόνης Μωυσιάδης: Θεωρία Πιθανοτήτων Ι, Κλασική πιθανότητα, μονοδιάστατες κατανομές, Θεσσαλονίκη, Εκδόσης Ζήτη, 1999.


PS. Wikipedia has an interesting article about the calculation of the Gaussian integral.

A Rotation in R3

In the Summer term of 2009 I worked at the Technical University of Berlin as a Tutor for the lecture "Mathematics for Physicists". In the 5th exercise sheet there was an interesting exercise (3b):

Let

 X = \left(\begin{array}{ccc}<br />
0&-1&0\\<br />
1&0&0\\<br />
0&0&0<br />
\end{array}\right)

Show that etX is a Rotation at angle t and give the rotation axis.

I had given two solutions for this exercise. A non intuitive but elegant one and a technical one. I still like this approach, that's why I publish it here (in German).

For the first solution I was inspired by the idea, that multiplication with eti for real t is a rotation by angle t, and that i is represented by the matrix X but without the 3rd row and column in the set of all matrices of the form

 \left(\begin{array}{cc}<br />
a&-b\\<br />
b&a<br />
\end{array}\right)


with real a and b, which is isomorphic to the set of complex numbers (see the above link about this isomorphism).