More Remarks on Corona

What have we learned so far? The numbers are not clear, nor are the consequences. Moreover, there is a lot of distorted information, some can even be called fake. Let me tell you what I took from all that.

  • The most important problem that the new virus is causing is the exceeding number of cases of pneumonia and ARDS caused by fluid in the lungs. This seems to be a lot worse than in the ordinary flu or older Coronaviruses, probably because we are not yet immune. The virus is also causing other damages to your body, but lung problems are the most common ones. While the chances to need oxygen or ventilation are not known precisely yet and will depend on your age, the situation in the hospitals of the affected regions speaks a clear language. We need to concentrate on medical care and break the chains of infection to slow down the development of the disease!
  • The mortality seems to be high only in people of old age, especially those that are weakened by other diseases. We might observe that the overall mortality does not rise much at all. The number of deaths by COVID19 published so far all over the world supports this view. The conclusion is to protect the elderly as much as we can to spare them a painful and sudden death requiring intense medical care. We do that, again, by reducing social contact and keeping the number of infected people at any time as low as possible. By the way, hygiene and care should have been observed in contact with elderly people at all times.
  • The current economy is based on the production of consumer articles and marketing. Many countries have reduced or have been forced to reduce state activities that do not lead to direct profit, such as health care and education. This economy is not fit for any crisis that needs social solidarity. Since profit and interest are the driving forces and the main means of wealth distribution each recession throws us into a deep crisis. Politicians see that now and try to stabilize the situation. Essentially, this is a takeover and a regulation of the industry by financial means, which can be summed up as printing money. Even the idea of state control of key industries is revived. We should learn from that crisis that there must be a balance between the private sector of the economy and social needs covered by the state at all times. If that is out of balance, we are not flexible enough to fight a crisis like this one.

Keep healthy and stay at home for a while!

31. März 2020 von mga010
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Remarks on the Corona Pandemic

Here are some remarks that I noticed in these difficult times. Some are math-related, some are not.

  • Humans seem to be unable to image big numbers or grasp the concept of exponential growth. In fact, I cannot really do that myself. What I have to do is compute figures and set them in relation to the total. To help us understand the problem here is one example: Letting the virus spread in a small town of 10000 can easily mean that 1/4 of them are infected at the same time, i.e., approximately 2500. About 1/20 of those need intensive medical treatment, and at least 1/100 need ventilation. We end with a number of 25. This is impossible to do for any countryside hospital. Those small numbers can be understood and they should frighten you to the point that you understand how necessary it is to break the chains of infection. Of course, the same computation for the complete US or German population is even more frightening.
  • Most of us do not understand how bad it is that the world population will grow to eleven billion in the near future. We should have stopped that by education, sharing of wealth and women’s rights thirty years ago. Moreover, we have destroyed local support chains by globalism and our excessive capitalistic system. This falls on our feet now. In African countries, the population has to live in slums at close contact and is depending on the global food chain, and also on the medical support of „first world“ countries. This is a recipe for disaster. Even we are affected by the problem because our support with masks and other medical aids is broken since China is no longer delivering. My only hope is that we come to rethink our world after that crisis.
  • We all cannot grasp probabilities properly. I am now over 60 years old. That means that my non-corona chance of dying in the next year exceeds the corona related chance, even if I catch the virus. I am more likely to die of a heart attack, get brain or lung cancer, get involved in a car accident, or get usual pneumonia during that year. We cannot avoid death. It is never a good idea to think about the chance to die too much. Remember the words of Epicurus: „The death is none of your business. When he is here you aren’t. When you are here he isn’t.“
  • The press and TV are now running on full speed to shovel numbers onto the public and present the drama of single cases. Unfortunately, this agenda together with a lack of social contact may cause a lot of harm to some individuals. If only we would take other causes of unnecessary deaths with the same degree of attention! I would prefer a more scientific attitude in the media, and more reticence to dramatize individual fates. This only leads to a distorted view of reality.

24. März 2020 von mga010
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Mathematics of an Epidemic Disease

Since we are currently fighting the Corona epidemy I thought it might be a good idea to simulate the spreading of such a disease in Euler Math Toolbox. The numbers here are fictitious and can only deliver qualitative results.

The mathematical model I took is quite simple.

  • The disease has n1=7 days of incubation. During this time the infected person can infect others.
  • The disease has n2=7 days of severe sickness. I have assumed that there are no further infections during that time.
  • We start with the a-immune population plus 10 infected cases.
  • The chance for infection depends on the probability to contact an infected person during the incubation time and the number of contacts each day that could lead to an infection. I assume that a person has nc=0.2 to nc=0.3 contacts each day which could lead to an infection.
  • In the end, the population is either in the non-immune or the immune group.

So the total population is mapped into a vector x of groups as

  • x[1] : non-immune
  • x[2] to x[1+n1] : incubation
  • x[2+n1] to x[1+n1+n2] : sick
  • x[2+n1+n2] : immune

The formula for the number of new infections in x[2] (first day of incubation) is

\(x[2] = x[1] \left(1-(1-\frac{S}{P})^{n_c}\right)\)

where S is the number of incubated people and P is the total population. So 1-S/P is the chance that the contacted person is not infected, and thus the factor computes the chance for infection. Note that this factor is very small at the start since S/P is very small. Yet the exponential growth does its job over time.

Here is a typical plot of the epidemy with nc=0.3. This is equivalent to contacting 9 other people in a month in a way that can spread the disease. We see that almost 90% of the population has to go through the disease in this case.

Development with nc=0.3

Here is another plot with nc=0.25, equivalent to contacting 7.5 other people in a month.

Development with nc=0.25

That reduces the number of cases a lot and limits the peak number of sick persons from 37% to 22%. For the medical system, this is an enormous relief.

The code of Euler Math Toolbox to do these plots is listed below. You can cut and paste it into EMT to try yourself.

>n1=7; // 7 days of incubation time
>n2=7; // 7 days of sickness
>pm=0.01; // mortility rate
>pop=50000000; // population
>nc=0.3; // number of infectious contacts
>function oneday (x,n1,n2,pm,pop,nc) ...
$xn=x;
$xn[2]=floor(xn[1]*(1-(1-sum(x[2:1+n1])/pop)^nc)); // new infections
$xn[1]=x[1]-xn[2]; // remaining uninfected
$xn[3:2+n1+n2]=x[2:1+n1+n2]; // progress one day 
$xn[-1]=xn[-1]+x[-1];
$return xn;
$endfunction
>x=zeros(2+n1+n2); x[1]=pop; x[2]=10; // start with 10 cases
>function sim (x,n1,n2,pm,pop,nc) ...
$X=x;
$repeat
$   xn=oneday(x,n1,n2,pm,pop,nc);
$   X=X_xn;   
$   until sum(xn[2:1+n1+n2])==0;
$   x=xn;
$end;
$return X;
$endfunction
>X=sim(x,n1,n2,pm,pop,nc);
>Xs=X[,1]|sum(X[,2:2+n1])|sum(X[,2+n1:1+n1+n2])|X[,-1];
>plot2d(Xs'/pop,color=[green,orange,red,blue]); ...
>labelbox(["immune-","infect+","sick","immune+"], ...
>  colors=[green,orange,red,blue],x=0.4,y=0.2,w=0.35):

We can also understand the problem a bit better if we study the number of new infections in x[2], the first day of incubation. It starts with our chosen number of initial infections. But those 10 induce 3 new infections and the 13 new take another 3. Then we have 16 and they infect 4, and so on exponentially.

New Infections

If the chance S/P of an infected contact is small we have

\(\left(1-(1-\frac{S}{P})^{n_c}\right) \sim \frac{n_c S}{P}\)

Thus

\(x[2] \sim n_c S\)

exactly as observed.

13. März 2020 von mga010
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Another Youtube Problem

Recently, I found the following problem on Youtube:

\(x+y+z=1, \\ x^2+y^2+z^2=2, \\ x^3+y^3+z^3=3, \\ x^n+y^n+z^n= \,?\)

I did not continue to watch the video. So I have no idea what solution is presented. I decided to treat that as a challenge and to observe my steps, including the failed ones.

The problem sounds like a normal set of three equations with three unknowns that we can solve by eliminating. After all, this is the standard technique that we learn in school. But it is actually not that easy. Eliminating x from the second equation and solving for y yields an expression in y which contains a square root. Inserting everything in the third equation results in a mess.

Let us try numerically. We use Euler Math Toolbox here.

>function f2(x,y) &= 2-x^2-y^2-(1-x-y)^2
 
                        2                2    2
                     - y  - (- y - x + 1)  - x  + 2
 
>a3 &:= 3
 3
>function f3(x,y) &= a3-x^3-y^3-(1-x-y)^3
 
                        3                3    3
                     - y  - (- y - x + 1)  - x  + 3
 
>plot2d("f2",levels=0,r=2);
>plot2d("f3",levels=0,>add):

We see, that the equations do not have a real solution. But they have, of course, a complex solution by the main theorem of Algebra.

The second idea is to try some tricks on the equations. E.g., we could multiply the first equation with x, y, and z and add the three results. Because of the first and the second equation, we get

\(xy+yz+xz=-\dfrac{1}{2}\)

That is a new equation. But is not clear how this is going to help.

After a while, the mathematical instinct kicks in to try something more generic and simple. Pattern matching inside the brain brings a known trick to the mind. Any sequence

\(1,x,x^2,x^3,\ldots\)

has an interesting property. If x is the zero of a polynomial

\(p(t)=t^3+at^2+bt+c\)

It satisfies the interesting property

\(x^n+ax^{n-1}+bx^{n-2}+cx^{n-3}=0\)

Thus the sequence above satisfies a recursion formula. Now, if x,y,z are the three zeros of the polynomial p(t) all three satisfy the same recursion formula, and so does the sum of the three zeros.

\(s_n = – (as_{n-1}+bs_{n-2}+cs_{n-3})\)

for

\(s_n = x^n+y^n+z^n\)

So we know from our three equations and the trivial equation

\(s_0 = x^0+y^0+z^0=3\)

that

\(3 = -(2a+b+3c)\)

But we also know that

\(p(t) = (t-x)(t-y)(t-z) = t^3 – (x+y+z) t^2 + (xy+yz+xz)t -xyz\)

Once we see that, we get a=-1 immediately from our problem. Having played around with tricks, we also know b=-1/2. From that, we now have

\(a=-1, \quad b=-\dfrac{1}{2}, \quad c=-\dfrac{1}{6}\)

We can verify that with Euler Math Toolbox.

For that, we compute the zeros of the polynomial and check the equations numerically. As expected, two of the zeros are complex.

>s=polysolve([-1/6,-1/2,-1,1])
 [ -0.215425+0.264713i,  -0.215425-0.264713i,  1.43085+0i  ]
>sum(s)
 1+0i 
>sum(s^2)
 2+0i 
>sum(s^3)
 3+0i 

The expression for the real zero of p(t) is terrible and involves a third root.

\(z=u^{1/3} + \dfrac{5}{u^{1/3}} + \dfrac{1}{3}, \quad u=\dfrac{\sqrt{13}}{9 \cdot 2^{2/3}}+\dfrac{11}{54}\)

Thus, we get a recursion for the sum and thus for the equation sign in our problem.

\(s_n = s_{n-1}+\dfrac{1}{2}s_{n-2}+\dfrac{1}{6}s_{n-3}\)

Check with EMT.

>fraction sequence("x[-1]+x[-2]/2+x[-3]/6",[3,1,2,3],10)
 [3,  1,  2,  3,  25/6,  6,  103/12,  221/18,  1265/72,  905/36]
>fraction real(sum(s^[0:9]')')
 [3,  1,  2,  3,  25/6,  6,  103/12,  221/18,  1265/72,  905/36]

Can we make a problem with an easier solution? Sure! We only have to start with simpler zeros.

\(x+y+z=0, \\ x^2+y^2+z^2=14, \\ x^3+y^3+z^3=-18, \\ x^n+y^n+z^n= \,?\)

The answer is now

\(s_n = 1 + 2^n + (-3)^n\)

This kind of mathematical thinking looks like a touch of genius. But it is not. Not at all. It is just the summary of long experience in mathematics. The tricks involved are well known. They are just combined in the right way. Mathematical problem solving is just pattern matching using the patterns that have been exercised before. Maybe some stubbornness is needed, and a lot of trial and error. But no genius.

It does help very much to have a numerical and symbolic program like EMT available to check the computations.

02. März 2020 von mga010
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The Two Envelopes Problem

Recently I came across a Youtube video by „vsauce“ explaining the two envelope paradox. The problem is at least half a century old, and most likely a lot older. It can be found in a 1953 book by Maurice Kraitchik which is the oldest citation in Wikipedia.

Someone places money into one of two identical envelopes and double the money into the second. We select one of the two envelopes without knowing if we have the first or the second one. We open it. Should we keep that money or return it and opt for the other envelope?

(Let us assume that the value in the first envelope is a rational number. So we cannot conclude with certainty which envelope we have opened. If we selected from integer numbers we would know the envelope in the case of an even value.)

The naive analysis goes like this:

Let X be the amount we have.  Since we have chosen the envelopes at random, the other envelope contains X/2 and 2X with the same probability. So our expectation after switching is

\(\dfrac{X/2+2X}{2} = \dfrac{5}{4} X\)

So switching sounds like a winning strategy. It gets more striking if we consider switching before even opening the envelope. Switching back and forth drives the expected value to infinity. This is clearly nonsense.

This paradox is a confirmation of my principle:

To analyze a probabilistic problem we need an experiment that we could, at least hypothetically, implement in a computer. Equivalently, we need a probability space to work with.

The two envelopes problem does not satisfy this principle. The reason is that it is impossible to select a rational number at random from the set of all rational numbers. You cannot even select an integer evenly from all the integers. We need a probability distribution for that. Think of how you would test the switching strategy on a computer. You will have to tell the machine how to select a number. It can select a number with equal probability from all the numbers it can represent. But it cannot do so from all, infinitely many numbers.

Now that we know that the original problem makes no sense, we can start to analyze the problem by assuming an a priori distribution for the value M in the first envelope. By doing so, we get a bunch of very interesting questions. (By the way, it is always easy to create questions in mathematics. Only a few questions have nice answers, however.)

Let us start by selecting M evenly from 1 to n. Clearly, we switch if the opened envelope, containing X, is odd. We will keep the money only if X>n. For n which is a multiple of 4, we have the following cases:

  1. M<=n/2: We will switch no matter if X=M or X=2M. Thus the average outcome is 3/2 M.
  2. M>n/2: We get 2M in every case.

It is an exercise to compute that our average result is

\(E = \dfrac{15n+14}{16}\)

This is a blog about Euler Math Toolbox too. So let us simulate the strategy in EMT. In this case, a loop would be a good idea, maybe written in C or Python. However, it is also possible to do this elegantly with the matrix language

>n=10; N=1000000;
>M=intrandom(N,n); // select a number between 1 and n
>sel=intrandom(N,2); // select 1 or 2 envelope
>X1=M*sel; X2=M*(3-sel); // X1 is in the selected, X2 in the other
>sum((X1>n)*X1+(X1<=n)*X2)/N // X1 for X1>n, X2 else
 10.253055
>(15*n+14)/16 // expected outcome
 10.25

We could also try a distribution on the positive real numbers for M. A typical candidate is the exponential distribution. We decide ourselves to keep a value greater than some a>0.

In this case, we switch for all M<a/2 and M>a, getting 3/2 M on average. In half of the cases between a/2 and a, we get 2 M. In the other half we still get 3/2 M. Thus

\(E = \dfrac{3}{2}+\dfrac{1}{2} \int\limits_{a/2}^a x e^{-x} \, dx\)

The best choice is a = 4 ln(2) with

\(E \sim 1.68\)

We can simulate that too.

>n=10; N=1000000;
>M=randexponential(N); // select a number between 1 and n
>sel=intrandom(N,2); // select 1 or 2 envelope
>X1=M*sel; X2=M*(3-sel); // X1 is in the selected, X2 in the other
>a=4*ln(2); sum((X1>a)*X1+(X1<=a)*X2)/N // X1 for X1>n, X2 else
 1.68097393969

Note that switching all the time or never both expect E=3/2.

You might ask if there is a distribution for M which ensures that it does not matter to look inside the first envelope. But that is not the case. Such a distribution does not exist.m Knowing the distribution and looking at the first envelope will always yield an expected overhead for us.

03. Februar 2020 von mga010
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Two Tilted Rectangles

I found this problem from the wonderful Math with bad Drawings site in my news feed. They cite it from Carolina Shearer. Her twitter account contains more such nice problems. It is the kind of problem which seems only adequate for advanced students. Sometimes, you can solve them by looking at it in the right way. Most of the time, you will start a lengthy computation. Often, you will notice in retrospect that the solution was quite easy and you could have guessed it.

For this problem, I did not see the solution. The problem, of course, is to fit two equal rectangles into a square in the shown manner. For a start, try to find a reason why the point 1/2 on the lower side solves the problem. You can use the general fact that a=b+c in the figure below.

 

I admit that I did the computations using Euler Math Toolbox. That works. However, there is an elementary fact that can be used to prove a+b=c. For a hint, consult the following image. If the green lines are equal the red lines are equal.

 

05. November 2019 von mga010
Kategorien: C.a.R. - Z.u.L., English, Euler | Schreibe einen Kommentar

Euclidea App

Geometry is not as popular nowadays as it was. So I was very surprised to learn about a game that features geometry puzzles. It poses problems of Euclidean geometry in the plane, e.g. constructing the tangent to a circle. To solve the problem fully, goals have to be completed. One goal (L) is to do it with the minimum number of steps with all tools available (which include composite tools like the middle perpendicular). Another goal (E) is to do it with the minimum number of elementary tools, which are circles and lines. The third goal (V) is to construct variations of the solution, like reflections of the solution. The goal (E) is often very difficult even for elementary constructions. I show two examples below.

The app is done by some Russian programmers. I could find no information on the authors at all. Actually, there are lots of Russian programmers that make money with their skills in different areas. Besides gaming, an example is flight simulation where Russia programmers made some of the best addons. The programmers are not necessarily skilled geometry geeks. There are old books buried somewhere in libraries that contain lots of geometry problems and solutions. Those are the reminders of the old times when geometry was still popular.

The app is extremely nicely done and shows dedication and skill with user interfaces. Some details could need perfection and I will talk about that later. The app is free. But you can buy more problems. It is a nice touch that you get these for free if you have solved all the previous goals. However, I am considering to donate to these programmers.

Here is an example of a problem. I present the problem with my own program C.a.R. This program could also pose problems with restricted toolsets, called assignments. However, there is no game feature and I never cared about the number of steps taken. It was a program to help to teach geometry in schools.

Your job is to construct the green line given the black lines. In the app, the given objects are not fat and the user cannot set colors. That makes it more difficult than it should be to keep track of the progress.

Of course, it is quite easy to do the job with two given tools, the line tool, and the perpendicular. Moreover, every geometry student should know the following solution.

It takes 5 elementary steps. It is a typical mathematical solution because it is based on another solution to another problem. Mathematics often does that. We know how to construct the middle perpendicular. Thus we know how to construct a perpendicular to a line through a point on the line. It remains to see the tangent as perpendicular to the radius. But the goal was to do it with three elementary steps, not 5. My second solution was the following.

It takes 4 elementary steps still and is based on another well-known geometry fact, the Thales circle. Observe the brown circle (2) to see why this works. It is still not the optimal solution. I now tell you a secret: You can find all solutions on Youtube. Nowadays, going to Youtube is the first step that students take.

The following is the solution with three steps. Unfortunately, I was given that solution before I could find it. Why it works is by no means obvious.

The only proof I could find is a computation with angles.

By basic equivalences, we get that the angles with equal color are equal. By the circle chord theorem, the angle of a chord of a circle with respect to a point on the circle is half the angle with respect to the midpoint of the circle, thus black=2*blue. We also have black+red=90°. Thus black+2*blue=90°.

There should be a simpler proof to this, but I have not yet found it.

All in all, this is an extremely well-done app. I would like to use colors while constructing. I would also like to use the given segments as lines. This is important in the following example, where you have to construct the center of the inscribed circle with only 6 elementary steps. Note that the usual construction takes 4 elementary steps for each angle bisector.

That is already very complicated. The first step is to construct one angle bisector using the blue circles. The second bisector is found with the mysterious red circle which is constructed by one of the blue circles. Its second intersection with the same blue circle yields the second angle bisector. I found this by trial and error. After all, you can construct the angle bisector in C.a.R. so see where it should be. In the app, you can show the targets too. Did I tall you that you can also find the solution on Youtube?

Does anybody have an easy proof?

10. Oktober 2019 von mga010
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A Geometry Problem

The internet, and especially Youtube, are a vast resource for mathematics now. You find all sorts of problems, explanations, and tutorials, usually done by very talented people. I like to dig around in that pile and soon find some pearls like the following one. The problem has been presented by a guy named Presh Talwalkar. Some of his problems I knew already, but this one was new to me.

The problem is to determine the area of the quadrilateral with the question mark. Of course, no angles or side lengths are given. It took me a while to figure out a simple solution.

The first idea is to put all this in algebraic terms. To make things simpler, we would put the lower-left corner to (0,0) and the lower-right one to (0,a), and the third one to (b,c). The points on the left and right sides of the triangle can then be determined using algebraic geometry (depending on a,b,c) and thus the area in question. This looks tedious but is a really nice exercise. Do not forget the determinant formula for the area of a triangle. Let’s do it!

Euler Math Toolbox (EMT) happens to have a geometry package which can use Maxima to compute symbolically with geometric objects. It turns out to get more involved than expected. Trying to do this by hand seems to be really hard work.

>load geometry
 Numerical and symbolic geometry.
>A &= [0,0]; B &= [a,0]; C &= [b,c];
>P1 &= u*C+(1-u)*A; P2 &= v*C+(1-v)*B;
>&solve(areaTriangle(A,P2,B)=6,v), P2v &= expand(P2 with %)
 
                                    12
                               [v = ---]
                                    a c
 
 
                           12 b   12      12
                          [---- - -- + a, --]
                           a c    c       a
 
>&solve(areaTriangle(A,P1,B)=7,u), P1u &= expand(P1 with %)
 
                                    14
                               [u = ---]
                                    a c
 
 
                                14 b  14
                               [----, --]
                                a c   a
 
>lv &= lineThrough(A,P2v); 
>lu &= lineThrough(P1u,B); 
>D &= lineIntersection(lv,lu)
 
                      2
                   7 a  c + 84 b - 84 a     84 c
                  [--------------------, -----------]
                       13 a c - 84       13 a c - 84
 
>sol &= solve(areaTriangle(A,D,B)=4,[a,b,c])[1]  ...
>  with [%rnum_list[1]=s,%rnum_list[2]=t]
 
                             168
                        [a = ---, b = s, c = t]
                             5 t
 
>&expand(areaTriangle(A,C with sol,P2v with sol)) - 3
 
                                   39
                                   --
                                   5
 
>&B with sol
 
                                 168
                                [---, 0]
                                 5 t
 
>&C with sol
 
                                 [s, t]
 
>&P1u with sol
 
                                5 s  5 t
                               [---, ---]
                                12   12
 
>&P2v with sol
 
                             108   5 s  5 t
                            [--- + ---, ---]
                             5 t   14   14
 

The problem is that there is more than one solution. The triangle is not determined by those three areas! However, the area in question is always the same. This is quite a surprising fact if we start algebraically as above. On the other hand, it is not surprising if we assume that the problem can be solved at all. For any transformation (x,y) -> (kx,y/k) keeps the areas intact. Moreover, any transformation (x,y) -> (x+c,y) keeps the given areas intact. So if we have one solution, we can effectively get another solution where C is placed anywhere.

I took the values into C.a.R. (with sliders for s and t, and B,C,D,P1,P2 according to the expressions above) and here is one such triangle.

 

The computations seem to work, and moving the sliders shows that 7.8 is indeed the solution to our problem.

Isn’t there an easier and more intuitive solution? I have not watched the linked video yet. But upon discussion with a friend, the solution occurred to me. Have a look at the following sketch.

The trick is to look at the triangle ABP1 and to see BP1 as its base side. Then divide the triangle into ADP1 and ABD. Those triangles have the same height upon BP1 and since the area is height times base over 2, we get (solving for the height)

\(\dfrac{4}{DB} = \dfrac{3}{DP1}\)

Now the same can be done in the CBP1. Thus

\(\dfrac{y+2}{DB} = \dfrac{x}{DP1}\)

Thus

\(\dfrac{y+2}{4} = \dfrac{x}{3}\)

Now we do the same with the triangles over the baseline AP2.  We get in the same way

\(\dfrac{x+3}{4} = \dfrac{y}{2}\)

Now we have two equations for x and y. The solution is

\(x = \dfrac{21}{5}, \quad y = \dfrac{18}{5}\)

So

\(x+y = \dfrac{39}{5}\)

To find such solutions is a matter of practice, combined with trial and error, plus a grain of stubbornness. The brain needs to have enough math tricks to look for, simply. Then you need to look around if anything looks familiar to you.

18. Juni 2019 von mga010
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Avoiding CO2 Emissions

Sometimes I wished people (including myself) would think a bit more before they write. Looking around for CO2 sins, it is fairly easy to point the finger to flying. Indeed, aviation has tax privileges that hide the true costs of this means of transportation. This should be stopped. But in terms of CO2 output, modern airplanes are not that bad. Only the railway and especially travel buses are better. A car must be packed with at least 3 persons to beat the most modern airplanes. And airplanes should be used on distances only where cars or travel buses cannot compete at all.

The arguments usually continue with a claim for more railway connections. This, of course, does only help if the railways are (1) driven by regenerative energies and (2) replace other means of transportation. If those conditions are not met more and cheaper railway connections only add to the CO2 footprint of mankind. The most efficient way would be to stop driving our own cars and using travel buses. We should not think that individual traffic can go on like now in the future world of eleven billion people. An option is to reduce traveling before we are forced to do so. The means to achieve this are local economies and the net.

Moreover, traveling in total adds only 15% to the global CO2 emissions if my numbers are right. And flying is only a small part of traveling. By far the major emissions come from the industry, power plants and housing. And this will be rising due to the eleven billion people that will need basic resources to live. We should have stopped that growth earlier by education, fair chances, equal rights for women and shared welfare. But now my kids have to find a way to get the planet through this sickness called „mankind“. And there are ways to do it. We need new thinking and much more science, and a new awareness of the challenges we are facing. We don’t need a brutal business that ignores all warnings.

And we do not need simple solutions which ignore all statistical facts.

17. Juni 2019 von mga010
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Mathematische Software – Geogebra

This is the Android Version of GeoGebra. It is a lot more cumbersome to use than the desktop version, but you can get used to it. Although I will be criticizing it a bit below it is a very nice software.  It can do far more than an average student would ever need. If you do not know the power of GeoGebra, have a look at its home page. My own programs Euler Math Toolbox or C.a.R. and others in the same category may be more powerful on many areas and more versatile. But GeoGebra offers an all-in-one package with teacher support and a world community. Over the years, they completed the program and added missing features. They even included 3D constructions and augmented reality. For this, we had to use specialized programs like Archimedes 3D or Cabri 3D, both payware. Moreover, they added JavaScript support on webpages so that constructions can still be embedded after Java has been killed in the browser. This free package is to be recommended.

Instead of writing a review, let me point out the shortcomings of CAS with regard to the learning process. No, I won’t be arguing that software hinders the process of acquiring mathematical skills and math has to be done with pencil and paper. Rather the contrary is true. I will be arguing for more software usage. But it needs to be used in an intelligent way. For that, we need intelligent teaching.

Have a look at the example in the image above. It was the first example I tried on my Android device. We are discussing the function

\(f(x) = e^{5x} \, \sin(x)\)

We want to learn its behavior. It is very difficult to get a really good impression on the Android device. You can try to zoom in and out. But without further information, you will not be able to grasp its structure. The software uses the nicest feature of touch screens, the pinch zoom. So you can zoom right into the interesting region as in the image above. Even then, it looks as if the function was zero left of -1.5.

If you zoom out further you see the following.

One can only understand this image if one knows how the two factors look like and have ever studied a dampened oscillation before. So the plot does not really help without the mathematical background. But, on the other hand, if you have the background the plot can be a huge help in asserting the knowledge and confirming it.

Next, I tried to find the first local minimum on the negative axis. You can solve that numerically in the program by touching the graph in the minimum. The software will then display one of these black dots showing all special points of the plot, and you can read the coordinates below the plot. I think this is a very nice way of grasping math and something EMT cannot do that easily. Of course, you can do it on the command line.

>function f(x) &= exp(5*x)*sin(x)
 
                               5 x
                              E    sin(x)
 
>plot2d(f,-1,0.2);
>xm=solve(&diff(f(x),x),0.2)
 -0.19739555985
>plot2d(xm,f(xm),>points,>add);


But let us talk about the CAS aspect of this solution. GeoGebra produces a very interesting solution to this problem.

\(\{ x = 2 \tan^{-1}(\sqrt{26}+5), \, x=2 \tan^{-1}(-\sqrt{26}+5) \}\)

There is a switch to evaluate this numerically. If it is pressed four surprising values appear: -191.31°, -11.31°, 168.69°, 348.69° (rounded). The degrees can most likely be avoided by setting the program to radian mode. The values are correct.

>fzeros(&diff(f(x),x),-200°,360°); %->°
 [-191.31,  -11.3099,  168.69,  348.69]

What do we make of all this?

First of all, the computations, algebraic or numerical, do not make much sense without proper explanations and without the mathematical background. The zeros of the sine function and consequently of the function f simply are the multiples of 180°. Between each zero, the function has at least one extremum, alternatingly a minimal and maximal value. So much is easy to see with the naked eye. In fact, there is exactly one extremal point in each interval. This is harder to see, however. By computing the derivative, we get the extremal points as the solutions of

\(\tan(x) = – \frac{1}{5}\)

Every book about trigonometry contains a plot of the tangent function like the following.

I added the line y=-0.2. So you can easily see that the extremal points repeat in distances of 180°. Problem solved.

Why does GeoGebra produce such a complicated answer involving the square root of 26? That is a mathematical problem in itself and well above the capabilities of high school students. The reason may be connected to replacing sine with 1 minus cosine squared and solving a quadratic equation.

And why does it only show four of the infinitely many solutions? I do not know.

We learn from all this that numerical or algebraic software or plots can be useful. But without mathematical background they are useless.

26. Mai 2019 von mga010
Kategorien: Uncategorized | Schreibe einen Kommentar

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