Interview with Dr.Abrahamson

The Tortoise and Achilles

Zeno of Elea (c. 450 BCE) is credited with creating several famous paradoxes, and perhaps the best known is the paradox of the Tortoise and Achilles. 

This paradox ca be only applied in the branch of mathematics, because its complexity cannot be understood in other ones, for instance, physics.

The paradox consist on a race between Achilles and a tortoise. At the beginning, the Tortoise asks Achilles to give her a small head start. Achilles, laughing, answers 10 meters, knowing for sure he will win even though the tortoise begins "x" meters ahead, in this case, 10. 

Once the race was going to start, Achilles, annoyed by the tortoise´s laugh, asked her: "How can you be so happy when you will lose for sure?" "Because I won´t, and I´m sure about it" answered the tortoise. "How come?" replied Achilles "Well look, if I start, for instance 10 meters ahead, How long will take you to catch me?" asked smoothly the tortoise." One or two seconds, I mean, almost nothing" answered in a brave way Achilles."but... think about it, in these 2 seconds, I will not be there anymore, I´ll be ahead again." "Well, then I will catch you again in less than a second""but again... I will not be there, maybe 1 or less than one steps from you, but still ahead; therefore, I´m sorry Achilles, but you will never catch me" Achilles, surprised, answer"Yeah... but... but... oh no, you are right, as always" replied in a sad way.

This paradox obviously cannot happen in real life, but demonstrates that mathematics rules applied in non-mathematical situations can give us strange results; in this case, it give us one of the principals of an infinity limit.

Curiosities; Meal with Álvaro

    2^n is the formula used for this problem knowing that any kind of papers cant be folded more than 7 times. If we fold it 8 times, the solution will be 256 sheets of paper (thickness) which cant be impossible to fold knowing that it is made from wood. 

Curiosities, Meal with Alvaro

San Petersburg paradox

Mathematics, as well as this blog has been demonstrating, is related to all aspects in our life; this includes probability. However, real life is not always adjusted to what mathematics say. This is the case of the San Petersburg Paradox; this paradox uses probability as its tool. By definition, probability is a measure or estimation of likelihood of occurrence of an event and takes place every time in our day to day, for instance, in every gaming we can find in casinos (roulette, slot machines...). With this post , I am trying to demonstrate why, even though mathematics try to explain one of these games and the chances we have of winning (San Petersburg) it is not adjusted to reality.

• This game consists on betting chains. First of all, you have to bet two euros. A coin will be dropped; If the result is heads, the profit will be the double of the previous bet, in this case, 4 euros. In case you continue obtaining heads, the profit will be continuously multiplied by 2 (2nd round you get heads, profit will be 8, 16 for the 3rd, 32 the forth etc...) On the other hand, (if you get tales), the results will be the same as explained with heads but negative (first of all you will lose 2 euros, then 4, 8 etc...). 
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• This game has been analyzed by mathematicians in order to obtain a "winning formula", but formulas cant describe reality. In fact, maths say that, by minimun the probability is, you can keep winning forever. But real life, as well as we all know, says that there is a moment where luck will turn its back. 
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• In order to find a finite profit (finite value, not infinite as probability would say), the mathematician Bernoulli described this game with the following formula, assigning a finite value to the profit: 

This formula gives an implicit relationship between the gambler's wealth and how much he should be willing to pay to play (specifically, any c that gives a positive expected utility).

Bernoulli

In conclusion, the San Petersburg paradox demonstrate that mathematics cannot be applied for everything in real life, even though it is the most important tool we have nowadays.

Betrand Russell; "Beauty of mathematics"

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music." —Betrand Russell

Bertrand Russell

Bertrand Arnold William Russell, British philosopher, logician, mathematician,historian, and social critic from the XIX-XX century, described mathematics with the quote we have above, and he could not be more successful with this expression. As well as it is described in the introduction of the present blog, mathematics are everything in our lives; but on the other hand, it is difficult to imagine it... then, what could be better than a video showing it? After researching and looking for examples thanks to the media I have, I was able to find a video which concentrates different examples of our everyday life and how it is related to math in just 1:40 minutes. It bring us different examples: Mathematics involved in computers, in our footprint or even in the area of each snowflake.

Then, let´s go straight to the point:

Beauty of Mathematics

Brilliant, right? The video show us how everything can be calculated using maths; probability of the dice, equilibrium of the top, noise made by the doors of the metro etc... Therefore, using mathematics, we are supposed to "predict" what is happening and what is going to happen, and that is why mathematics are so important in our lives.

Everything around us is Math. Perhaps, we could say Maths is the most useful tool the humankind has; it is used in our day to day for absolutely everything, even though we do not notice it: From the very beginning of each one (our birth), the amounts of products our mother has been taking to have a successful pregnancy were calculated by this tool, and during the rest of our lives, each act we do, (taking a bus, make a phone call or even eating at a restaurant) are possible thanks to the comfort the technology is giving us, which obviously is possible thanks to mathematics.

Therefore, in the this blog I have created, I am willing to show and share mathematics in itself, we different kind of posts, from interviews to videos where I would like everyone to enjoy and understand a little bit more how is math intruding in our world, and why it is that important in our day to day.

University of the student: http://politecnica.universidadeuropea.es/

Youtube channel: http://www.youtube.com/user/AulaUE

"A little bit of history; Calculus"

By definition, "Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations". But apart from definitions... How come Calculus appeared in our knowledge? Who created it? Who noticed Calculus should be a new branch in this area? To explain these and more questions, we have to return to our past, reminisce those times we did not live in and try to find the logical aspects of calculus, to make it "understandable" for us. This reasoning can be explained with more than one example... For instance, how can we defend a country without knowing its past? or how can we try to solve a problem just by applying "rules and the theory" if we do not even know what are we doing? Therefore, let´s review Calculus and its past.

It can be said that Calculus was mainly created by Isaac Newton and Gottfried Wilhelm von Leibniz. Both of them developed new systems independently, where Newton considered variables changing with time and Leibniz thought of the variables x and y as ranging over sequences of infinitely close values.

Isaac Newton

Gottfried Wilhelm von Leibniz

During the development of Calculus over the years, not everyone was agree with either Newton or Leibniz´s models; for example, Lord Bishop Berkeley made serious criticisms of the calculus referring to infinitesimals as "the ghosts of departed quantities". This made the community of mathematicians react, and thanks to these "criticisms", new names with new ideas for the development of the area appeared: Cauchy, Weierstrass, and Riemann reformulated Calculus in terms of limits rather than infinitesimals.

Thanks to all these names, we know Calculus as it is in our days, one of the most useful tools we can use.