General Overview, Algebra

For this post, I am going to focus on a personal reflexion about how has been this course 2013-14 referring to the subject of algebra. Furthermore, this is a way of saying good bye to this academic year and closing the blog.

Algebra, as far as I could see, is another mathematical world which is ruled by different laws, systems, equations… What do I mean? Well, as well as I remember, our teacher Guillermo Castilla asked us the very first day: Who has ever studied Algebra before? To that question, some people of the class raised their hands. Funny, he said. Now seriously, who has ever studied real algebra, and not the “algebra” you study in school which in fact is not the subject itself? No one said anything. And as he said, this algebra was different. Forget about everything you know, focus on beginning from zero, they said… and it was right. In order to understand this subject, it is important to understand it from the minute zero,  because if not you get lost as son as you do not even imagine.

As I could observe, the complexity of this subject states on understanding the concepts, because once you do it, you can move through it and you can be sure you never fail. In mathematics this happens as well, but not in that range. In algebra, if you know were you are going, then you do it right. How you change from bases? How you transform matrixes? How can you move from one base to a reciprocal one? I repeat, what I could learn from this subject is that knowing the method, you can get any of the previous questions.

Furthermore, I would like to remark that this subject also aids in the fact that you will structure your mind… and even though people may believe this is unreliable, it is not. Clear ideas in algebra are the meaning of right answers, so no disorder is permitted. Finally, in order to close this blog, I would like to underline how important was this blog for us, the students, since we found a way of expressing our ideas, our thoughts… such as in this report.

History of matrices

The history of matrices returns to times before christ. Chinese texts from the 300 BC show that with small matrices. But the first matrix considered in our days as a "contemporary matrix" that appeared was invention of, first of all, SeKi Kowa, Japanese mathematician, and also a little bit later in Germany by Gottfried Leibnitz, german mathematician. 

Seki Kowa

Olga Taussky Todd

Later, in around 1800, it was found different methods from ancient times to solve simultaneous equations as well as Carl Gauss would discover later. This make us think that matrixes are not things from the present, but from long time ago. In fact, the term matrix was introduced for the first time in 1850, by James Joseph Sylvester. After, in XX century, it would be Olga Taussky Todd the mathematician who developed matrices into its final state, the one that showed us how to work with them. Also, she expanded their uses to other fields, such as using them for analyzing vibrations on airplanes in World War II.

James Joseph Sylvester

Thereby, since their first appearance in China, matrices have been an important tool for humanity even though they were rescued a few centuries ago. From their very beginning, were they were used for calculating simple things such as commerce, etc... to our days where as well as it is said before, they started being used in World War II for measuring the vibrations on airplanes.

Interview; What´s mathematics and algebra for you?

For the present interview, I could ask a 4th year student in mathematics and informatics double degree student of the Polytechnic university of Madrid about his career and how he sees things inside it. The name of the student is Alejandro Ortega de los Ríos and he offered himself to respond all the questions I had about his passion: mathematics.

Me: Hello Alejandro, first off I will like to thank you because of giving me your time in order to answer my questions.

Alejandro: No problem.

Me: Well so first of all, I would like to ask you why mathematics.

Alejandro: Since I was a little kid, mathematics were a field that impassioned me so much. Of course I am not going to lie, I had doubts since we only live once and I did not want to commit to something I would not like in the future... that is why I had to decide between engineerings and mathematics... but at the very end, I let my heart decide and I studied what I love, and not thinking about the future... and so far I think I chose correctly.

Me: Why you think mathematics are no so interesting for kids of our days?

Alejandro: How mathematics are taught in schools... I do not think is the way to make them appreciate it. You go to a student, tell him: You have to do this or you will not pass, and repeat the course and see you next year... it is not the way to do it. I am not saying I got the key, but when I was in school, I had good and bad teachers. This last ones made me think this subject was hateful, but the first one made me think the opposite. Mathematics has to be taught as a tool for life... something we all need to use if we want to success, not just as a tool for passing or failing exams, because this is not the way to learn. Making kids feel mathematics is the alternative way.

Me: What would you do in order to change this?

As I said, I do not have the key... but I repeat, making kids feel mathematics, showing them that it is an useful tool for life... just as an example, I´ve been watching a series called "Breaking Bad" for the last months. In this series, the main character is a school chemistry teacher. During the episodes, he shows how he has the biggest weapon no one has: knowledge in the fields of mathematics and chemistry, which lets him get whatever he wants with simple formulas he applies. He is able to fabricate artificial bombs, batteries for cars, electrical systems... this is the way of teaching kids, showing how useful it can be for them in the future.

Me: Well, and now let´s get deeper into the topic. What´s algebra for you?

Alejandro: Algebra is just a branch of mathematics... let´s call it the advanced mathematics. For me, algebra is the key for understanding absolutely everything, from simple mechanisms to the laws that rule in the universe.

Algebra Picture.

Me: Why is it so important knowing about algebra?

Alejandro: Without algebra, the world we know nowadays would not be the same. Computers for instance are programmed with algebraic equations and systems. Mobile phones, structures... absolutely everything, even those investigations scientists do out of the Earth to learn about the outer space... I repeat, absolutely everything is algebra, and is the tool humans use for developing.

Me: How you see yourself in the future=

Alejandro: Buff... not really sure, but hope involved in big projects.

Me: Ok Alejandro, thanks you for giving me your time.

Alejandro: Do not worry, it has been a pleasure. Bye.

My video about absolute value of complex number

And what is algebra for?

Algebra as well as explained in previous posts, works for many different fields, and are the key for human development in the 21st century. In this post, I could find two different videos that explain both whats algebra for and how we feel it in our day to day:

For both videos, it is important to put english subtitles since both of them, from the same channel, speak in spanish.

As well as we could see in both videos, algebra is technology, computers, advance... everything around us is constructed through algebraic structures.

The history of algebra

Until the 19th century, Algebra was simply the theory of equations, but not until before the algebra we know in our days was the same. In fact, it is a branch of mathematics, that began being known as the work of an expert mathematician of the 16th century, François Viète.

François Viete.

But how did interest for Algebra started?

In Ancient Egypt,  people learned to solve linear (ax = b) and quadratic (ax² + bx = c) equations. They also could solve some indeterminate equations, which give us an answer for the previous question.

From the past, we can observe that ancient civilizations wrote out algebraic expressions using only occasional abbreviations, and by medieval times, Islamic mathematicians were able to talk of the unknown “x”, and work out the basic algebra of polynomials.

Back again with the 16th century, mathematicians such as René Descartes wrote Book III of La géometrie (1637)which looks much like a modern algebra text. This and references we can obtain from François Viète show us that algebra started being founded at around this century.

René Descartes.

By the time of Gauss, algebra could be finally named as in modern state. Attention mathematicians had focused on polynomial equations changed into studying the structure of abstract mathematical systems who’s axioms were base don the behavior of mathematical objects i.e.: complex numbers, algebraic groups…

Later, XIX-XX century, algebra shifted again with the discovery of quaternions; complex numbers of the form a + bi changed into a + bi + cj + dk. After this discovery, mathematicians started investigating vectors, and this bring us to our days, where algebra, in a continuous develop, has occupied importance and an role in every field, and not only in mathematics.

Algebra Introduction

Short time ago, I read an article about why is Algebra so important in our days if we do not see it, we do not feel it, we do not know how it exactly works. Well, in this article, algebra was named as the gatekeeper subject. Used by professionals ranging from electricians to architects to computer scientists. It fills every field, explains the relationships of our universe: algebra are de advanced maths we need to success. "It is no less than a civil right", says Robert Moses, founder of the Algebra project, which advocates for math literacy in public schools.

Algebra is more than what we study in university after preparing for around 18 years in primary school. Is a way of reordering our mind, and finding different ways of thinking. In conclusion, Algebra is, in short, the gateway to success in the 21st century. What's more, when students make the transition from concrete arithmetic to the symbolic language of algebra, they develop abstract reasoning skills necessary to excel in math and science.

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...). 
• 
  
• 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. 
• 
  
• 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.