I have always liked very much the mathematics but the courses of the life pushed me towards the physics. I remember especially my first steps in the arithmetical calculation, to do operations at the biggest possible speed and of correct form; the years of baccalaureate, when I experienced the impact of the differential and integral calculus. Unfortunately, on having come to the university, that youth admiration disappeared like the smoke of a cigaret in a gale. The mathematics subjects in the physics career were not what I was waiting for myself: I went so far as to hate the topology, with his open and closed balls, the adhesions and the closing ceremonies; the distinguishing equations did not seem such explained by mouth of my teachers, very much theorem and demonstration and at the time of truth they were the physicists those who had to teach us to obtain the solutions that we were interested in. With the algebra the thing was not very different: vectoriales spaces, dual bases and all that paraphernalia that I have ever never used again in my professional performance like physical. I suppose that if the previous lines are read by some mathematician one will throw the hands to the head (as always) and he will say to me that if the importance of the rigor, that if this, that if other, but the truth is that I am completely against that the mathematics subjects in the careers that are not the same Mathematics career give them the mathematicians. I try neither to generate debate nor to open any discussion, simply I give my opinion, mistaken or not. End.
Very well, I have got entangled as it is usually habitual in me and I still have not said to you that this post that I write today (not, you have not been wrong of blog) takes as an assignment to take part in an initiative proposed by José Antonio, the author of the marvelous blog Tito Eliatron Dixit, named a Carnival of Mathematics, which target is to spread this so exciting science and that produces so many headaches to million students of the whole world and all the epochs.
For this first edition I have prepared a simple, but nice thing, to return to the origins, to the first steps that happen when they begin studying mathematics in the school. I will center on the geometry and on the arithmetic and will dedicate this entry to the youngest people, which they are of course those who with any more steadfastness need not to feel defrauded, disappointed, for this wonderful human invention that there are the MATHEMATICS.
Many students (even mine in the university), increasingly, unfortunately, remember with difficulty certain expressions as there are the area of a sphere, the volume of a cone, the square of the sum of two numbers and similar things. That of things that one is in an examination...
And if there was some form simple to remember expressions as the previous ones? Since it turns out that yes they are. I am going to tell you, for example, how to remember with help of the geometry the value of the square of the sum of two numbers. That you should have good memory you will remember that this square one can determine addend to the double of the product of both numbers the squares of each of them. But let's see this otherwise. Let's call to and b to the two previous numbers and let's draw a side square (a+b). Let's measure the length to on a horizontal side and let's do the same with a vertical side. Let's plan from each of these points one straight line up to the opposite side of the square. We will have now our original square and in him inscribed a side square to, other of side b, y two sides rectangles to and b. Well, now without much ado that to remember that the area of a square is the square of the value of one of his sides, there is not any more that to add the areas of the two smallest squares and that of two rectangles (a2, b2, ab, ab). And it is already: (a+b) 2 = a2 + b2 + 2ab.
It is possible to make something similar to find the square of the difference of two numbers. In this case, a side square shows to. Next the length measures itself on his perpendicular sides b (we will suppose that b < a) y se trazan dos líneas rectas, igual que en el caso anterior de la suma. Tenemos ahora inscritos dos cuadrados, uno de lado (a-b) y otro de lado b, además de dos rectángulos idénticos de lados (a-b) y b. Lo que queremos es expresar el área del primer cuadrado inscrito (la sombreada en la figura) y esto se puede hacer restando al área del cuadrado de lado a (el más grandote, el que contiene a todos los demás cuadrados y rectángulos, para entendernos) las áreas del cuadrado de lado b y las de los rectángulos de lados (a-b) y b. Total: (a-b)2 = a2 + b2 - 2ab.
Since my teacher's condition does not forget me not for this special occasion, I am going to propose to you, to suggest, to design the necessary geometric method to remember the value of the product of the sum for the difference of two numbers, that is to say, what was saying that "the sum for the difference is the squares difference" ((a+b) (a-b) = a2 - b2).
But the most curious thing comes now. It turns out that the latter expression can take advantage to carry out the calculation of squares of entire numbers of simple form. How is it possible to do this? Very easy. Suppose that you want to know the square of 12. What do you do? Since to add and to reduce to the previous number the necessary quantity so that a number finished in zero is obtained. In this case, there joins and is reduced 2 (let's call this one the magic number). This way, we have 12 + 2 = 14 and 12 - 2 = 10. Next we multiply the obtained numbers (14 * 10 = 140) and to the result we add the square of the magic number (22 = 4). Whole: 140 + 4 =144, which is the looked square of 12. What is the relation between all this and the formulita of the previous paragraph? Attentive: the number to es 12, which we add and remain b (the magic number), having (a+b) and (a-b). Since we want a2 we do not have any more that to clear, with which to the product of the sum for the difference it is only to add to him the square of the magic number: a2 = (a+b) (a-b) + b2.
Have you liked? I hope to make it better in the next edition. Ah, and I apologize to the mathematicians if I have not continued the rigor that so much they love. After all, I am only a physicist...
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