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Enigmatic Serranno numbers

The intriguing property of the Serranno series

Alejandro Serrano

A not-so-well extended myth confers magic  properties on Serranno numbers. These numbers make up the so-called Serranno series, which starts with two ones and subsequent numbers are calculating by adding two times the previous one to the next-to-previous one (e.g., 3 = 2 x 1 + 1, 7 = 2 x 3 + 1) . Therefore, the first ten elements of the series are:

1, 1, 3, 7, 17, 41, 99, 239, 577, 1393

An intriguing property of this series is that the ratio between two consecutive numbers is closer and closer to the so-called silver ratio. The silver ratio is usually denoted by σ (the Greek letter corresponding to ‘s’, after silver and Serranno) and its value is  σ = 1 + √  2 = 2.4142135623731…  For instance, dividing the 20th by the 19th element of the series yields a number with thirteen identical digits: 9,369,319/ 3,880,899 = 2.4142135623730….

The magic nature of Serranno numbers stems from this outstanding property, since the silver ratio is described as one of the most mysterious numbers in nature.

It is believed that Serranno followers use a drop of water as a secret symbol to identify themselves. The drop of water is made up by combining a circle and a right triangle whose edges are tanget to the circle (see figure below.)

The drop of water was chosen because it hides the silver ratio. Indeed, dividing the distance between the top corner to the bottom of the drop by the radius of the circle exactly yields the silver ratio!

The silver ratio is also related to beauty in nature. From small to large water drops, the silver ratio is supposed to confer beauty on the objects that contain it.


Order your personalized “Enigmatic Series” here: @SerranoAlej. The kit includes a new series nominated after your name, your own golden-type ratio (choose among several precious metals), a personal geometric shape, and an entry in wikipedia.



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Fibonacci numbers: not that magic

An explanation of the “magic” property of the Fibonacci series

Alejandro Serrano

A well extended myth confers magic  properties on Fibonacci numbers. These numbers make up the so-called Fibonacci series, which starts with two ones and subsequent numbers are calculating by adding the two previous ones. Therefore, the first ten numbers of the series are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. An intriguing property of this series is that the ratio between two consecutive numbers is closer and closer to the so-called golden ratio (the golden ratio is usually denoted by φ and its value is   (1 + √ 5)/ 2 = 1.618033989… ) For instance, dividing the 20th by the 19th element of the series yields a number with eight identical digits: 6,765 / 4,181 = 1.61803396…,

The magic nature of Fibonacci numbers stem from this outstanding property, since the golden ratio is described as one of the most mysterious numbers in nature (see Mario Livio’s cite below.)

“Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

(Mario Livio. The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number, p.6

It is believed that the Pythagoreans used a 5-pointed star, called a pentagram, as a secret symbol to identify themselves. The pentagram was chosen because it hides the golden ratio everywhere.

The golden ratio is also related to beauty in nature. From sunflowers to shells, the golden ratio is supposed to confer beauty on the objects that contain it. Moreover, several dimensions in human creations are also driven by the golden number. For instance, many of the proportions of the Parthenon in Athens exhibit the golden ratio. A contemporary example is the (golden) ratio between the length and the height of a credit card.

The truth of the matter is that this “magic” property of the Fibonacci numbers can be explained with a bit of math. In fact, the numbers in the Fibonacci series can be calculated using the following expression

For instance, if n=4:

The eigenvalues of the matrix on the right-hand side of this expression turn out to be (1 + √ 5)/ 2 and  (1 – √ 5)/ 2 , i.e., φ and -0.61803… respectively. Note that φ > 1 and 0 < -0.61803… < 1. That is why, after a number of iterations, the impact of the second eigenvalue on the vector on the letf-hand side is negligible . Therefore, for n sufficiently large, only the first eigenvalue, φ, is relevant when calculating the following element of the series, and then

which explains the supposed-to-be “magic” nature of the Fibonacci numbers.

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Spaniards still suck at speaking English

Where Spain stands and how it can break the current vicious circle it is trapped in

Alejandro Serrano

In a recent study by EF Education First (www.ef.com.es/epi), the English level of Spaniards was ranked 17th out of 19 European countries, representing 92% of the total European population excluding the UK and Ireland. Only Russia and Turkey obtain worse results than Spain in the ranking.

The graph below shows those 19 countries, given their gross domestic product (GDP) at PPP per capita and their level of English, as measured by the EF EPI index. As you can probably guess, Spain is the red diamond. There is a strong positive correlation between level of English and GDP per capita. English level of countries to the left of the diagonal line, including Spain, is worse than what the GDP of the country would suggest.

Source: own elaboration based on data from www.ef.com and the world bank

Three years ago, the Spanish prime minister, Mr. Rodríguez Zapatero, announced a plan aimed at reaching an ambitious goal: “in ten years, Spaniards leaving high school will speak fluent English,” he solemnly asserted. But reality is quite stubborn and gives today plenty of anecdotal evidence that this will not be quite the case. If you have visited or lived in Spain and tried to communicate in English in the street (no matter the age, old or young,) you know what I am talking about.

It is true that an effort is being done in private and public schools to put a remedy to this situation. A small, but increasingly portion of curricular time is already taught in English, and bilingual education is little by little gaining momentum in many schools. However, a huge problem to implement this plan is that there are not enough native teachers to teach English, thus students do not learn how to speak. Try this: tell a 12-year-old good student to recite a list containing fifty irregular verbs (eat-ate-eaten and the like.) He or she will do it by heart, but will be incapable of correctly pronounce half of them. Let us face it: non-native teachers in Spain do not devote enough time to conversation in class (my group is too large is the usual excuse) and, even if they do, they lack the appropriate phonetics skills; even many of them speak English with an (often strong) Spanish accent.

In the business space, this unfilled educational gap leads to a clear situation of competitive disadvantage when Spanish firms try to compete with other firms in the international arena. It is not a coincidence that Spanish multinationals sell much more in Latin America than its European counterparts, who mainly sell in Europe, a much more natural market in terms of geographic distance. There is a huge language barrier, another one to add to the long list of Spanish barriers to achieve competitiveness.

How to break this vicious circle of low-skilled teachers and low-skilled students without spending enormous amounts of money on élite schools or “imported” teachers?

First, you have to start from the very beginning, taking care of the youngest by not translating cartoons on TV into Spanish  and movies for kids in theaters. Then you do the same for teenagers (first, the Harry Potters, then the twilights, then all the American movies, then…)

Second, the curriculum in schools has to be turned upside down. Start teaching only oral English in elementary school. Use cartoons, movies, and the internet to expose kids to native English. Start teaching writing skills to sixth or seventh graders, when they are ready to absorb grammar easily.

Finally, prevent students from digressing too much by learning other languages, such as German, or Italian, or French. Unless they have a clear vocation towards languages, English will be the only one they will most likely need at work. It is better to speak fluent English than intermediate German, English, Italian, and French.

Following these simple guidelines, in fifteen years, teenagers will have acquired the speaking skills that they lack today. The solution is not that original; see what the blue diamonds in the top-right part of the chart above (Norway, Denmark, Belgium, The Netherlands) have been doing for many years.

These are my two cents; it seems to me like a simple recipe for success. It is going to take a while, so the sooner they start, the better.

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Un colegio inglés internacional

Alejandro Serrano – Heraldo de Aragón, Oct 2011| España

La larga crisis económica en la que continuamos inmersos continúa enviándonos señales sobre qué fórmulas ya no funcionan en nuestra particular búsqueda del crecimiento económico, como prolongar el gasto público en construcción o tratar de atraer inversión extranjera ofreciendo al inversor salarios competitivos. El consenso generalizado es que el nuevo modelo económico debe basarse en la I+D y la innovación. Esto es más fácil de decir que de hacer, y quizá por eso nuestra comunidad no ha tenido demasiado éxito hasta ahora tratando de atraer empresas internacionales en estas áreas.

La verdad es que para atraer a Aragón a los Googles y los Amazons de este mundo (ese, no menos, debe ser el objetivo), es preciso crear las condiciones apropiadas para que las multinacionales decidan venir. Esto incluye un aspecto esencial, y es que los directivos de estas empresas también deben tener los incentivos para venir. Y ahí, admitámoslo, es donde comienzan nuestros problemas. Zaragoza no es Madrid o Barcelona, no tiene las ventajas de una gran metrópolis, y éste es un inconveniente que no puede ignorarse.

Es cierto que se han hecho mejoras significativas en este sentido: Zaragoza está bien comunicada por autovías, podemos llegar por AVE a Madrid o Barcelona en menos de noventa minutos y volar directamente a varias ciudades europeas, aunque sea à la Ryan Air. Además, debido parcialmente a la EXPO 2008, Zaragoza es bonita de ver, está bastante limpia y es una ciudad muy segura.

Pero aún hay mucho camino por recorrer. Cuando las multinacionales deciden dónde poner sus cuartes generales regionales o sus centros de I+D, una de las preguntas clave es si la ciudad candidata tiene un escuela internacional. Una respuesta negativa descalifica inmediatamente la propuesta. Un problema que Zaragoza tiene hoy es que carece de una escuela internacional en inglés. Lo más cercano que existe son los llamados colegios bilingües, pero estos no están bien vistos por las empresas porque los alumnos deben aprender español, que es un requisito que no siempre los padres ejecutivos perciben como deseable. Lo que necesitamos es la versión inglesa (quizá pública, quizá privada) del colegio Molière.

El nuevo colegio crearía además demanda local, dado que un número cada vez mayor de padres se da cuenta de la importancia de que sus hijos hablen un nivel muy alto de inglés, algo todavía inusual en Aragón. Esta demanda local hará la inversión aún más atractiva, pero debemos tener en cuenta que esta escuela debe crearse pensando en el lucro cesante actual, es decir, en el dinero que no está llegando a nuestra región por carecer de ésta.

¿Queremos realmente atraer a las mejores empresas del mundo a Aragón? Entonces pongámosles las cosas fáciles. Los nuevos consejeros de Educación, Tecnología y Economía deben recoger esta iniciativa y ponerla en marcha. ¿Difícil de conseguir? Quizás, pero si la DGA ha decidido contar con un alto cargo irlandés y este periódico ha publicado este artículo también en inglés, significa que algo está cambiando en Aragón, y hay razones para creer en la creación de un colegio internacional en nuestra comunidad, que nos lleve a dar un paso de gigante en la dirección adecuada.

Para saber más sobre cadena de suministro www.zlc.edu.es

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Why Zaragoza needs an international English School

A necessary condition for economic growth

Alejandro Serrano

The long economic crisis we are going through keeps sending signals about what formulas do not work any longer regarding our particular quest for growth, such as prolonging government expenses on construction or attracting foreign direct investment based on lower wages. The consensus of opinion is that our new economic model should be grounded on R&D and innovation. Easier said than done, and thus it is that our community has kept struggling trying to attract international companies on these areas.

The truth of the matter is that drawing the Googles and the Amazons of the world to Aragon (that, and not less, should be the goal) requires creating the appropriate environment for companies to come. A key issue, which can make or break a deal, is that the executives working for those firms should also have the appropriate incentives to live in our region. And, let us admit it, here it is where our challenges start. Zaragoza is not Madrid or Barcelona; it has not the advantages of large metropolis, and this is a major drawback that cannot be ignored.

It is true that some significant improvements in this regard have been made: today we can drive almost everywhere from Zaragoza along highways, reach Barcelona or Madrid by AVE in less than ninety minutes, and fly directly to some of the large European cities, even if it is à la RyanAir. Furthermore, and partially due to the International Exposition of 2008, these days Zaragoza looks very nice, is quite clean, and happens to be extremely safe.

But there are still some gaps to fill. When corporations make decisions about where to place regional headquarters or R&D centers, one of the key questions to ask is whether the city under scrutiny has an international school or not. Only if the answer is in the affirmative may the city remain on the short list. As of today, a major issue is that Zaragoza does not have a true English international school. The closest we have is what we call bilingual schools, but these are assessed as inadequate by companies, because children are required to know or learn Spanish, a requisite which, admittedly, is not always perceived as desirable or even feasible by their parents. What we need is the English version (maybe public, maybe private) of the Aragonese Molière School.

Such a new school will also create local demand, since an increasing number of parents are aware of how important it is for their children to speak fluent English, an oddity still today in Aragon. This local demand will make the investment even more attractive, but we should keep in mind that the focus has to be on the current opportunity cost, i.e., on recovering the huge amount of money that is left today on the table for not having such a school in the region.

Do we really want to attract top-notch companies to Aragon? If so, then let us pave the path for them to come. The new consejeros of Education, Technology, and Economy should pick up the baton and move this initiative forward. Difficult to achieve? Maybe, but if an Irishman can become an official at the DGA and this newspaper can publish this article in English, it means that something is changing in Aragon, thus there is some hope that an international school becomes a reality and our autonomous region be moved to the next level.


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