domingo, 18 de mayo de 2014

Graphical representation of the domain and range of real functions

A function specifies a rule by which an input is converted to a unique output.

More precisely:

A function $f$ is a rule that assigns to each element $x$ in a set $D$ exactly one element, called $f(x)$, in a set $E$.

The domain of a function is the set of all possible $x$ values that can be used as inputs, and the range is the set of all possible $f(x)$ values that arise as outputs.

It's helpful to think of a function as a machine (see Figure 1). If $x$ is in the domain of the function then when enters the machine, it is accepted as an input and the machine produces an output according to the rule of the function. Thus we can think of the domain as the set of all possible inputs $x$ and the range as the set of all possible outputs  $f(x)$.

Figure 1.
The most common method for visualising a function is its graph; which consists of all points $(x,y)$ in the coordinate plane such  that  $y=f(x)$ and $x$ is in the domain of $f$.

The graph of a function $f$ gives us a useful picture of the behaviour of the function. Since the $y$-coordinate of any point $(x,y)$ on the graph is $y=f(x)$, we can read the value of $f(x)$ from the graph as being the height of the graph above the point $x$ (see Figure 2).

Figure 2

The graph of $f$ also allows us to picture the domain of $f$ on the $x$-axis and its range on the $y$-axis as in Figure 3.

Figure 3
For the domain of the function we need to ask: What is the set of all the valid inputs $x$?

Meanwhile, for the range of a the function we need to ask: What is the set of all the valid outputs $f(x)$?

In the next applet you can see a graphic representation of the domain and range of functions. In this case, the green horizontal line represents the domain and the salmon vertical line represents the range. The function is represented with the dotted curve.

Type your function and see how the domain and range change. Selecting the asymptotes will show you particular cases where the function you typed is whether defined or not, in particular values of $x$.

Some particular cases: x^2 for $x^2$, exp(x) for $e^x$, abs(x) for $|x|$, 1/(x^2+1) for $\frac{1}{x^2+1}$, ln(x) for $\ln x$ and sqrt(x) for $\sqrt{x}$.

Follow the next link for opening the applet in an external window:

Derivación de la fórmula para calcular las raíces de una cuadrática

En varios sitios de internet se puede encontrar la derivación de la fórmula
para resolver la ecuación cuadrática $ax^2+bx+c=0$.

Por ejemplo:

El siguiente procedimiento me parece menos rebuscado:

                  Sean $a, b$ y $c$ números reales con $a\neq 0$. Consideremos la ecuación
                  Entonces tenemos
                  Multiplicamos por $4a$ en ambos lados
                  Ahora sumamos en ambos lados $b^2$
                  Lo anterior lo podemos escribir como
                  Entonces, aplicando raíz cuadrada resulta
$$2ax+b=\pm \sqrt{b^2-4ac}$$
                  Despejando $x$ obtenemos la expresión:

Al parecer este procedimiento data del siglo IX y se suele atribuir un matemático de la India llamado Sridhara Acharya.


- O'Connor, J. & Robertson, E. F. Sridhara. The McTutor History of Mathematics Archive, University of St Andrews. Disponible en:

A geometrical approach to the concepts of speed and acceleration (Part I)

1. Introduction

1.1 The concepts of speed and acceleration

How good is your understanding of speed and acceleration? Would it stand up under cross-examination in court? Here are some questions that you might be asked by a court prosecutor to test your credibility as an expert witness:

1. Speed is understood as distance divided by time. What distance is being referred to here? What time is being referred to here?

2. Is this what your car’s speedometer is measuring when it registers, for example, 60 km/hr? To what distance and time would it be referring?

3. Until the introduction of digital technology in the 1980s, cars used eddy current speedometers (you can google this). Did these measure any actual distances and times? What did they measure?

4. How would you measure the change in speed of a vehicle?

5. What do you think acceleration means? (‘Getting faster’ isn’t an expert answer and would reflect badly on your credibility. You would be expected to give a numerical definition.)

1. 2 Galileo Galilei and the study of free fall motion

The renaissance Italian scientist Galileo Galilei (1564 – 1642) introduced the modern notions of speed and acceleration in the context of trying to understand the motion of falling objects. He described the motion of a dropped object by saying that the distance of its fall was proportional to the square of the duration of its fall. In his world:
  • there were no clocks or stop watches as we know them,
  • there were no graphs, formulas or functions, 
  • there was no algebra or calculus.

So what could he have possibly meant by his description? In the light of the dot points above, you should aim at a purely geometrical interpretation. Can you devise an animation that realises it? Are Galileo’s words really a description of the motion of the dropped object or just some limited aspect of it? Is there some better way to describe the motion of objects? To answer the above questions it may be helpful to talk about one of the most famous experiments conducted by Galileo, known as the inclined plane experiment.

2. The inclined plane experiment

We all experience motion in our daily lives. In particular, everybody has felt or witnessed motion due to gravity. The next situations are examples of what we all know as Free Fall Motion.
  • Throwing a ball straight up, 
  • Dropping a coin from the top of a building
Free fall motion has been studied in all kinds of ways since ancient times. However, at the dawn of the seventeenth century Galileo introduced a decisively new approach which led him to his remarkable discovery about the relationship between the distance of fall and duration of fall of a dropped object.
Galileo made this discovery indirectly, because it was impossible to measure accurately how long an object took to fall using the technology of his day. He observed, however, that a falling object was an extreme case of a ball rolling down an inclined plane – in the case of a falling object, the plane is effectively vertical.

A replica of Galileo’s inclined plane (Source: Museo Galileo gallery)

So Galileo investigated the relationship between the distance of travel and duration of travel of a ball rolling down a plane at a range of inclinations that were not too steep, where the necessary time measurement could be made with reasonable accuracy. 

Water clock
To measure the time it took for a ball to roll down the inclined plane, Galileo used a water clock. This device consisted of a large vessel of water placed in an elevated position with a small hole drilled in the bottom connected to a pipe of small diameter. A thin stream of water flowed into a container below the vessel. To start the clock, he allowed water to flow into the container. To stop the clock, he stopped the flow of water. To reset the clock he emptied the container. By weighing the amount of water in the container, he could then compare the times it took the ball to travel each distance. For example, if twice as much water (in weight) filled the container, he knew that the time measured was twice as long. 

Although not used by Galileo, a variation on the water clock is to measure the height of the water collected in a tube or pipe, instead of weighing water collected in a container. This measures time as a length, which is more attuned to the way we represent it these days.

Thus, you can compare the distance covered by the ball and the height of the water in the container. In the next section you will find a dynamic representation of Galileo’s experiment.

3. A dynamic representation

We can emulate the inclined plane experiment (without considering friction) to gain deeper insights into Galileo’s thought processes and the development of mathematical ideas. 

3.1 Galileo’s experimental setup
The applet on the next page shows a geometric representation of Galileo’s experiment. The inclined plane is represented with a right-angled triangle. The ball is represented with a circle. Finally, in this case, the water clock is represented with a rectangle depicting the tube or pipe that is collecting running water. We are to compare the height of the water with the distance covered by the ball.

The goal of this first applet is to let you familiarise yourself with Galileo’s experimental design. The ball rolls down the plane, which can be set at different inclinations.

General instructions:

To set the ball in motion, push the Start button. Push the Reset button to begin the simulation again.
The slider, with the tag Base (horizontal segment at the bottom), changes the angle of inclination of the plane.

Instructions for interacting with the applet:

The above simulation allows you to manipulate the angle of inclination of the plane. The point is to get familiar with the applet in preparation for the next activity. What shows you that it takes less time for the ball to roll down planes that are more steeply inclined?

For opening in a new window click in the next link:

Galileo's inclined plane experiment. Fresco by Giuseppe Bezzuoli, 1841. Tribuna di Galileo, Florence. (Source: Wikipedia)

3.2 Galileo’s discovery

The next applet adds features to the previous one to describe Galileo’s experiments. These features are:
Time Intervals: You can choose the number of intervals of the same length into which the time (container) is divided. On the right side of the container is displayed the number of complete time intervals, when the ball has covered a particular distance.

Time: Move this option to see where on the plane the ball would be at that time.

The distance covered by the ball is divided into a number of intervals of the same length, which depends on the number of time intervals. Under the ball is displayed the number of complete distance intervals covered by the ball.

Instructions for interacting with the applet:

1. The applet starts with one Time Interval, this means that the time is not divided. Divide the time into two intervals and observe the motion of the ball rolling down by pushing the Start button or using the slider Time. How many intervals does the ball cover in the first time interval? And in the second?

2. Choose another number of Time Intervals and observe what happens in each case. For example, if the time is divided into three intervals then how many distance intervals does the ball cover in the first time interval, in the second one, and in the third one?

3. Find a relationship between the number of time intervals and the number of distance intervals covered by the ball and observe whether this relationship is still maintained when you change the angle of inclination of the plane. What is the relationship between the distance covered by the ball and the elapsed time?

For opening in a new window click in the next link:

A work made in collaboration with J. Rice and K. E. Matthews.

Una versión en ibooks para ipad saldrá disponible en los próximos meses (espero pronto). Mientras tanto he aquí una versión en pdf:

miércoles, 2 de abril de 2014

Breve Tabla Cronológica de la Historia de las Matemáticas (Actualizada)

El documento que aquí comparto contiene información actualizada con vínculos a sitios relacionados con la historia de las matemáticas y también he agregado algunas referencias.

Importante: Si has descargado este documento y encuentras algún vínculo que no funciona, por favor envíame un mail para corregirlo.  

jueves, 20 de marzo de 2014

La suma de todos los números naturales es -1/12 (Curiosidades del infinito)

Qué resultado obtendríamos  si realizamos la suma de todos los números naturales
Por supuesto, nuestra respuesta sería que el valor de la suma es infinito, lo cual concuerda  con nuestra experiencia y con las reglas matemáticas que hemos aprendido en la escuela. Entonces podemos afirmar que: La suma de todos los números naturales es infinita.  Esto lo podemos escribir de la siguiente forma
$$1+2+3+4+5+\cdots =\infty$$
Pero qué pensarías si, utilizando un método matemático, podemos obtener la siguiente expresión
$$1+2+3+4+5+\cdots =-\frac{1}{12}$$
Suena un tanto ilógico... bueno, si te interesa saber cómo se puede llegar a este resultado, te invito a leer el documento Curiosidades del Infinito (muestra abajo). Es mi primer intento de recopilación de ejemplos matemáticos que muestran ciertas anomalías cuando se involucra el concepto del infinito. Lo estaré actualizando cuando tenga tiempo con más referencias y por supuesto, con más ejemplos paradójicos. Mucha razón tenía Borges al mencionar que: Hay un concepto que es el corruptor y el desatinador de los otros. No  hablo del Mal cuyo limitado imperio es la ética; hablo del infinito.

Árbol fractal, (Facultad de Ciencias de la Educación, Universidad de Colima) 
Foto by JCPC©