Infinity – The Most Abstract Creation in Mathematics

Infinity is a thrilling mathematical representation of something that our human mind can hardly imagine. When we think of infinity, we should think that:

  • It is not big;
  • It is not even huge;
  • It is neither something tremendously large;
  • Nor is it something enormous;
  • It is just endless!

Yes, infinity has no end. In our world, we have nothing like this, and this is why we cannot imagine it, although many of us accept it. Because it is endless, infinity can never grow. In fact, it does not have to do anything, but just be.

What Is Then Infinity?

Some might think that infinity is a number, a real number perhaps. It is no such thing, it is just an idea, an idea about something that does not have an end. There is no way to measure infinity. Those who say that faraway galaxies would be infinite are wrong, because a galaxy always has limits, while infinity does not, and nothing can compete with it. Unlike finite things that surround us, infinity is simple. Just because it has no limits, we do not have to define it, while finite things have to be defined as size, weight, form, color, and so on.

To illustrate this, think of simple geometric representations of a line. According to its definition, a line has infinite length in both directions. A line segment, instead, has a length and we have to mention it. To take another example, the row of natural numbers is infinite. We count: 1, 2, 3, and could never stop, because no number is the last in this row.

However, we can have the same thing represented in two different manners. Let us consider a fraction like 1/3. This is certainly a finite number – the third part of a whole. But what happens when we write it as a decimal number? We should write 0.3333333…, with an infinite number of 3s, but this is not possible, because we could never stop. This is why we say “0.3 repeating” and write it like this: 0.(3).

Man has always tried to reach infinity, hoping that there was a limit to it. This is how huge numbers appeared, like a “googol”, which can be written as a 1 followed by 100 zeroes, or a “googolplex”, which is a 1 followed by a googol of zeroes. However, these numbers are also finite, no matter how much time it would take to write them down.

In the meantime, infinity keeps being an idea of something that never ends.

The Wonderful World of Fractals – The Artistic Side of Mathematics

Mountains, ferns, broccoli and the stock market… What do they all have in common? A single answer to this question really establishes a link between them: fractals. This modern branch of geometry is the link between the regularity and repeatability of the fern and the market risk on Wall Street. Fractal geometry is a new science, not yet well understood by common people, but already used in researches like that of the variations in heartbeats.

What Are Fractals?

They are geometric patterns that repeat at every magnification level, infinitely. Benoît Mandelbrot, the “father” of fractals, described them as geometric shapes that represent “a reduced-scale version of the whole” when they are separated into parts.

To be more illustrative, you should imagine the Russian nesting dolls, although a fractal is a more complex thing, because nature is more complex, being so simple in its essence. The comparison with the Russian dolls is only to imagine how the same pattern can be repeated at lower scale. There are a few features that characterize a fractal:

  • Any segment detached from a fractal reproduces the whole;
  • The shape and colors of the whole are reproduced infinitely by inner structures;
  • It is impossible to say which substructure is the last, because there is no such thing – every substructure is composed of new elements resembling it perfectly.

Fractals can be found in nature almost everywhere. If you look at a branch of broccoli, you will be amazed to observe that it is identical in structure and color to the whole it was detached from. So is the lining of human lungs. It has a pattern that makes possible the absorption of more oxygen.

Fractal geometry is the science that can express real-world processes in equations. To everyday persons, fractals look breathtaking. To scientists and mathematicians, they are an infinite world to explore.

Why Fractals Are So Important

Fractals help scientists study and understand important concepts, from patterns in snowflakes to bacteria growth or brain waves. For instance, who would believe that cell phone antennas make use of fractal patterns to pick up signals better?

Scientists say that anything that has a pattern or rhythm could be a fractal structure and, as a consequence, described in mathematical equations. Although fractals were known long before Mandelbrot, only the development of the computers in the last part of the 20th century made them easier to study and represent graphically.

What Can Chess Do to Make Us Love Mathematics More

There is no need to say that mathematics provides the foundation that every child needs throughout his/her life. We could think that mastering the basics means knowing how to add, subtract, multiply and divide numbers.

We should reconsider this, because we are living in an information age, and information is reported to double at an incredible rate of less than two years. As a consequence, mastering the basic skills today means:

  • Knowing how to make a decision;
  • Being able to solve a problem;
  • Possessing critical thinking, deductive and inductive abilities of reasoning;
  • Having the ability of making proper judgments and good estimates.

If we, maybe, do not love mathematics, we surely love to play games, and this is where chess comes to complete the picture.

Is Chess a Tool That Improves Our Math Abilities?

Math includes problem solving, but playing chess games also involves solving problems. Could we conclude that playing chess makes us better problem solvers? There are arguments that support this theory.

Chess requires a mental workout, good thinking ahead, planning, systematizing and determining the outcome of our or our partner’s moves. We cannot memorize all the moves that can be made in a chess game. We can remember a limited number of positions, but not more. In mathematics as well, memorizing does not make any sense, only our thinking skills are what really matters.

Researchers have studied the impact of chess playing on students, related to their ability of improving their math problems solving skills. What they found out is really interesting. Not only have their math skills improved, but so have their social habits. As a result, chess is considered a powerful educational tool that should be used in every school.

According to recent research, these are the abilities that chess is expected to improve:

  • Visual memory;
  • Attention;
  • Skills related to spatial thinking;
  • Prediction skills and capacity of anticipating consequences;
  • Ability to formulate and use criteria to make decisions and evaluate alternatives.

Can the Repulsion for Mathematics Be Overcome through Chess?

Note that the vast majority of people would feel embarrassed if you asked them how good they are at math. Most of them are not willing to check a restaurant bill because they think the math behind it is too complicated. This happens because math involves too many rules and formulas to remember, as they say.

But what about playing games? Do they not have rules and procedures? Why do we love games and not mathematics? An answer could be that, if mathematics were treated like a game, more people would learn to be better problem solvers.