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Symmetry in Mathematics


Our readers can reflect with us on how the natural numbers {0, 1, 2, 3,…} arise in the natural problem of counting, how naturally we can proceed to investigate their properties and how naturally this inquiry leads us to new symbolic worlds. .

One of the ways forward in further investigation of the properties of numbers is the path of equations. We ask: what number added to 1 results in 3? The answer is very simple:

if x +1 = 3, then x = 2.

But the scenario changes a lot when we ask:

what is the number that added to 1 results in 0?

We now have the equation:

x + 1 = 0.

The reader realizes that it is not possible to find a solution to this equation in the world of natural numbers. Here comes the symmetry: why there would be no solution to this equation, why the privilege of certain numbers no in making the equation

x + 1 = n

have solution?

The above equation would be asymmetric with respect to certain values ​​of noif it could not be resolved. This is how negative integers arise. They are the hidden symmetry of the above equation. Moreover, they also solve the problem of lack of symmetry that we notice in the world of natural numbers when we see that this world has a beginning (0) and no end, that is, there is no largest natural number. Now with negative numbers the world of numbers becomes a symmetrical world because this new world has no beginning either:

{… , -3, -2, -1, 0, 1, 2, 3,… }

All equations of type x + m = n now have a solution. No more the privilege of certain numbers over others that the above equation has a solution for them and not for others.

The reader has an inevitable question: But in the world of practical life, how can this symmetry of positive and negative integers be useful?

Well, the reader must then ask whether in practical life there are important phenomena or situations involving opposing objects that can be thought of as positive and negative. If so, it is necessary to ask whether the addition and multiplication of relative integers also makes sense for such objects.

It would be a good idea for the reader to consult a physicist, chemist, or biologist on the issue of whether there are, in these researchers' areas, situations or objects that are susceptible to quantitative description within a world of positive, negative, or null quantities. Encyclopedias can also be helpful in presenting knowledge that has been gained by applying the concept of negative and positive quantities to the real world.

We have seen that negative integers {…, -3, -2, -1} solve the problem of asymmetry of natural numbers {0, 1, 2, 3,…} that have a beginning and no end. By adding the negative integers, the new number system becomes symmetrical about zero.

How can we represent negative numbers? That is, can we imagine a figure that gives us a good idea of ​​the integer system?

One of these ways is to represent integers as points on a line. We pick a point and associate it with the number 0. Then we mark 1 to the right of 0, keeping a certain distance. At this same distance we set -1 to the left of 0. Then 2 to the right of 1, -2 to the left of -1, and so on. The point -no is therefore the point to the left of 0 at a distance that is no times the distance from 0 to 1.

The reader cannot avoid the following question: What to think about the points that are between the points used to represent the integers? For example, could the dot exactly between 0 and 1 match some new type of number? The reader will immediately say: Well, wouldn't fraction 1/2 be the new number occupying the average position between 0 and 1? That's right, the fractions will play the role of filling the “vacuum” between the entire points marked on a line.

Again we find the scientific situation of the need to fill in a space or broaden an idea that seems to indicate a hidden symmetry, or an unexplained privilege. Here we could say that the privilege that only whole numbers can be represented as points on a line is strange, unexplained.

The symmetry hidden here is the idea that all points, equally, should represent numbers, not just the points used to mark integers.

The reader again finds himself wondering if Nature has processes that can be described in non-integer quantities. Mathematically it is natural to imagine the numerical system from a line, choosing any point to represent zero. Well, at least for us, three hundred and so years after René Descartes, the French philosopher and mathematician who introduced the idea of ​​number representation into a line, creating Analytic Geometry, that sounds natural.

Each new idea we introduce to solve a problem of hidden symmetry or some other mathematical problem naturally leads us to other unexpected ideas that make sense and whose development eventually reveals new mathematical truths.

The new mathematical truths, in turn, reveal to us descriptions of previously unexplained, or even unknown, processes of Nature.

When we mark the fractions on a line we get the impression that all points on the line could be occupied. But Pythagoras had already realized that the square root of 2 is not a fractional number. Thus, we should also be able to find a point on the line for the square root of 2. By the end of the 19th century, mathematicians already knew that the number pi, which appears in the circumference length formula 2 pi R, is not fractional either. therefore it should also correspond to a point on the line.

The number pi also appears in many other mathematical situations. We cannot enumerate all occurrences of the number pi in mathematics and even in mathematical models that seek to represent phenomena of nature. Numbers such as the square root of 2 and the number pi are called irrational numbers because they are not fractional, that is, they cannot be represented for reasons between integers. A natural question then arises: How many are these irrational numbers? Are there points on the line available to also represent these irrational numbers?

In the late 19th and early 20th centuries, mathematician Georg Cantor discovered that there are many more irrational numbers than fractional numbers! The way Cantor demonstrated this truth was a big surprise in the mathematical world. Another time we will address the issue of showing that there are many more irrational numbers than rational or fractional numbers.

The conception of the geometric line as a numerical line, that is, each point corresponds to a fractional (rational) number or an irrational number, was a great innovation in mathematics.

About 300 years ago, Renée Descartes probably did not imagine that most points on the geometric line would correspond to irrational numbers. But another question immediately arises: are the real numbers gone? Or is there still some kind of real number that will also require a point on the line to accommodate it?

This problem of whether rational and irrational exhaust the points of the geometric line is the problem of completion. Mathematicians know today that the number line is complete. That is, there are no gaps between any two real numbers. But does that end the problem of knowing what all the numbers are? It is interesting to note that not yet! If on the one hand we can already see the real numbers as a continuum of points, that is, a straight line without holes, on the other hand, we still cannot solve the problem of finding a number whose square plus one is zero.

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