If we have a certain number of digits as an answer, it is no longer an infinite number. The really important thing, when we talk about infinite numbers, is that they are infinite. So, if we say, for example, one million digits, that number is already limited because we can always think of adding one more digit. But the question has many meanings and has raised various paradoxes throughout the history of mathematics.
The concept of infinity is relatively new and is, in part, due to the numbering system we have. Civilizations such as the Egyptian or Aztec, with non-positional numbering systems, never considered quantities greater than certain values, because they did not have symbols that allowed them to represent such numbers and, therefore, the same thing happened with the concept of infinity. However, infinity completely underlies positional systems like our number system, and the way we represent quantity is the key to creating an intuitive idea of infinity. In the 20th century, the German mathematician David Hilbert said that infinity cannot be seen in reality. He argues that it is not possible to divide matter indefinitely and that infinity may be a necessary notion in our thinking, even if it does not exist in reality. The idea of infinity seems to be tightly defined, but it continues to cause controversy and controversy.
Currently, infinity distinguishes two meanings in mathematics. The first of them, infinity is taken as infinity, which always continues and which in mathematics we call potential infinity. The second, infinity is considered as a whole, a finished process with its limits reached, thinking about the set of all the numbers without thinking about each of them, which we call current infinity. But you should know that some of the great mathematicians like the French Augustin Louis Cauchy or the German Carl Friedrich Gauss denied the existence of this infinite current.
Coming back to your question, as I said, the current numbering system allows us to contemplate the concept of infinity. In this case, if we have a number of specific numbers, whatever they are, we can always imagine a number with another number, so it is not infinite. In other words, there is no such value.
But just because that number doesn’t exist doesn’t mean infinity doesn’t exist. When we talk about numbers we can add one more digit so that it is not infinite. That is, for example, the conception of Gauss. But since the end of the 19th century, the concept has changed. At that time, the current infinity was proposed and consisted of defining infinity as a whole, as limits. This step allows us to relate infinity to the limits of functions or sequences. For example, if we think of a sequence with even numbers: 2, 4, 6, 8, 10… That sequence grows forever and we always think of a bigger number. The limit of this range is unlimited. But if we consider a sequence that is: 1, 1/3, 1/4, 1/5… That sequence is decreasing even though it doesn’t decrease to minus infinity, it decreases to of 0, but it does not reach 0. If we can put infinite numbers in the denominator of that fraction we can reach 0, but we can only reach it in the limit, that is, the sequence has a limit which is 0 if the denominator tends to infinity.
And in functions it is similar; If with sequences we talk about natural numbers, with functions we talk about real numbers. Real numbers are those that allow us to represent all values in a straight line, an infinite line with negative and positive numbers and include, among others, natural numbers . This is what is used to count the elements: 1, 2, 3, 4… and so on to infinity.
Mónica Arnal Palacián has a degree in Mathematics and a doctorate in Education. Professor at the University of Zaragoza, he researches the teaching of mathematics.
The question was sent by email by Ángel Gael Romero
Coordination and writing: Victoria Toro
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