If we had a certain number of digits as an answer, it would no longer be an infinite number. The really important thing when we talk about infinite numbers is that they will have no end. So, for example, if we say a million digits, that number is already finite because we can always think of adding another digit. But the question makes a lot of sense and has led to various paradoxes throughout the history of mathematics.
The idea of infinity is relatively new and is partly due to our number system. Civilizations such as the Egyptians or the Aztecs with non-positional number systems never considered quantities larger than certain values because they did not even have symbols that allowed them to represent these quantities, and therefore the same thing happened with the concept of infinity. However, infinity implicitly underlay positional systems such as our number system, and the way quantities are represented is key to developing an intuitive conception of infinity. In the 20th century, the German mathematician David Hilbert discovered that there is no infinity in reality. He argued that it was not possible to divide matter indefinitely and that infinity could be a necessary idea in our thinking even if it does not exist in reality. The concept of infinity appeared to be strictly defined, but continued to be a source of controversy and paradox.
Currently, infinity in mathematics has two meanings. The first of these is infinity, understood as something that has no end, that can last forever, and that in mathematics we call potential infinity. The second option is to consider infinity as a whole, as a completed process whose limits have been reached, thinking about the set of all numbers without thinking about each of them individually, which is what we call current infinity. But you should know that some of the great mathematicians such as the French Augustin Louis Cauchy or the German Carl Friedrich Gauss denied the existence of this present infinity.
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Coming back to your question, as I said, the current numbering system allows us to think about the concept of infinity. In this case, if we have a number of specific digits, whatever they may be, we can always imagine a number with one more digit, so it is not infinite. That is, 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 always add another digit so that it is not infinite. This is, for example, the idea that Gauss had. But since the end of the 19th century the concept changed. It was then that the current infinity was proposed and consisted of defining infinity as a totality, as limits. This step allows us to relate infinity to the limits of functions or sequences. For example, if we imagine a sequence with even numbers: 2, 4, 6, 8, 10… This sequence grows to infinity and we can always imagine a larger number. The limit of this sequence is infinite. But if we think of a sequence that is: 1, 1/3, 1/4, 1/5… This sequence decreases, although it does not decrease to minus infinity, but decreases towards 0, but never reaches 0. If we could put infinite numbers in the denominator of this fraction, we would reach 0, but we only reach it at the limit, that is, the sequence has a limit of 0 as the denominator tends to infinity.
And it’s the same with functions; If we speak of natural numbers for sequences, we would speak of real numbers for functions. Real numbers are those that allow us to represent all values on a straight line, an infinite line that contains negative and positive numbers and includes, among other things, natural numbers. These are the ones used to count elements: 1, 2, 3, 4… and so on to infinity.
Monica Arnal Palacian She has a degree in mathematics and a doctorate in education. She is a professor at the University of Zaragoza and conducts research in mathematics education.
Question emailed from Angel Gael Romero
Coordination and writing:Victoria Toro
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