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Miloslav Ciz 2024-03-15 22:32:12 +01:00
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@ -11,7 +11,7 @@ The term *infinity* has two slightly distinct meanings:
It could be argued that potential infinity is really the reason for the existence of true, high level mathematics as we know it, as that is concerned with constructing mathematical [proofs](proof.md) -- such proofs are needed anywhere where there exist infinitely many possibilities, as if there was only a finite number of possibilities, we could simply enumerate and check them all without much thinking (e.g. with the help of a [computer](computer.md)). For example to confirm [Fermat's Last Theorem](fermats_last_theorem) ("for whole numbers and *n > 2* the equation *a^n + b^n = c^n* doesn't have a solution") we need a logical proof because there are infinitely many numbers; if there were only finitely many numbers, we could simply check them all and see if the theorem holds. So infinity, in a sense, is really what forces mathematicians to think.
**Is infinity a [number](number.md)?** Usually no, but it depends on the context. Infinity is not a [real number](real_number.md) (which we usually understand by the term "number") because that would break the nice [field](field.md) structure of real numbers, so the safe implicit answer to the question is no, infinity is not a traditional number, it is rather a concept closely related to numbers. However infinity may many times behave like a number and we may want to treat it so -- for example the result of computing a [limit](limit.md) may be a real number but also infinity; so ultimately everything depends on our definition of what number is and we can declare infinity to be a number in some systems, for example there exists so called *extended real number line* which consists of real numbers and plus/minus infinity, which ARE treated as numbers.
**Is infinity a [number](number.md)?** Usually no, but it depends on the context. Infinity is not a [real number](real_number.md) (which we usually understand by the term "number") because that would break the nice [field](field.md) structure of real numbers, so the safe implicit answer to the question is no, infinity is not a traditional number, it is rather a concept closely related to numbers. However infinity may sometimes behave like a number and we may want to treat it so -- see for example [transfinite numbers](transfinite_number.md) that are used to work with infinite sets and the numbers can be thought of as "sort of infinity numbers", though they mostly live in a separate realm from the traditional numbers. Also for example the result of computing a [limit](limit.md) may be a real number but also infinity; so ultimately everything depends on our definition of what number is and we can declare infinity to be a number in some systems, for example there exists so called *extended real number line* which consists of real numbers and plus/minus infinity, which ARE treated as numbers.
An important term related to the term *infinite* is **[infinitesimal](infinitesimal.md)**, or *infinitely small*, a concept very important e.g. for [calculus](calculus.md). While the "traditional" concept of infinity looks beyond the greatest numbers imaginable, the concept of infinitely small is about being able to divide (or "zoom in", see also [fractals](fractal.md)) without end; for example in the realm of [real numbers](real_number.md) we may start at number 1 and keep moving closer and closer towards zero without ever reaching the "smallest nonzero number", as no matter how close to zero we are, we may always divide our distance by two. A term also related to this is [limit](limit.md), which helps us explore values "infinitely close", "infinitely far" etc.