system
Given the following post, respond as if you're commenting on Reddit:
human
The imaginary unit "i" satisfies i^2 = -1, are there any other numbers invented to satisfy otherwise impossible criteria?
gpt
Sometimes, you start with a number system and see what you can do in that number system.  Sometimes there are things that can't be done (they are impossible *with respect to that number system*), and so you define a new extended number system for which that operation is possible.

Suppose you start with the whole numbers 0, 1, 2, 3, 4, 5, etc.

Addition and multiplication of whole numbers always produce whole numbers.  Subtraction sometimes does (5-2) but sometimes it is not defined (4-5).  However, we can define the number -1, so that 4-5 is defined.  And now, if we see what addition and multiplication lead to with this number added, we wind up with the integers (..., -3, -2, -1, 0, 1, 2, 3, ...).  So by adding negative numbers, we can now solve any subtraction problem among the whole numbers and, in fact, among the integers.

Fractions appear similarly.  12/4 is defined if you're just using integers, but not 2/7.  So we add the rational numbers so that division becomes possible for any integers *a* and *b* (except for dividing by 0).

But still some equations don't have solutions when you have the rational numbers.  For example, we can't solve x^(2)=2 just using the rational numbers.  So add sqrt(2), and now, if we allow addition, multiplication, subtraction, and division, we'll have to include numbers of the form *p*+*q* sqrt(2), where *p* and *q* are rational numbers.  We can similarly add roots of other numbers to make other equations soluble.

Real numbers are introduced so that a certain class of limiting operations always produces a value.  Other limiting processes lead us to define objects called the p-adic numbers.

In short, notions of what can and cannot be done depend on your number system, and we have regularly extended our number systems to make more things possible (that is, to make it so that a given operation always yields a result in the number system in which you started).