All kinds of numbers and other mathematical entities are abstractions. Math works by establishing abstract concepts and rules and following them consistently to find out what they lead to. The usefulness of a particular branch of mathematics comes from real-world problems it can be apply to, and it’s precisely abstraction and generality what enables math to discover patterns showing up in very different kinds of problems. Entire branches of mathematics remained a purely theoretical game for centuries, and suddenly practical applications for them were found, especially since the advent of computers.

*Natural*, *integer*, *rational*, *real* and so on are just names used to tell different abstract concepts apart. Nothing makes any of them any more *real* than another. The only thing you need to work with them is to be consistent. There are scientific theories making use of imaginary numbers, like classical electromagnetism and quantum mechanics, and their predictions agree with empirical evidence. If they were wrong for using imaginary numbers, the computers we all are using to post here wouldn’t work, having been designed according to those theories.

I think most of the difficulty people find in mathematical concepts comes from trying to read into them more than there actually is to them.

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The red lake has been forgotten. A dust devil stuns you long enough to shroud forever those last shards of wisdom. The breeze rocking this forlorn wasteland whispers in your ears, “Não resta mais que uma sombra”.