This answer shows the “beauty” of mathematics thinking and, at the same time, the “ugliness” of mathematics communication and abstraction.
The author simply explains the concept of transcendental numbers not by hiding behind a definition but by walking the reader through the intuition behind it. I believe that his way to explain this concept is how most people end up to understanding transcendental numbers.
At the same time, mathematicians are very prone to hide this learning process and to present their understanding from “first-principles” or in an axiomatic way. This leads new learners to rely on these axioms to “understand” a concept instead of the intuition behind them. In this example, other responders using the definition of polynomial equation to define transcendental numbers. There is nothing wrong with this approach when building up the “mathematics cathedral of knowledge”, but it requires new learners to follow along multiple years of study to understand a concept that could also be introduced in a more intuitive way.
I thought an equivalent definition of algebraic numbers is “numbers you can build using +, -, *, /, exponents, and radicals”. But TIL there are polynomials whose roots can’t be written this way (one commenter gives x^5 + x^4 = 7 as an example).
x^5 + x^4 = 7
Is there a way to play this “get to zero” game backwards, so you can talk about how to build these numbers rather than how to break them down?
I’m not sure how you would play the “multiply by x” step in reverse, since if you’re starting from zero you don’t know the destination x yet.
Being transcendental is just another word for being Thoreau.
I can’t take credit for it, it’s from dialog in Fallout 4…