Why are Algebraic Equations (i.e. relation between variable and 0) special?

Question

I'm reading the book, "Annotated Turing" in trying to understand what computation really means. It has a section about algebraic equations and trancendental numbers. I'm quiet not understanding why they are special.

Answer 1

Short (and perhaps too general) answer: Algebraic numbers are special because they are solutions to polynomial equations, and polynomial equations are special because they are essentially the only equations you can make from just addition and multiplication.

The availability of different structures with addition and multiplication leads to a great interest from mathematicians in general to study polynomial equations, their properties and their applications. As an extension of that, we have a lot of theory about their solutions. So we know a lot about algebraic numbers and thus the distinction between algebraic and transcendental becomes important.