1. 'algebraic' and 'analytic'

How do you go defining these two terms ('algebraic' and 'analytic')? Do definitions overlap, or are they mutually exclusive? Is it even possible to precisely describe these two terms (both by their own definition and by their interrelation)?

I'm asking, for there are times when a textbook goes like "solve algebraically" or "solve analytically").

Keep away from (too much) champagne.

2. Originally Posted by courteous
How do you go defining these two terms ('algebraic' and 'analytic')? Do definitions overlap, or are they mutually exclusive? Is it even possible to precisely describe these two terms (both by their own definition and by their interrelation)?

I'm asking, for there are times when a textbook goes like "solve algebraically" or "solve analytically").

Keep away from (too much) champagne.
An algabraic solution means that your solution is not written in terms of numbers nor functions. It is written in terms of algabraic variables.

Example:

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ is an algagraic solution to the equation $ax^2+bx+c = 0$.

An analytic solution is when your solution is written in terms of elementary functions.

For example, solution to the equation:

$I = \int x^2e^x dx$ is $I = e^x(x^2 -2x+2)+c$

And then you have a numerical solution which would be, the solution to equations

$2x^2+4x-3=0$

or

$I = \int_0^1 x^2e^x dx$

which are numbers.

Thus, algebraic and analytic are not mutually exclusive, or better are they in any sort of other possible relations (and if so, what kind of relations are those)? Further, are there any other kinds of solution(s)?

4. Originally Posted by courteous
How do you go defining these two terms ('algebraic' and 'analytic')? Do definitions overlap, or are they mutually exclusive? Is it even possible to precisely describe these two terms (both by their own definition and by their interrelation)?
Both of these terms have well defined meanings in particular areas of mathematics, which you can find by googling for them, but those are not the meanings you need as we see from:

I'm asking, for there are times when a textbook goes like "solve algebraically" or "solve analytically").
Here the terms are being used almost as synonyms. Here the meaning is solve with pencil and paper with an answer that is an expression containing the variables and constants of the problem and common functions.

If there is a difference it would be that "algebraically" might suggest that only algebraic means of combining terms is permitted (combinations by addition, subtraction, multiplication, division, (rational)powers and roots, with brackets as required), while "analytically" might suggest that elementary transcendental functions are permitted.

But generally with this usage they mean the same thing.

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5. Often mathematics can be usually broken up into three categories: algebraic, topological, and analytic. If you have a book on Complex Analysis it will first talk about the algebraic properties of the complex numbers, then talk about the topological properties of the complex numbers, and then finally the rest of the book shall be devoted to the analytic properties of the complex numbers. The word 'algebraic' means to be related to algebra, algebra here is not high-school algebra, but something that is sometimes referred to as abstract algebra. All properties of the complex numbers that are based on abstract algebra are therefore referred to as algebraic. While analysis (like calculus) is concerned with studying limits, derivatives, and integration and other properties. Anything that falls under this category is called analytic.

6. Originally Posted by ThePerfectHacker
Often mathematics can be usually broken up into three categories: algebraic, topological, and analytic. If you have a book on Complex Analysis it will first talk about the algebraic properties of the complex numbers, then talk about the topological properties of the complex numbers, and then finally the rest of the book shall be devoted to the analytic properties of the complex numbers. The word 'algebraic' means to be related to algebra, algebra here is not high-school algebra, but something that is sometimes referred to as abstract algebra. All properties of the complex numbers that are based on abstract algebra are therefore referred to as algebraic. While analysis (like calculus) is concerned with studying limits, derivatives, and integration and other properties. Anything that falls under this category is called analytic.
You forgot to say what topological is! Things under the topological category would have to do with spatial and geometrical concepts, things such as curves and deformations would fall under this category.