# What does it mean to solve a problem 'analytically'?

Also, when you solve a problem or prove something without the usual of pictures/visuals, what's that called? Is it called 'geometrical' if you DO use visuals?

chiro
Also, when you solve a problem or prove something without the usual of pictures/visuals, what's that called? Is it called 'geometrical' if you DO use visuals?

Of the field of geometry, two of the most important attributes are distance and angle.

Lots of people do associate geometry with something that is more or less a visual representation, but the word geometry actually means 'measure of the earth' and it has evolved to the point where we can measure things (metric, norm) in terms of distance on objects that are not flat, but instead have positive or negative curvature (like the surface of a circle or a horse saddle).

Distance relates to metrics and norms, and angles relate to inner products and the two of them can be related to each other by using standard definitions of inner products and norms.

Solving something analytically usually means finding an explicit equation without making approximations. When solving differential equations, analytic solutions can be difficult and some times impossible. The power of computers have made even the most difficult differential equations (that relate to reality) a lot easier to approximate numerically. The classic example is the three body problem in physics.

Do you know Newton's Method for finding roots to polynomial equations? That is an example of a numerical approximation method that can be used if there is no good formula for finding the root (analytic solution).

In my experience (only undergrad) there is no such thing as a proof of anything significant by using visuals. Often diagrams are used to clarify an explicit proof, but the proof would stand on its own. Of course there are problems that concern a physical question, but even then, the proofs rely on non-visual arguments.

Second Chiro's response. Geometry go far beyond what can be visualized or drawn.

Deveno