What is a map projection?

As you probably know, the Earth is not flat; it's more like a sphere. To be precise, it has a specific shape that it's not 100% accurately represented by any mathematical model. To add on it, Earth's surface is constantly changing and has way too many details to be exactly mathematically modeled. And here, we talked about 3D modeling. Is it easy to make a 2D model from the 3D irregular object? Of course not. On the other hand, GPS navigation is working very precisely, maps are more and more accurate, and people are rarely lost due to poor navigation. I would say map projection is the process of representing Earth on a 2D surface. The more detailed way to define map projection (Wikipedia definition) is to say that it is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane.
 
How many map projections are there, and which is the best?

A lot. And there is not the best one-each projection has it's cons and pros, so the right question isn't which one is the best but for which purpose map is needed? Between geographers, it's common to use metaphors in order to explain the problem-imagine you have to peel an orange or an apple. You can't peel it equally without any mistake-some part will always have a small error-you'll cut too deep or too shallow, or you will squash the fruit a bit. Anyhow, since 1569, Mercator projection made by Flemish cartographer Gerardus Mercator is the most popular one. Why? It's good for navigation.

Mercator projection

Different versions of Mercator map projection are used for teaching in schools, navigation, and the military. It is also commonly used by street map services. Companies like Google, Yahoo, and OpenStreetMaps widely use it. It was originally made for sailors, and its main strength is preserved up today-it makes navigation easy. Parallels and meridians on the Mercator projection are straight and perpendicular to each other-linear scale is constant in every direction around any point, thus preserving the angles and the shapes of objects. Another advantage is that the Mercator projection shows the whole Earth's surface on the flat, rectangular surface-it's practical to put it in the book, brochure, on the a4 paper, print plenty of wall maps out of it and so on. As said earlier, it's impossible to make a perfect projection. Mercator projection has a significant flaw. It makes areas far from the equator looking much bigger than in reality. For example, on Mercator projection, Greenland appears the same size as Africa, when, in reality, Africa is 14 times larger.

(author: Daniel R. Strebe, https://commons.wikimedia.org/wiki/File:Mercator_projection_Square.JPG)
To conclude and get a grasp of the math behind the process, here is one formula used in the Mercator projection. Equations place the place the x-axis of the projection on the equator and the y-axis at longitude , where  is the longitude and   is the latitude. Ln (ln) is natural algorithm, and sinh-1 stands for inverse hyperbolic sine function.

What about your country, is it presented correctly on the Mercator map? Which other projections are suitable for navigation? What do you think about their pros and cons?
Feel free to write down your findings, questions, and comments.