What can I do to learn modern mathematics? Calculus and discrete mathematics were made over a hundred years ago. They are ancient. So what books do I need to read to learn modern mathematics?
Ancient they may be but they are still current today. Before anyone can advise you on learning materials, we need to know about your current state of education. While waiting for that - try the MIT OpenCourseware series of lessons ... these are college courses. If you get stuck, you'll have some idea what you need to know first.
I studied how to add two numbers in pre-school. It was about 10 years ago. Arithmetic was invented more than 10000 years ago(I guess). It is ancient but without it, mathematics as a field would not be there.
Do you want to learn modern mathematics because it was invented less than 100 years ago? What do you mean with modern mathematics anyway? Anything that was invented 100 years ago is fine? It will however likely have prerequisites that are more than 100 years old and that you will need to know.
Mathematics isn't like physics. There hasn't been a 'quantum' revolution in math in the last hundred years like with physics. which has upset all math done in earlier days. Most of the plane and solid geometry taught in high schools and colleges today would be recognized by Euclid. The only new twists unfamiliar to Euclid would be the development of analytic geometry (17th century) and non-Euclidean geometry (19th and 20th centuries). Algebra, trigonometry, calculus, analysis, etc. haven't been 'modernized' because their basic principles are still valid and useful. As you age and grow in experience, you'll learn that not everything newer than next week is necessarily better than what has come before. On those occasions where something truly revolutionary comes down the pike, people will stand up and take notice of it.
I'm not saying that those branches are useless, I am simply wanting to learn new subjects in mathematics.....
But why not learn new subjects that are 200 or 100 years old? Studying something recent (like 50 or 25 years old) is going to be virtually impossible right now, since you need to know the previous subjects really well. For example, something "recent" like noncommutative geometry won't make a lot of sense without first learning differential geometry, which is already 200 years old now.
Ok, so I learned algebra, trigonometry, differential and integral calculus, and linear algebra. Can you provide me some resources that I need in prerequisite to the "recent" mathematics?
What micromass is trying to tell you is that you're approaching this the wrong way and with a wrong attitude, as well. The age of a theory in science is really not important. Surely, we all want to work with the most up-to-date knowledge, (thing not always possible) but to get there one cannot jump the subjects which universities call prerequisites. Linear algebra + calculus is fine. The standard prerequisites for advanced mathematics. One can go from here to several directions: functional analysis, abstract algebra and representation theory, differential geometry. These 3 vast subjects go under <modern mathematics> nicely. So make your pick and enjoy.
Right. The primary motivation should be "What do I enjoy and want to know more about?". This can be something rather modern like functional analysis, or something more older like differential geometry on curves and surfaces. Unlike other sciences, math knowledge doesn't get outdated. It always remains true. Sure, every now and then everything gets expressed in a more abstract language, but that doesn't mean the old things aren't worth reading. In fact, the old stuff can be a very good motivator for the modern mathematics. It's not like biology (for example), where things that are 200 years old are often wrong now. Newer does not mean better in mathematics.
like I've already said before, I'm not saying that those branches are useless, I am simply wanting to learn new subjects in mathematics.....
To see what the current modern topics in mathematics are about, for those who have already learned the "ancient" mathematics I recommend T. Gowers (ed), The Princeton Companion to Mathematics