Math used to be fun. The hardest thing about it was counting up to 100. The numbers also had weird shapes, so that also added to its difficulty. Then, we learned how to combine numbers in different ways: addition, subtraction, multiplication, and division. With these simple operations, you would think that it was all there was. Next thing you know, letters are added with the numbers, which was also known as Algebra. As you progressed through higher math and encountered weird symbols, which are still as unknown to me like the Egyptian Hieroglyphics, eventually, you encounter the highest level of math by the average student: Calculus, which helps enable you to solve real-life situations. After your last Calculus class, you would think there is no more math to learn. Nope. The math after Calculus is basically math problems that you will almost never encounter and use in your daily life. An example would be computational science.
Computational science is "concerned with constructing
mathematical models and quantitative analysis techniques and using computers to analyze and solve
scientific problems." It involves different theories to approach certain problems. Since math can be tedious to do by hand, computers are used to analyze mathematical models. Programs are made just so models can be studied by inputting different parameters. Some examples of different real-life models used include
graph theory and
mathematical optimization.
Numerical analysis is a very common method used in computational science. It is "the study of
algorithms that use numerical
approximation (as opposed to general
symbolic manipulations) for the problems of
mathematical analysis." Numerical analysis usually involves getting results that are not exact, since it is not always possible. It is more about obtaining approximate solutions. An example would be
; you would not get an exact number.
Related to computational science, there is also
symbolic computation, also known as computer algebra. It is "a scientific area that refers to the study and development of
algorithms and
software for manipulating
mathematical expressions and other
mathematical objects." The solutions are usually exact numbers with expressions that contain variables with no given value. An example would be the equality symbol:
Even though we almost never use higher level math, it is still applied in some fields or in some real-life situations without us even knowing it. Since we are in a computer-related field, it would be a plus if we can understand some simple mathematical models.
I agree with you that if you become an engineer or a C.S major and have to run simulations all of the math up to and including calculus becomes useless; in practice finding exact answers to a solution becomes really expensive.
ReplyDeleteWhen you said "The math after Calculus is basically math problems that you will almost never encounter". I argue that the math after calculus is the only type of math you WILL use because calculus becomes too inefficient to carry out on a computer. An example is in linear algebra/calculus where you rationalize a matrix via Gram-Shmidt method. This methods is good for teaching students the concept of orthogonality, but if you implement it on a computer your computer will run out of memory before you can run any type of simulation. It is only in math classses after you take all calclus/linear algebra that you learn about how to solve problems efficiently.
Ryan, the post is very cool. I really like your blog about Mathematical Analysis. I agree that running simulation solves most common problems of an issue because analysis is all about statistics and numbers. However, to get the exact number for any solutions could be too expensive in some cases. Currently, most algorithms are designed to get the best results in shortest time for a problem. That is why mathematical analysis can really help scientists to build up a good baseline for those algorithms.
ReplyDeleteIn short, this is a very informative post. Keep doing this good stuff, Ryan.