complex number-Differential equations

In the following lecture, the professor reintroduced the idea of complex numbers and its application in differential equations, which I found very intriguing.

Complex number, for those who are not familiar to the concept, is a number that does not exist in any visible or “real” world. It is not used in daily life, because it does not really “exists”. Many of the beginner level math students are taught that there can be no square root for any negative numbers, nor can there be any negative squared numbers. When they are asked what is the square root of -1, they would say: “There is none”.

That is not necessarily true. As the conundrums in math approaches those in life, it is inevitable that at some point, the square root of a negative number is needed. To remedy the situation, the concept of “complex number” was created. It is assumed that the number “i” is the square root of -1 (i2=-1), and then every square root of every negative number can be expressed.  For example, the number -9 can be considered as the product of -1 and 9, and the square root of -9 would be the product of the square root of -1 and the square root of 9, which is 3i. picture from http://world.mathigon.org/Real_Irrational_Imaginary

I always imagine the complex numbers as a different set of numbers (just like real number), but from a different dimension. It is like something from 5th dimension (3rd dimension is space, 4th is time, 5th is reality, meaning there is different realities going on at the same time, time is like a tree instead of something linear), another reality that cannot be seen, yet very much real.

I was mostly fascinated by the application of such concept in advanced math like calculus. The existence of complex number just makes everything clear.

We always talked about one single problem in Calculus 1adv: “∫e-xcosx dx”, it was a very hard problem to solve for us, because no law except integrating by parts was applicable for it. In a word, integrating by parts just means to keep taking the derivative of one component, in this case “cosx”, and to keep taking the integral of the other component “e-x”, then combine them together when any of them cannot be integrated or differentiated. The problem is, it is no end for the derivatives for cosx (it becomes –sinx, then –cosx, then sinx, then cosx again), and the integral for e-x is always -e-x or e-x (which again will loop back to itself). In calculus, there is a very long and complicated process involving using integrating by parts twice (to see the specifics click here), but it is much simpler to use complex number to solve it. By Euler’s formula (eix=cos(x)+sin(ix) for specifics click here), we can view the e-x as it is, and cosx as the “real” part of eix, and just change the original problem e-xcosx into e(-1+i)x by adding an invisible sin(ix), which can be integrated instantly, and all that has to be done is to take off all the component with “I”, because they are supposed to be invisible. By doing that, a clean solution is reached, with no complication at all.

I can see the beauty of seeking simple truth in complex things here.

To quote the professor: “What’s beautiful in math, people ask; Something complicated turns into something simple, that’s beautiful.”

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