Engineering at Dordt College

In my Differential Equations class (MATH 204) we end up working with complex numbers in connection with underdamped second order systems. We just got to that part of the course this week. Most students have seen complex numbers in a high-school math course or elsewhere, but a refresher is helpful. Although complex numbers are just incidental to the main topic (second order underdamped differential equations), it includes some really interesting equations, like the one above. The equation is a consequence of “Euler’s Formula.” (”Euler” is pronounced “oi’-ler.”) The equation above ties five remarkably important numbers into one relationship, which makes it very beautiful.

1.) The number e is the base of the natural logarithms. The advantages of using natural logarithms over base ten logarithms become apparent when you study calculus, or differential equations. In particular, the derivative of f(x) = e^x is the function itself, which is a very useful property. (The number e is about
2.71828. . . . It is irrational. As a decimal number it goes on forever with no repeating pattern.)
2.) The number pi is of course the ratio of a circle’s circumference to its diameter. (The number pi is about 3.14159. . . . It is also irrational.)

3.) The number i is the “imaginary number.” It has the property that
i² = –1.

4.) The number 1 is the multiplicative identity.

5.) The number 0 is the additive identity.

All five of these numbers have interesting histories. The proof of Euler’s Formula goes beyond a blog like this, but again, includes beautiful relationships. When we say, “God Created the heavens and the Earth” we need to remember that these relationships of remarkable complexity, which we have been able to model in mathematical formulae, are part of his general revelation to us. By doing mathematics like this, just for the sake of trying to understand what we observe in the creation, we get a closer and more meaningful understanding of our Creator.