Infinite series are up there with some of the coolest functions in math. They’re at the heart of Calculus with Reimann sums and integration, and it’s no wonder they appear everywhere in analysis because of it. But what really cements their status in physics and other applied math is power series, sums of terms with […]
Convergence and Complex Taylor Series Read More »
Why don’t we just use vectors instead? This question is thrown out all the time when discussing the applications of complex numbers, usually because of a mix of unfamiliarity and bad marketing on part with the whole “imaginary” thing. Of course, a fanboy of complex numbers like me might launch the opposite question: complex numbers
Vectors vs. Complex Numbers: Intro to Complex Differentiation Read More »
If you’ve been following along with my earlier posts on the complex number system then you’ve gotten a taste of the useful applications that in all its convoluted glory, presents in trigonometry, algebra, and more. But while you might be on board with treating complex numbers as a numerical trick that happens to have practical
Algebra Needs Complex Numbers Read More »
All math is connected. But as mathematicians go deeper and deeper into the weeds the results they find grow increasingly remote from their roots in reality. But these links still exist no matter how strange or well-hidden they may be, and complex numbers are a perfect demonstration of this. So, after uniting our complex operations
Imaginary Numbers and Real Math Read More »
What is the sum of 3 apples and 4 oranges? This classic statement highlights the difficulties in comparing the incomparable. At least, incomparable in one regard. Because, of course, it’s commonplace and perfectly correct to reply that the answer is “7 fruit.” In this context of different fruits, adding together apples and oranges makes perfect
Imaginary Apples and Oranges Read More »
Think about when you learned algebra for the first time. On that seemingly uneventful day of middle school, the letter “x” is carefully scrawled onto the class whiteboard, and all of a sudden the world of functions, systems of equations, and incomprehensible word problems smashes its way into your mind like a battering ram of
The Mystery of the Complex Plane Read More »
