Cardan’s Problem

“The shortest path between two truths in the real domain passes through the complex domain.” – Jacques Hadamard

So last week I was doing a quick review through the history and discovery of complex numbers, when I came across the story of Gerolamo Cardano (Cardan) and his pupil Rafael Bombelli. I won’t go too deep into the details, but essentially, mathematics had come to a standstill. There is a principle in mathematics called the Gauss’s Fundamental Theorem of Algebra which states that “every non-constant single-variable polynomial with complex coefficients has at least one complex root.” Essentially, this means that every generic equation f(x) = mx + b must have at least one solvable value for x when f(x)=0. In even more different words, if an equation (say….y) operates as a function of another variable (say…x), then one should be able to solve for at least one value of x.  Even further, the theorem states that however many powers (n) that x is raised to, there will be n solutions for x.


y = x2 + 1 (shown below)

So according to Gauss, since x is raised to the second power, there should be two solutions for x when y = 0.. But, if you know how to read this graph, it is clear that the curve does not overlap the x-axis (where y =0) anywhere. 

“So what is the deal???”-Cardan

Anyways, CArdan came upon this problem, and died with it. Fortuenately, he passed his thoughts on his pupil, Rafael Bombelli. You see, Cardan has spun his wheel in circles to try and get an answer to this problem. Even after many attempts, and approaching the equation from several angles, he continued to come up with a solution with the √(-121).If you have taken an algebra class, you will know that it impossible to take the square root of a negative number.

Anywho, eventually, Bombelli came up with a solution. He essential left the √(-121) part alone and ended up solving the equation, for x, getting a positive integer real number as the solution. Eventually, this discovery led to the coining of the terms imaginary and complex numbers! I bet you are so glad you know that now.

How does this relate to brain injury?

            In many senses, our current ideas on how to treat a traumatic brain injury are similar to Carndan’s understanding of the geometry of complex numbers. We understand that the brain is plastic.  We also have seen some people recover miraculously from TBI. Unfortunately, there are still millions of cases worldwide in which the most passionate, hardworking physicians and scientists in the world simply shrug, and scratch their heads.

            The point I’m trying to make lies in the quote up top. Many times, in order to solve complex problems in the real world, we must take a step back from our current idea of reality, accept what we don’t understand, reassess the situation with new eyes, and then attack the problem head on. Only then can we come up with a solution that is practical and applicable in the real world.

            Now, you can take that literally, and think that in order to understand TBI the scientific community may need to redefine some of our paradigms around the way we treat TBI. This may very well be true. But I want to make a more practical point. When you are struggling through complex issues such as eye fatigue, insomnia, chronic fatigue, vertigo, etc, the solution to the problem may only come when you quit focusing so much on the central roadblock, and move forward while accepting the roadblock, while addressing other smaller issues first. I have found that many times, when I was struggling with fatigue, depression, or sleeping well my solution isn’t in sleep hygiene, or nutrition. My solution is often as random as needing to call my mom, hang out with friends, or even (hold your breath!!!), going out to have fun with friends.

            So whatever you are struggling with today, understand that many have struggled through similar things before, and many have found solutions.  If you focus myopically on your problems, you may never see your answers. Open up your mind a little bit, and tend to the part of the garden that you can touch.

Thanks for taking the time to read guys and gals. Love you all!