The more I think about math, the less I realize I know.
Learning about complex numbers has called me to re-evaluate how I think of negative numbers, or even natural numbers. I have to say the experience has been frustrating and has caused me to be skeptical about literally everything I think I know about math. Nevertheless, is pondering questions about the existence of these number systems useful? Can there be some reward in this? Do we even know that we're right in extending our number system the way we have?
It seems we extend whenever we can't solve;
- To solve $0=x+1 \hspace{5mm}$ we extend to the negatives
- To solve $1 = 2x \hspace{10mm}$ we extend to the rationals
- To solve $x = \sqrt{2}\hspace{9mm}$ we extend to the reals
- To solve $x = \sqrt{-1}\hspace{6mm}$ we extend to the complex
Whats stopping me from extending to solve
- $x = \frac{1}{0}\hspace{4mm}$ ?
Just as $x^2 = -1$ seemed meaningless pre-complex numbers, so does $x = \frac{1}{0}$. It seems this is the last type of 'equation' to solve for which we havn't invented some number system. So why can we extend to solve the previous equations, but not this one. Why do we think what we've done is even right?
I know this is a really soft question. But at the same time, the philosophy tag doesn't exist (lol) for no reason. Thanks,