Why Aren't Counting Numbers Enough

My goal with this blog is review the evolution of mathematics from a conceptual perspective.  For most people, through out their lives, all they seem to need are counting numbers.  By counting numbers, I mean a finite set of whole numbers that can be organized as simple decimals, fractions, or approximations.  I'm talking about the numbers that can be handled on a nonscientific calculator.  So, why do we need anything beyond counting numbers?  I will answer this question by detailing the problems and discoveries that led mathematicians to go beyond the counting numbers.  

I should also point out that I am not a mathematician.  By profession, I am a software development manager.  My interest in mathematics stems from my fascination with patterns and my love of history.




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$e$: $\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ = $\lim\limits_{n \to \infty}\sum\limits_{i=0}^n\left(\frac{1}{i!}\right)$

$\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ = $\lim\limits_{n \to \infty}\sum\limits_{i=0}^n\left(\frac{1}{i!}\right)$ since: (...