Why does the Fibonacci Series start with 0, 1? The Fibonacci Series is based on the principle that the succeeding number is the sum of the previous two numbers. Then how is it logical to start with a 0? Shouldn't it start with 1 directly?
One key number-theoretical reason for starting the sequence $(0,1)$ instead of $(1,1)$ is that it makes the divisibility property of the Fibonacci sequence more straightforward to state; i.e., that $F_k$ divides $F_{nk}$ for any $k,n$.  If you start with $F_0=1$ instead of $F_0=0$ then this breaks down (for instance, in that numbering $F_2=2$ but $F_4=5$) and a lot of results have to be presented with indices shifted.  This has to do, roughly with the representation of $F_n$ as $\frac{1}{\sqrt{5}}\left(\phi^n-\varphi^n\right)$ (with $\varphi=\frac{1}{\phi} = \phi-1$); the fact that the exponents 'match up' with the index leads to straightforward arguments for the various divisibility properties.
$0$ should not be considered as a Fibonacci number. Accordingly to Zeckendorf's theorem, every positive integer can be uniquely written as the sum of one or more distinct non-consecutive Fibonacci numbers. Taking $0$ as a Fibonacci number is therefore like taking $1$ as a prime number!

By the way, the divisibility property still holds taking $F_1=F_2=1$, so I don't agree with the argument given in the first answer.