In decimal expansion of real number, why the number of digits at the left side of decimal point is finite? I know the decimal expansion of a real number can be written as:

$r=\sum_{i=-\infty}^{N}a_{i}10^{i}$

The question is why it has the upper limit $N$ of the summation? Why instead, we write a real number as 

$r=\sum_{i=-\infty}^{\infty}a_{i}10^{i}$

I understand that if the series (at the right side) is infinite in both direction, it can not be convergent and can not be deal with, so as an infinite big real number.
Besides above consideration, is there any rule comes from the definition of real number forbid $N\rightarrow\infty$? Or, is it really necessary to define a real number to be a one we can deal with by current tool? 
 A: I don’t know whether the following helps, but, just dealing with positive numbers first:
Step 1: Every integer has a representation as a finite decimal.
I think you can grant this, it’s what you did in the very first years of schooling.
Step 2: Every real number between $0$ and $1$ has a representation as a decimal, extending potentially infinitely to the right.
(The fact that some reals have two different-looking representations need not bother us.)
It’s one of the properties of the real number system that if $\lambda>0$ is a real, there is an integer $n$ with $n\le\lambda<n+1$. This is the Archimedean property of the reals, mentioned by several of the commenters. If a system does not have this property, then it certainly is not the system of real numbers.
Step 3: Now take any positive real number $\lambda$, find that integer $n$ mentioned just above, and represent $n$ by a finite decimal expansion (Step 1), and represent $\lambda-n$ as a potentially infinite expansion (Step 2). Put the two expansions together, and get your decimal expansion of $\lambda$.
As several others have said, the fact that this procedure doesn’t give you an infinite string of digits to the left comes from Archimedes.

But: I would be remiss if I did not mention that there is a self-consistent system in which there are decimal expansions going infinitely to the left. They do not go infinitely to the right as well, however. This system is called the “$10$-adic numbers”, and it is of relatively minor interest for mathematicians. If, on the other hand, you write your integers in base $p$, where $p$ is a prime number, and allow the expansions to go infinitely to the left (still forbidding infinitely many digits to the right), you get the $p$-adic numbers $\Bbb Q_p$, which are of extremely great interest in more advanced number theory. But that is a story for another day.
