Can a number have infinitely many digits before the decimal point?

I asked my teacher if a number can have infinitely many digits before the decimal point. He said that this isn't possible, even though there are numbers with infinitely many digits after the decimal point. I asked why and he said that if you keep adding digits to the left of a number it will eventually approach infinity which is not a number and you could no longer distinguish two numbers from one another.
Now this is the part of his reasoning I don't understand: why can we distinguish two numbers with infinitely many digits after the point but not before it? Or is there a simpler explanation for this?

Answer

The formal way to understand this is, of course, using the definition of real numbers.
A real number is "allowed" to have infinite digits after the decimal point, but only a finite number of digits before. (http://en.wikipedia.org/wiki/Real_number)

(if it interests you, there are numbers that have infinite digits before the decimal point, and only a finite number after. Take a look at http://en.wikipedia.org/wiki/P-adic_number . just to have some fun, know that the $10$-adic expansion of $-1$ is $\color{red}{\dots 99999} = -1$)

If you want to get some intuition about this, first think that, as your teacher said, said number would approach infinity, which is not a real number. This is reason enough.

About the comparing two numbers part: if I give you

$$1234.983...$$ and $$1234.981...$$
you know which one is bigger, it does not matter what the other digits are.

But with

$$...321.99$$ , $$...221.99$$ you don't, because the information relies in the "first" digit. Of course nobody know what the first digit is, since there is no first digit.

But as I said before, this is to gain some intuition; the correct way to think about this is using the definition (which is not trivial)