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?

Let me just remark that I find it curious that the question exclusively talks about "numbers", and the answers, while sometimes referring to p-adics as something else to think about, exclusively consider the question to be about "real numbers" and/or measures.
Incidentally, the branch of mathematics called "number theory" is very little concerned with real numbers except as an analytic tool.
When we are talking "n-adics" and take, say, a base of 10, then something like
...666666666666667
can be considered a representation of 1/3 since multiplying it by 3 yields
...000000000000001
It turns out that things work out more systematically if you don't take a base of 10 but rather of a prime number.  Either way, one can develop theories around such constructs and rules to work with, and calling "real numbers" numbers but such things not is purely arbitrary.
If you choose to do it that way, "number theory" is barely about numbers any more...
