I am amateur and not having much knowledge of mathematical terminology so i try to explain what i think with my words:

Term "number" feels like something static. Something what simly is. Something which you can hold. Something what is stored somewhere. I tend to think that number is data. But something's wrong here.

Any real number i can thing of is at some point generated by specific algorithm or more generically some form of computational process which generates stream of digits and that is far from being static. For example what we calls "number PI" is result of execution of specific algorithm(s).
That sequence: 3.1415926 - thats not PI - thats only part of it. Part of what? Write that number on paper please... 

You cannot. Because PI is output of algorithm which never terminates. Still you can calculate with it like it is finite number. So if you have infinite output of algorithm and algorithm itself who of these two deserves to be called PI? Primary is algorithm i think. So is number PI algorithm? (Or rather group of algorithms - there are different algorithm which generates same result - PI)

But algorithm can be number only if it have some special properties. This algorithm dont needs to be finite but needs to be in principle "streamable"
its output breakable to finite sequential parts (i dont know how to call this property) so you can use output of infinite algorithm as input to other algorithm in finite time. Number PI have this property. Have all real numbers this property? Can they be all "streamed"?

I am little bit confused. If you understand can someone explain this and make it clear?