Is term "real number" equivalent to "group of algorithms generating stream of digits"? 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?
 A: What you mean is equivalent to computable numbers. There are also definable numbers.
You can also be interested in this question. 
A: In standard mathematics, there are many more real numbers that there are algorithms, and therefore many real numbers that are not specified by algorithms. Even so, it's not often you come across a concrete example of a number that does not have an algorithm -- Chaitin's constant is one example.
This depends on the fact that every algorithm can be written down as a finite sequence of letters (this is implicit in the concept of an algorithm), and there are only countably many finite strings of letters. Not all strings of letters encode algorithms, of course, but those that do can be listed in order of increasing length and alphabetically within each length -- and then we can use Cantor's diagonal argument to find a real number that is not computed by any of the algorithms.
The only possible snag in this argument is whether it is really true that the strings that describe algorithms "can be listed". As a matter of practical fact, we cannot actually list all the algorithms and only those -- because it is not generally possible to see given a string whether it's an algorithm (which keeps producing digits forever without entering an infinite loop) or not, due to the halting problem!
Still, if you accept that there is a set of exactly those strings that describe algorithms, then the rest of the the argument above goes through and can't be stopped -- and there will be more real numbers than algorithms.
On the other hand, you may deny that such a set exists. In that case you're not doing standard set theory anymore, but instead constructive mathematics, which is a respectable field of study as long as you don't confuse it with the standard non-constructive mathematics everyone else uses. Within the particular framework of constructive mathematics it is indeed possible to say that every real number is computable. On the other hand, this doesn't necessarily mean that that the real numbers are countable, because the algorithms cannot be listed.
