Reaching for that Number Theory book (Part I)

I now turn to number theory, because this field seems like a solid foundation for mathematics and physics. I start with Chapter 1 of the esteemed book, Theory and Application of Infinite Series [1].

The title of Chapter 1 is “Principles of the theory of real numbers.” Section 1 is called “The system of rational numbers and its gaps.” I have wondered about numbers and how they relate to the number line encountered in math classes. Hopefully this section clears up some of my confusion, but I don’t want to get my hopes up too high…

I do not intend to rewrite the entire book — that would be ludicrous! Instead, as I read I will take note of topics that I deem important. Alright, here it goes.

The author first mentions certain numbers such as $ \sqrt{2}$ and claims that $ \sqrt{2}$ has never been written down in decimal form by any person. In other words, apparently $ \sqrt{2}$ does not have a finite amount of digits. But how can this be known for certain? I would be interested in seeing a proof that shows $ \sqrt{2}$ cannot be written down in decimal form. I have heard that $ \sqrt{2}$ is an irrational number, but that is just a label. Could irrationality be a consequence of the base 10 number system, or is irrationality independent of the number system that one uses?

Another question I have is, can a number such as $ 3$ be written as $ 3.0…$? Here, $ …$ represents an endless amount of zeros. If so, then by the same logic could one say that no person has written $ 3$ in decimal form because it can also be written with an endless amount of digits? These ideas will remain in the back of my mind, and I do not expect them to be answered any time soon. But at least the author shares a similar sentiment as me as he writes “we should have to enquire into the whole significance or concept of the natural numbers 1, 2, 3, …”

References