# derive-it.com

## 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 .

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

 Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.