Heavy tools to study apparently simple structrures or objects?

[tgt]

We all know that a lot of machinery goes into study the set of natural integers such as algebraic number theory and analytical number theory. Why are the natural integers so special? What other mathematical objects involve so much attention in terms of machinery used to study them?

 

[lurflurf]

The integers are likely the most studied structure. So no other mathematical objects involve so much attention in terms of machinery used to study them. Integers are special because they are the numbers used for counting and because they contain many important properties that make answering questtions about them nontrivial. Since Integers are the prototype of many other stuctures, if some stucture derived from them (such as Gaussian integers, rational numbers, or algebraic integers) is interesting they got it from the integers. Algebraic and analytic number theory are not limited to integers, but one of there applications is the study of the integers.

 

[tgt]

Algebraic and analytic number theory are not limited to integers, but one of there applications is the study of the integers.

What other structures can they be used to study?

 

[lurflurf]

What other structures can they be used to study?

I already mentioned Gaussian integers, rational numbers, and algebraic integers.
Algebraic number fields are natural object to study in algebraic number theory.
Dedekind rings, finite fields, real numbers, complex numbers,...

 

[tgt]

I already mentioned Gaussian integers, rational numbers, and algebraic integers.
Algebraic number fields are natural object to study in algebraic number theory.
Dedekind rings, finite fields, real numbers, complex numbers,...

Actually, I just realized that the natural integers is based on a base 10 system which isn't special. What happens if we have base 4 or something?

 

[lurflurf]

Actually, I just realized that the natural integers is based on a base 10 system which isn't special. What happens if we have base 4 or something?

The integers are not based on a base 10 system. The base is just a means of representation, a nice way to warite out and work with numbers. Sort of like measuring a distance in inches or centimeters. A good example is divisability, for example it is very easy to tell if b^n divides a number is it is written in base b, but the truth of that fact is not changed in a different base.
15890 (base 10)=64220 (base 7)
This number is divisible by 10 and 7
in base 10 it is obviouly divisible by 10 less obviously 7
in base 7 it is obviouly divisible by 7 less obviously 10