Heavy tools to study apparently simple structrures or objects?

In summary, the natural integers, which are the numbers used for counting, are likely the most studied structure in mathematics. They contain many important properties that make answering questions about them challenging. Other structures that involve a lot of machinery in their study include Gaussian integers, rational numbers, algebraic integers, algebraic number fields, Dedekind rings, finite fields, real numbers, and complex numbers. Despite being commonly thought of as being based on a base 10 system, the integers are not limited to this base and the choice of base does not affect their properties.
  • #1
tgt
522
2
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?
 
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  • #2
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.
 
  • #3
lurflurf said:
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?
 
  • #4
tgt said:
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,...
 
  • #5
lurflurf said:
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?
 
  • #6
tgt said:
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
 

1. What are some examples of heavy tools used to study apparently simple structures or objects?

Some examples of heavy tools used in scientific research include electron microscopes, X-ray crystallography machines, nuclear magnetic resonance (NMR) spectrometers, mass spectrometers, and particle accelerators.

2. How do these heavy tools provide more detailed information about simple structures or objects?

These tools use advanced technologies and techniques to analyze and image structures and objects at the atomic and molecular level. This allows scientists to see and understand the intricate details and properties of seemingly simple structures or objects.

3. What are the advantages of using heavy tools in scientific research?

The use of heavy tools in scientific research allows for more accurate and precise measurements, as well as the ability to study structures and objects that are too small or complex for traditional methods. They also provide valuable insights into the fundamental properties and behavior of matter.

4. Are these heavy tools accessible to all scientists?

No, heavy tools are often expensive and require specialized training and facilities to operate. They are typically only accessible to researchers working in well-funded institutions and industries.

5. How have heavy tools advanced our understanding of the world?

Heavy tools have played a crucial role in advancing our understanding of the world by allowing us to study and manipulate matter at the atomic and molecular level. This has led to groundbreaking discoveries and developments in various fields such as medicine, materials science, and environmental science.

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