Would someone tell me about the importance of number theory?

[AdrianZ]

well, I've recently found myself interested in the subject, I hadn't studied the subject in high school and I haven't taken the course in university yet but since I've read Herstein's abstract algebra book I have become familiar with some congruence equations and other simple stuff. Right now I'm studying Gauss's Disquisitiones Arithmeticae and It seems so fascinating to me. Somewhere I read a comment made by Gauss that says 'Mathematics is the mother of all sciences and number theory is the mother of mathematics', so, If a giant like Gauss believes number theory is that important in mathematics, then it must be! The thing is I don't know why number theory is that important. I understand that many concepts in group theory and abstract algebra are in fact generalizations of the properties of integers that have originated from the works of 18th and 19th century mathematicians in number theory but I don't know how number theory is researched these days. It seems that modern number theory is way more complicated than the naive number theory.
Would someone give a short introduction to number theory here? Why It is so important as Gauss says? I've heard that all great mathematicians were good number theorists, from Euler, Gauss, Eisenstein, Galois and other great 19th century mathematicians to contemporary mathematicians like Emil Artin, G.H. Hardy, Paul Erdos, all of them were strong mathematicians in number theory. so If someone tells me about number theory more, I'd really appreciate that.

 

[lostcauses10x]

AdrianZ
There are a great many sources on the internet for what you seek.
Number theory was basically developed due to division, and the result of prime integers.

A few books I could suggest are
Invitation to Number Theory
Number theory and its history
There are quite a few books that can be got worth the read, out there.

It is a great subject and worth learning about.

 

[mathwonk]

I don't know much about number theory, but my impression is that it is considered important first of all because it concerns some of the most absolutely basic mathematical objects, the integers, and yet is very challenging to understand. Secondly it has fascinated the best mathematical minds of history for centuries so belongs to a lengthy and honorable scientific tradition in our culture. Technically, the research done in this area by geniuses like Euler and Riemann, has shown that it is not only beautiful and fascinating, but intimately connected with many other deep mathematical subjects like analysis, algebraic geometry and even topology. Finally it has been shown in recent times to have extremely useful applications that were not foreseen, like cryptography, e.g. internet security.

 

[homeomorphic]

Number theory is a big gap in my knowledge, but I will say, according to Bressoud in one of his radical approach to real analysis books, part of the impetus for the development of modern analysis was dealing with nasty functions that arose in number theory.

Evidently, number theory is used in computing Whitehead groups, which is important in topology. I mention this, not because it's important (unless you do surgery theory or something like that), but just to make the point that it can pop up in unexpected places. It's not too surprising that any given piece of mathematics is important because there are so many interconnections between different subjects.

I don't think one needs to know number theory to be a great mathematician these days. In the past, it was easier to be good at many fields, but these days, more specialization is necessary due to the sheer volume of math out there.

 

[20Tauri]

I've had two quarters of elementary number theory, and it's a lovely discipline. I can't speak much to more advanced fields like algebraic or analytic number theory, but you are correct that many concepts of algebra originated in number theory. Additionally, the field has enjoyed new relevance with the advent of computers because of its cryptoanalytic applications. For example, the RSA public-key encryption algorithm comes out of modular arithmetic.

 

[SteveL27]

I've had two quarters of elementary number theory, and it's a lovely discipline. I can't speak much to more advanced fields like algebraic or analytic number theory, but you are correct that many concepts of algebra originated in number theory. Additionally, the field has enjoyed new relevance with the advent of computers because of its cryptoanalytic applications. For example, the RSA public-key encryption algorithm comes out of modular arithmetic.

Hardy would have said that number theory is the most beautiful subject in math precisely because it's so useless. He is quoted as saying Nothing I have ever done is of the slightest practical use.

http://en.wikipedia.org/wiki/G._H._Hardy

It's only in the past couple of decades that public key cryptography has made number theory "relevant." That is the LEAST important factor in determining the value of a branch of mathematics :-)

 

[Anti-Crackpot]

Without Number Theory you don't have String Theory.

Suggested: Symmetry & the Monster (Mark Ronan)

It goes a bit deeper than just Public Key Cryptography...

 

[chiro]

With number theory, it might help you to realize that number theory helps us define periodic properties of systems that are concerned with whole or rational numbers.

When you are doing a Modulus operation (Mod n), you are basically dealing with a periodic function that goes from 0 to 9 and then repeats itself all over again.

It turns out that this periodicity has interesting properties when you study when you look at results like Eulers Theorem up to things like the totient function and applications to cryptography.

If you have trouble thinking about periodicity, then picture a sine-wave or if you want a better idea of what the modulus function looks like thing of a saw-tooth wave and instead of the saw-tooth wave being continuous up until the next cycle, think of that part looking like a stair-case function at each integer.

Now imagine this kind of behaviour amplified in all of the complex ways that it could be extended to and think about the idea when instead of just wanting to 'calculate' something (Mod n), that you want to find a solution to some expression where (Mod n) must equal something and you're beginning to see the kinds of things that are considered in number theory.

 

[Anti-Crackpot]

With number theory, it might help you to realize that number theory helps us define periodic properties of systems that are concerned with whole or rational numbers.

When you are doing a Modulus operation (Mod n), you are basically dealing with a periodic function that goes from 0 to 9 and then repeats itself all over again.

It turns out that this periodicity has interesting properties when you study when you look at results like Eulers Theorem up to things like the totient function and applications to cryptography.

If you have trouble thinking about periodicity, then picture a sine-wave or if you want a better idea of what the modulus function looks like thing of a saw-tooth wave and instead of the saw-tooth wave being continuous up until the next cycle, think of that part looking like a stair-case function at each integer.

Now imagine this kind of behaviour amplified in all of the complex ways that it could be extended to and think about the idea when instead of just wanting to 'calculate' something (Mod n), that you want to find a solution to some expression where (Mod n) must equal something and you're beginning to see the kinds of things that are considered in number theory.

Well said Chiro.

And in regards to periods, imagine you're a physicist working with waves trying to figure out how those waves might "fill" the circumference of a circle (ala' de Broglie's reinterpretation of Bohr's model of the atom...).