# Value of studying number theory?

1. Aug 13, 2015

### Mirero

It seems that no matter how unrelated two subjects of mathematics appear to be, there are always ways to use techniques from one area of math and use it to prove many useful results in the other, and vice versa.

However, from my (inexperienced) point of view, number theory seems to be the only exception to this. That is to say, many mathematicians use techniques from other fields to prove results in number theory, but I very rarely see number-theoretical techniques applied to other fields. I’m not sure if this is due to my unseasoned mathematical knowledge or if the statement has a degree of truth to it.

That being said, would it be a good use of my time to study, at the very least, elementary number theory? Or would my efforts be better used to build up my foundations in other areas of mathematics?

2. Aug 13, 2015

### Staff: Mentor

Personally, I would study all the math you can and not drop anything.

W. H. Hardy wrote the book "The Mathematician's Apology" stating that the math he worked on had no practical application and that that was the most beautiful math of all. However, it was later used in cryptographic analysis and code development.

I think all math is like that, an answer in search of a problem so one shouldn't discard it because we don't yet know what to do with it.

https://en.wikipedia.org/wiki/A_Mathematician's_Apology

3. Aug 13, 2015

### mathman

It very much depends on your interests. I am a retired mathematician - I never studied number theory.

4. Aug 13, 2015

### micromass

Yes, it is a good idea to learn number theory if you want to learn pure math. Why? Because number theory has had a very important role in the history of pure math. Many of the techniques and concepts we use today come directly from number theory.

I always feel it is important to know how a certain concept was invented and what its original uses were. This will help you understand mathematics on a much deeper level, and it will make certain important math concepts be less unmotivated. I think this is a very good way of learning things. If you want to learn this way, then knowing number theory is essential. For example, concepts like ideals and other concepts in abstract algebra come directly from number theory. A lot of techniques in complex analysis were also motivated by number theory.

So while number theory might not be strictly necessary to understand other math, it has a big motivational role.

5. Aug 13, 2015

### laplacean

6. Aug 13, 2015

### Mirero

Ugh, I was afraid you guys were going to say that, especially since number theory is a big wall for me.

Although I guess we all encounter walls at some point, anyway, and from what you guys say I guess I should try to get by it. Thanks for the perspective guys!

7. Aug 14, 2015

### mathman

I suggest that if you want to learn math, get through calculus. After that there are many directions to go - number theory is important for some, but for others it is irrelevant.

8. Aug 15, 2015

### thelema418

In terms of a variety of jobs I have had, I have used number theory more often than calculus. The most helpful topics have been finding solutions to diophantine equations and the concept of check digits. Number theory is particularly helpful for forensic accounting.

In my experience, number theory allows for the invention or engineering of mathematical solutions in a way that I have not experienced with other disciplines.