Is Number Theory Really That Useless?

[mathsuxhard]

Speaking as a CS student taking a Math minor. I have taken classes and done relatively well in most of the upper div Math courses I taken (ODE, Linear Algebra, Optimizatiion, etc etc)

Most of these subjects are quite useful in real life.

Linear Algebra? I use matrices fairly often.
Optimization? Definitely use it a ton.
Differentials? I might not use it a lot but I know a ton of engineers who do.

I just finished taking Numbers Theory, and it had to be the most useless course I ever taken. No, I do not know or care to prove 2+2=4. No, I do not know or care to prove that 2n+1 can give infinitely prime numbers. It was more useless than that Anthropology course about Asia I taken as a Freshman.

During the final, half the class walked out after they were told they could take a C without taking the final (as long as their midterm was a C). This is the first class I failed (won't be for much longer, going to complain to the professor to at least get a C in the class).

So number theorists out there, please explain to a math dummy like me how your subject is useful.

Mod note: edited out profanities

 

[Pyrrhus]

Not true,

I use Number theory in my estimation codes of econometric models to take draws from densities quite often.

 

[inknit]

Maybe this is what you feel about math in general. Have you taken real analysis or abstract algebra?

 

[mathsuxhard]

Maybe this is what you feel about math in general. Have you taken real analysis or abstract algebra?

If I hated Numbers Theory, what would make you think I would take Abstract Algebra after? lol

Like I said, I have respect for applied Math. Stuff that you can use in your everyday life. If I was hiring for some financial firm, I'm not going to hire the guy that knows prime factorization on the back of his hand...I'm going to hire the guy who knows stats.

@Pyrrus, what applications do you use from Numbers Theory?

Fermats Last Theorem? Pythagorean Triples? Name one useful thing one of these theories do and I'll give you a cookie.

 

[SHISHKABOB]

maybe you should be a bit more specific in your title and say "Numbers Theory is useless for me"

I'm sure it wouldn't be around if it didn't have SOME use

 

[micromass]

I certainly understand your feelings. Even I, as a pure mathematician, find number theory to be inelegant and useless. However, this is certainly not true if you think about it.

The most useful thing about number theory is in the study of prime numbers. This study is used in protection of websites on the internet. Being able to recognize large primes is essential in internet security and in coding theory.

When sending a satellite to Mars, then a lot of the transmission gets lost along the way. So we would like to reconstruct the transmission without much errors. This is done in coding theory and number theory is very important in there.

I think you were very unlucky. If it is true that your course only bothered with proving 2+2=4, then you had a bad professor. There are certainly applications of number theory, and it's sad that you didn't cover those.

This is probably a mistake in mathematics. We, mathematicians, find our theory beautiful in itself and do not really care for the applications. Our students often do need applications to appreciate the theory. So the professors should spend more time giving nice applications instead of theorem-proof.

Maybe next time you take a course, you could ask people who already took the class how it was. Or you can ask on this forum what to expect and whether or not to take it.

 

[dodo]

Hi, mathsuxhard,
some "pure" mathematicians, and maybe some theoretical physicists as well, would probably say that a subject being "useless" is a good thing in itself; it's pure science, as opposed to applied. (Sure enough, posts on number theory applications will follow, but that won't help.)

I'm sorry that you have suffered a subject that others do for fun. But my bet is, if someone is stuffed by force with chocolate cake, he'll have to hate it. Unfortunately that's how our educational institutions seem to work (no blame on teachers, as they suffer it just as much): cram as much as possible in as little time as you can ("Lockhart's lament" comes to mind). Anyway, as long as you find subjects that really motivate you, you will be alright. My 2 cents.

 

[Office_Shredder]

Rsa

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[Number Nine]

The reason you can use a credit card is, of course, because of the difficulty of factoring large numbers. Similarly, the reason you can't pickpocket a few credit cards and retire at 20 is because number theorists have not yet developed efficient factorization algorithms. Modern cryptographic techniques make use of number theory far more dense and abstract than anything you've ever encountered, and you can't understand any of it without understanding the material you're learning now.

Of course, all of this is irrelevant. Number theory is studied because it's interesting, not because it's useful.
Honestly, if you think number theory is bad, take a few courses in abstract algebra.

 

[Pyrrhus]

@Pyrrus, what applications do you use from Numbers Theory?

Fermats Last Theorem? Pythagorean Triples? Name one useful thing one of these theories do and I'll give you a cookie.

Sir, you need to control your attitude. I am not sure why you are being so condescending.

Generating Halton Sequence, Hammersley, and others low discrepancy sequences.

 

[Office_Shredder]

Like I said, I have respect for applied Math. Stuff that you can use in your everyday life. If I was hiring for some financial firm, I'm not going to hire the guy that knows prime factorization on the back of his hand...I'm going to hire the guy who knows stats.

If you're hiring for some financial firm, you probably aren't an undergraduate in college, so I don't see how your opinion really makes much difference here

 

[Robert1986]

I suggest getting Kenneth Rosen's Book Elementary Number Theory. It is chocked full of applications.

But, let me share this little anecdote. George Boole was a mathematician who did several things but one of them was to develop a system of algebraic logic. Now, at the time, it didn't seem like his ideas in this area had much "practical" value. However, computer architectures are essentially based on his ideas. At least at the logic gate level. Bools as data types are named after him because of his work in this area.

The moral: division of labour. Some people can prove theorems and/or come up with new ideas. Some can take those seemingly abstract theorems and use them. So, just because something is abstract and seemingly "useless" does not mean that it is.

 

[nonequilibrium]

Actually, it's just because number theory is so useful, that I'm not interested in it: as a theory I don't find it particulary beautiful, most of its merit is in its importance for applications.

On a side note, the OP could afford to be a bit more refined. "Name one useful thing one of these theories do and I'll give you a cookie." ...

 

[Robert1986]

Actually, it's just because number theory is so useful, that I'm not interested in it: as a theory I don't find it particulary beautiful, most of its merit is in its importance for applications.

Huh? I'm sorry, I will have to disagree with this 100%. I don't really know where you are coming from. Nearly all branches of math (I can't think of one that doesn't) has its roots in solving some practical real world problem. How it Number Theory any different? If it weren't for computers, number theory wouldn't be near as "useful" (at least not its current uses) as it is now, yet Number Theory was around long before computers.

Sorry, but I just don't see where you are coming from, here.

On a side note, the OP could afford to be a bit more refined. "Name one useful thing one of these theories do and I'll give you a cookie." ...

Now this is something I can agree with!

 

[nonequilibrium]

Huh? I'm sorry, I will have to disagree with this 100%. I don't really know where you are coming from. Nearly all branches of math (I can't think of one that doesn't) has its roots in solving some practical real world problem. How it Number Theory any different? If it weren't for computers, number theory wouldn't be near as "useful" (at least not its current uses) as it is now, yet Number Theory was around long before computers.

Sorry, but I just don't see where you are coming from, here.

I understand what you're saying. I admit I was overstating my case for "dramatic" effect, paving the way for misinterpretation, but in a more moderate restatement of my words I still stand by what I said: of course all math is (incredibly) useful, but I enjoy math because of its beauty, whereas I personally don't find number theory beautiful; but although I can "deny" the beauty of number theory, I cannot deny its importance. The OP suggested he would be interested in number theory if it were useful, and not just done for the sake of it, and I just wanted to make the case that I feel the reverse: I would appreciate it more if it more beautiful, and I cannot appreciate it just for its practicality.

Or even shorter, my point to the OP is that
1) number theory is incredibly useful;
2) ironically I would appreciate it more if it traded in some of its usefulness in return for beauty.

 

[R.P.F.]

Hmm...Do you know anything about cryptography?
I also find it funny how you called ODE, Linear Algebra, Optimization "upper div Math courses".

 

[R.P.F.]

2) ironically I would appreciate it more if it traded in some of its usefulness in return for beauty.

Algebraic number theory is beautiful.

 

[Robert1986]

I understand what you're saying. I admit I was overstating my case for "dramatic" effect, paving the way for misinterpretation, but in a more moderate restatement of my words I still stand by what I said: of course all math is (incredibly) useful, but I enjoy math because of its beauty, whereas I personally don't find number theory beautiful; but although I can "deny" the beauty of number theory, I cannot deny its importance. The OP suggested he would be interested in number theory if it were useful, and not just done for the sake of it, and I just wanted to make the case that I feel the reverse: I would appreciate it more if it more beautiful, and I cannot appreciate it just for its practicality.

Or even shorter, my point to the OP is that
1) number theory is incredibly useful;
2) ironically I would appreciate it more if it traded in some of its usefulness in return for beauty.

OK, now that I can see a little more.

 

[sunjin09]

Here is a relevant question I have long wondered though: does number theory have anything to do with special functions, e.g., hypergeometric functions whose Taylor series coefficients possesses certain patterns? I have been frustrated by equations without "analytic" solutions for many years.

 

[DeadOriginal]

Rsa

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Apparently the OP who is "CS" major has never heard of the RSA algorithm used in nearly all passwording features nowadays which takes extremely large numbers that are basically impossible to factor and uses it as an encryption key. You want to break into bank accounts? It's rumored that if someone ever finds the solution to the Reimann Hypothesis they will have the key to decrypting the RSA algorithm but its too bad. Number theory is useless so we should all stop studying it.

I feel like my tone was kind of sharp in that paragraph... It isn't meant to be that way.

If I hated Numbers Theory, what would make you think I would take Abstract Algebra after? lol

Like I said, I have respect for applied Math. Stuff that you can use in your everyday life. If I was hiring for some financial firm, I'm not going to hire the guy that knows prime factorization on the back of his hand...I'm going to hire the guy who knows stats.

@Pyrrus, what applications do you use from Numbers Theory?

Fermats Last Theorem? Pythagorean Triples? Name one useful thing one of these theories do and I'll give you a cookie.

If someone could factor a 15 million digit number on the back of their hands hell freaking yes I would hire him over the stats guy. He'd be able to crack any passcode known to man. Although some claim it is illegal I can already see in my mind the things I can do with that ability...

 

[Dembadon]

...

Like I said, I have respect for applied Math. Stuff that you can use in your everyday life.

What about those theorems that you freely use in your applied classes without questioning if they will work every single time you use them? They are the product of a rigorous process; a process of which you don't care to be a part, but incredibly useful to you whether you know it or not.

... If I was hiring for some financial firm, I'm not going to hire the guy that knows prime factorization on the back of his hand...I'm going to hire the guy who knows stats.

...

Your scenario is unrealistic; the person in question would be applying to the NSA, not your financial firm.

 

[dcpo]

Someone posted a relevant question on mathoverflow.

 

[20Tauri]

Did they not have a cryptography unit in your number theory course? Number theory has a lot of computational applications, RSA public-key cryptography, integer factorization, and primality testing (as people before me have mentioned) being some rather famous ones. I had a lot of fun with our crypto unit in number theory, but then again I liked pretty much all of it, useless or not.

 

[thrill3rnit3]

Don't feed the troll.

 

[DeadOriginal]

Don't feed the troll.

I would think you are feeding the troll by making a pointless post like that. Then again, I have just joined you.

Number theory is a beautiful thing. Everyone is entitled to their own opinions.

 

[Poopsilon]

Ya this is clearly just some guy who made an account for the expressed purpose of venting about failing his number theory class. But either way it has gotten people talking about why they like number theory and think it's useful.

I'm personally really amazed at the delightful way complex analysis almost doubles back on itself and ends up solving all these problems that seem at first to be purely arithmetic in nature. $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, for example. There's actually a really cool thread on mathstackexchange which gives like a bajillion different proofs of this identity.