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I often find myself facing this problem when asked by non-mathematicians about what I do. When the answer, "I'm a mathematician," doesn't suffice and I have to reply that I am a number theorist, I do not know how to explain to a nonspecialist what number theory is. The answer, "I study numbers," feels rather unsatisfactory to me at times like these. My thinking is that I'm probably not the only one who has encountered this. So my question is, how does one explain what number theory is to a non-mathematician?

For the purposes of this question I think it is reasonable to assume our audience possess a standard high school education.

EDIT: Some of the answers seem to go off tangent so I guess I should emphasize that the question really is, "how does one explain what number theory is to a non-mathematician?" and not how should a mathematician socialize with a group of non-mathematicians.

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It depends on what you do. "Number theory" is too vague. What is your specialty? –  Yuval Filmus May 23 '12 at 4:11
    
I think describing "specialty" is a problem even in a discussion between mathematicians, much less an outsider. So getting that in depth would be even more of a problem IMO. –  Eugene May 23 '12 at 4:29
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Although you say that "I study primes" seems unsatisfactory to you as an explanation, do you really think it is unsatisfactory to non-mathematicians (by which my sense is you even mean non-academics, e.g., not physicists)? Tell them you study primes and that understanding properties of prime and non-prime numbers is important in security in online commerce. Then move on to something more interesting to both of you. :) –  KCd May 23 '12 at 5:36
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This question reminds me of a story that a number theorist, who I won't name, told me last year. He was frustrated that while faculty from other disciplines at parties said they worked in "19th century French literature" or "17th century Italian music" and got polite nods, when he said that he worked in automorphic forms he'd get blank stares. As soon as he started saying that he worked in "late 20th century American number theory", he too got the polite nods. –  KCd May 23 '12 at 5:40
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You should start by letting them know you might be awarded $1.000.000. Then, they will be interested, and you can talk for a while. –  Pedro Tamaroff May 23 '12 at 21:30
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I think non-mathematicians are often more interested in the "why" and "how" than the "what", so I'm usually very light on mathematical details (unless they ask for it). I try to talk more about the motivations, history and fun facts of it, developping more or less detail depending the listener's (apparent) interest. My usual answer is along those lines :

In a nutshell, Number theory is the study of the integers. Although their definition is quite straighforward, there are still many unanwsered questions about them. If they ask for an example, I usually talk a bit about primes numbers : I explain we know stuff (that there are infinitely many of them, we can even roughly count them) but it's difficult to actually find them. Sometimes, I also mention that some seemingly simple problems like Fermat's last theorem took centuries to be solved.

One reason for this difficulty might be that there's a bigger picture that we still don't see. So being a number theorist is trying to get some perspective. Very often, this requires taking a detour, like creating and mastering sophisticated new objects which retrospectively sheds some new light on old ones. If they ask for an example, I talk about how some new numbers (negative, complex...) were created to solve equations, and mention that number theorists have invented a whole lot of other "exotic" numbers for other purposes (finite fields, algebraic integers, quaternions, $p$-adic numbers...).

Now number theory is interesting because integers are at the heart of mathematics, so understanding them might lead to advances in mathematics as a whole (which might lead to advances in science ?). Another reason is that we use it in cryptography because it provides problems that are difficult to solve. And on a more personnal note, my motivation is also that I find it beautiful and quite fascinating.

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+1 This 1. Does not sound like it was written for other math geeks and 2. Provides the fullest answer with an example. –  user23320 May 23 '12 at 12:40
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When I explain what I do to a nonmathematician, I usually briefly explain what a prime is (if they don't already know) and talk about how there are infinitely many of them.

If it turns out that they asked the question out of something besides cocktail-politeness, then I might continue to say that this is sort of like starting with the number $1$, adding $1$ a whole lot to get the sequence $1, 2, 3, \dots$ and seeing if that sequence has infinitely many primes. But we may consider other sequences too - it turns out that other sequences, like $2,(2 + 3),(2 + 3 + 3),\dots = 2, 5, 8, \dots $ have infinitely many primes too, while sequences like $2, 6, 10, \dots$ don't.

Let's be honest - usually people have tuned out by now. But let's suppose I'm talking to my girlfriend or something, and thus that she would feel awkward if she cut me off now (not saying this has happened or anything ;p). So I might continue to say that it turns out that one of the ways to understand primes in sequences is to understand what's called the Riemann Zeta function $\sum \frac{1}{n^s}$. This is sort of cool, and just a hint of the connections with certain special functions and the behavior of numbers.

Now I cheat a little, and mention something that people feel is a bit more approachable. The zeta function appears in other places too, like Zipf's law. And Zipf's law applies to many things besides languages - it tends to closely describe things like the major players mentioned in newspaper headlines. Some social sciences even claim it for their own (here, it is used to say that the size distribution of cities must fit a power law).

The binding idea here is that arithmetic functions, or other functions for that matter, sometimes have things to say about topics which they might not seem to describe. And as a number theorist, I look into some of these families of functions and how they behave.

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+1 "But let's suppose I'm talking to my girlfriend or something, and thus that she would feel awkward if she cut me off now (not saying this has happened or anything ;p)." I was worried you were about to start a proof by contradiction, then it became very realistic. –  000 May 23 '12 at 6:43
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Consider what people are looking for when they ask you this. People are looking to know about you, not for a lesson on prime numbers.

Suppose I meet folks in non-geeky jobs like finance or law or real estate. When I ask them what they do for a living, and even if I really want them to go into it, I'm not looking for financial methodologies or legal precedents or the way a home's value decreases over time while it's on the market. If you are trying to get to know a person (primarily) or what they do (secondarily), those things are tertiary.

When I am asking what someone does, I want to hear something like "My team is responsible for capital budgeting at GBX." They may tell me how excited they are about finance, or how they're studying for an MBA part-time, through GBX's tuition-reimbursement program. Or maybe they'd tell me it's just a job, and they're think of going elsewhere. People who are really good at this will have a few stories ready to share, and they're often not even directly related to their field. It may be a story about a coworker, or a client, etc., to engage your interlocutor for a few seconds.

So instead of explaining number theory, you're better off saying you're a professor or a consultant or whatever it is you do, and telling a fun anecdote. Mention that you're busy grading papers all the time, or tell something funny a study wrote on a test. The point is to tell people about yourself.

As @Auke and others pointed out, a lot of people don't understand what Mathematicians do. I don't mean that they don't understand what mathematicians study (which is probably true as well), but moreso, what their jobs looks like. When pressed on that subject, consider again avoiding the specifics of your area of research and speak to your tasks or assignments. For example, do you spend a lot of time teaching? Doing research? What does doing research look like? How do you collaborate? Or if you are working in industry, what is your role there--writing software? Doing some sort of analysis?

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The question is, "how does one explain what number theory is to a non-mathematician?" and not how does one introduce oneself to a group of people so I feel this answer isn't relevant. –  Eugene May 23 '12 at 21:14
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Most people know what a prime number is. You could explain that Number Theory is concerned with stuff like prime numbers and give an example or two of the kind of thing we know about primes.

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Summaries of four major number-theoretic directions of interest, to my mind:

  • Studying the internal structure of algebraic settings that generalize the integers but still retain fundamental arithmetic properties like divisibility, factorization, irreducibility/primality, and have a striking resemblance to the usual rationals. ($p$-adics, adeles, cyclotomic fields...)
  • Devising asymptotic bounds or statistical (even existence) results on patterns that encompass arithmetic properties again, like the distribution of primes or primes in arithmetic progressions.
  • General solutions to Diophantine equations, where they come from, when and to what extent integers can be written as certain types of sums (e.g. sum of four squares), etc.
  • Analytic objects - like $L$-functions or modular forms - whose functional equations, Fourier coefficients, or other features exhibit or encode arithmetic information.

Explaining what these mean, even simplistically, may take some time, but I imagine this would give a somewhat satisfying and rough account of what number theory is often "about."

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My typical answer is that I study prime numbers and their properties, and give some examples of open problems in the field. (There seem to be plenty of problems that are easy to state that have been difficult or are still open.) I'll also throw in some real-world applications of number theory, including cryptography, coding theory, music theory, and etc...

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It is quite hard enough to explain what a mathematician does, anyway, so why bother telling them your specialism?

People don't "get" math. Most people only know that they had to fill in formulas and apply the tricks they were taught at school - try explaining them axiomatic math instead. Non-scientists think that all mathematicians do is calculate things all day long - that mathematicians are somewhat like computers in that all we are allowed to do is execute simple calculations.

I don't really care what kind of microbes my bio friend is studying. I don't care what construction method my friend at civil engineering uses. I don't get the "core idea" of either, anyway, so I'd rather have them explain that instead.

I usually try saying something like "we simply assume certain simple things, such as a+b=b+a, and then try to deduce more knowledge". Even simple explanations like these are too vague for many, but it's worth a try. You can name a couple of theoretical systems, or slowly build up $\mathbb Z$ and $\mathbb Q$ - once they get that, number theory is just a more advanced problem.

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Thank you--this is the only answer that is not hopelessly socially awkward! Suppose I meet folks in non-geeky jobs like finance or law or real estate. When I ask them what they do for a living, and even if I really want them to go into it, I'm not looking for financial methodologies or legal precedents or the way a home's value decreases over time while it's on the market. If you are trying to get to know a person (primarily) or what they do (secondarily), those things are tertiary. You're better off saying you're a professor or a consultant, and telling a fun anecdote. –  Ray May 23 '12 at 10:04
    
I decided to make this an answer –  Ray May 23 '12 at 10:18
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If your bio friend can't say roughly what property of the microbe he or she is studying makes said microbe more interesting than the zillions of other microbes, do you not have reason to doubt the intellectual value of his or her endeavors? And if you do not bother to give him/her the chance to explain -- because "you don't care" about stupid microbes -- are you not altogether poorer for it, in the event that he/she actually has a very good reason why these microbes are interesting? I think it is very legitimate to take people at face value when they ask about your mathematics [ctd.] –  Sam Lichtenstein May 31 '12 at 4:15
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... and to make an honest effort to explain to them what exactly it is that you find interesting about the little part of the mathematical universe that you call home. I often fail when I make such an effort. But I strongly believe the effort it worthwhile. Here's an analogy. As a non-religious person, I have a hard time really "getting" where believers are coming from when they discuss their faith. But I hardly think this means I should avoid asking about the religious aspect of their intellectual and spiritual life. In fact, I would feel condescended to if they were just to say [ctd.] –  Sam Lichtenstein May 31 '12 at 4:18
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... "you wcouldn't understand; it's complicated", in response to an inquiry about their faith. Likewise, mathematicians need not patronize those who ask about their research. We may fail to convince a non-mathematician that our efforts are worthwhile. We will probably even fail to get across the basic idea of what we find challenging in our field. Yet we surely each have an answer to the question, "Why is my mathematics valuable or interesting?" We owe it to ourselves to share the answer with anyone who is curious enough to ask. –  Sam Lichtenstein May 31 '12 at 4:19
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