DanZ (![[info]](http://p-stat.livejournal.com/img/userinfo.gif){kind=small} fclbrokle) wrote,@ 2004-11-30 21:30:00 |
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Breaking into Zambian journalism
A new student group has come up at MIT called SciCom. They take articles written by MIT students and affiliates and get them published in a newspaper in Zambia that has a section on science. The group gets a full page each week. The idea is to offer some kind of educational resource to students in Zambia.
I offered to write an article, and immediately ran into some issues. First of all, I want to write about mathematics. But their formatting is very limited: I was told to stick to what I could do in Notepad. Right away, I need to find something without pictures and without notation.
I've finally come up with something. It's pretty basic, but I think it's interesting and comprehensible even if one has limited education. I'd really like to get comments on it; I'll be submitting it in a few days.
What does it mean for a set of things to be infinite in number? For there to be a never-ending supply of these things?
This is actually a deep question of mathematics, and I cannot hope to answer it quickly, though I can give you an idea. Take, for example, the positive whole numbers: 1, 2, 3, 4, 5, 6, and so on. I have to write "and so on" because there are infinitely many of them. This list can never end: if I ever end with the number n (for any value of n), I can always go on and then write n + 1. So, in some sense, I have a way to tell if a set is infinite: it is infinite if I can never make a list of all the members.
Mathematicians have done a lot with infinity: they have found and classified different kinds of infinities and looked at them in other ways. Rather than studying infinity, I just want to show you another infinite set, one that is more surprising. What follows is one of the most beautiful and important results of basic mathematics.
The result is important because it is part of the foundation of mathematics: most school children will learn it, and it is critical to understanding further mathematics. But why do I say beautiful? Why is that important?
The truth is that mathematics gets so abstract, so technical, that a mathematician can only find new results if he loves what he is doing. To study mathematics is to be always astonished by the proofs one sees; they are little pieces of insight into how we think, much like good poetry. Many times the usefulness of a result is not clear even to the mathematician that discovers it, but the beauty is always apparent.
The result I will prove is that there are infinitely many prime numbers. A number p is prime if it can only be divided by 1 and itself; for example, 7 is prime because it can only be divided by 1 and 7 (but not 2, 3, 4, 5, 6, or any other number). 1 is not considered to be prime. That there are infinitely many prime numbers tells us something fundmantal about numbers since any number can be written as a product of prime numbers in only one way. 30 is 2 times 3 times 5; no other prime numbers multiply to 30. Thus, in a way, our theorem about prime numbers tells us something about all numbers.
Moreover, "the infinitude of primes" is a rather beautiful result: it tells us that there are infinitely many of something, which is interesting. The proof is even more so.
This proof goes as follows (here, let * denote multiplication, so I can write 2 * 5 = 10). Suppose that there were not infinitely many prime numbers; then I could list them all out. Let's write the first prime as p1, the second as p2, and so on. If there were not infinitely many prime numbers, this list stops, so say that it has N primes listed. Then the complete list of prime numbers is p1, p2, p3, ..., pN.
What I'll show is that this list is not good enough: there's at least one prime that the list missed. But I'll be able to do this for any list, which shows that one cannot list out all the primes. But if there is no list of all the primes, there must be infinitely many!
All I have left to do is find a new prime that's not one of p1, p2, ..., pN. How do I create a new prime? First, I multiply them all together to get p1 * p2 * ... * pN. This is nothing new. But what if I add 1 to get p1 * p2 * ... * pN + 1? Let's write this as k, so k = p1 * p2 * ... * pN + 1.
Because 1 is not prime, any prime number is bigger than 1. So what happens when I divide k by p1? I get a remainder of 1: p1 does not divide k. Neither does p2, or any of the other primes in the list!
But any number can be written as a product of primes, and so either k is itself a prime, or some prime divides it. But none of the primes in our list divides k: then in either case we have found a prime not on the list.
This is it, so let's review what we've done. If there were only finitely many primes, we could list them all: p1, p2, ..., pN. Then we used these primes to make a new prime not on our list. We could take the new list and do this again if we wished, and keep going. The important part is that you cannot list out all the primes: there's always another one to put on the list, just like trying to list out all the whole numbers. Thus, there must be infinitely many primes.
I hope you found this logic as gorgeous as I did.
It is interesting that, although prime numbers are simple things (by mathematical standards), there are still many unsolved problems about them. Two of the famous such problems are Goldbach's conjecture and the Twin Primes conjecture. Goldbach observed that every even number he tested was the sum of two primes: 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, 10 = 5 + 5, 12 = 7 + 5, 14 = 7 + 7, and so on. Despite trying many millions of millions of numbers (using computers --- it's too boring for anyone to do by hand!), no one has found a single even number that's not the sum of two prime numbers. Goldbach conjectured (in a 1742 letter to a famous mathematician named Leonard Euler) that every even number is the sum of two prime numbers. Despite many attempts over the years, no one has been able to prove or disprove this conjecture.
The Twin Primes conjecture says that there are infinitely many "twin primes." Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), and so on. Again, no one has proved it or disproved it.
Why are these so hard to prove? In some sense, it's because the definition of a prime number is based on multiplication, but these results talk about addition. But this is just an excuse: we really don't know. They are, however, very interesting problems, and just working on them can provide new insights about the structure of numbers.
I hope I have demonstrated to you that mathematics is very much alive, and quite worth pursuing. The beauty we have seen, and have yet to see, is truly staggering. It is yet another wonder of mathematics that so many results that mathematicians study for their beauty turn out to be important and useful in many places in the real world. Gauss called mathematics "the queen of the sciences," and I must say that I think he was right.
A new student group has come up at MIT called SciCom. They take articles written by MIT students and affiliates and get them published in a newspaper in Zambia that has a section on science. The group gets a full page each week. The idea is to offer some kind of educational resource to students in Zambia.
I offered to write an article, and immediately ran into some issues. First of all, I want to write about mathematics. But their formatting is very limited: I was told to stick to what I could do in Notepad. Right away, I need to find something without pictures and without notation.
I've finally come up with something. It's pretty basic, but I think it's interesting and comprehensible even if one has limited education. I'd really like to get comments on it; I'll be submitting it in a few days.
What does it mean for a set of things to be infinite in number? For there to be a never-ending supply of these things?
This is actually a deep question of mathematics, and I cannot hope to answer it quickly, though I can give you an idea. Take, for example, the positive whole numbers: 1, 2, 3, 4, 5, 6, and so on. I have to write "and so on" because there are infinitely many of them. This list can never end: if I ever end with the number n (for any value of n), I can always go on and then write n + 1. So, in some sense, I have a way to tell if a set is infinite: it is infinite if I can never make a list of all the members.
Mathematicians have done a lot with infinity: they have found and classified different kinds of infinities and looked at them in other ways. Rather than studying infinity, I just want to show you another infinite set, one that is more surprising. What follows is one of the most beautiful and important results of basic mathematics.
The result is important because it is part of the foundation of mathematics: most school children will learn it, and it is critical to understanding further mathematics. But why do I say beautiful? Why is that important?
The truth is that mathematics gets so abstract, so technical, that a mathematician can only find new results if he loves what he is doing. To study mathematics is to be always astonished by the proofs one sees; they are little pieces of insight into how we think, much like good poetry. Many times the usefulness of a result is not clear even to the mathematician that discovers it, but the beauty is always apparent.
The result I will prove is that there are infinitely many prime numbers. A number p is prime if it can only be divided by 1 and itself; for example, 7 is prime because it can only be divided by 1 and 7 (but not 2, 3, 4, 5, 6, or any other number). 1 is not considered to be prime. That there are infinitely many prime numbers tells us something fundmantal about numbers since any number can be written as a product of prime numbers in only one way. 30 is 2 times 3 times 5; no other prime numbers multiply to 30. Thus, in a way, our theorem about prime numbers tells us something about all numbers.
Moreover, "the infinitude of primes" is a rather beautiful result: it tells us that there are infinitely many of something, which is interesting. The proof is even more so.
This proof goes as follows (here, let * denote multiplication, so I can write 2 * 5 = 10). Suppose that there were not infinitely many prime numbers; then I could list them all out. Let's write the first prime as p1, the second as p2, and so on. If there were not infinitely many prime numbers, this list stops, so say that it has N primes listed. Then the complete list of prime numbers is p1, p2, p3, ..., pN.
What I'll show is that this list is not good enough: there's at least one prime that the list missed. But I'll be able to do this for any list, which shows that one cannot list out all the primes. But if there is no list of all the primes, there must be infinitely many!
All I have left to do is find a new prime that's not one of p1, p2, ..., pN. How do I create a new prime? First, I multiply them all together to get p1 * p2 * ... * pN. This is nothing new. But what if I add 1 to get p1 * p2 * ... * pN + 1? Let's write this as k, so k = p1 * p2 * ... * pN + 1.
Because 1 is not prime, any prime number is bigger than 1. So what happens when I divide k by p1? I get a remainder of 1: p1 does not divide k. Neither does p2, or any of the other primes in the list!
But any number can be written as a product of primes, and so either k is itself a prime, or some prime divides it. But none of the primes in our list divides k: then in either case we have found a prime not on the list.
This is it, so let's review what we've done. If there were only finitely many primes, we could list them all: p1, p2, ..., pN. Then we used these primes to make a new prime not on our list. We could take the new list and do this again if we wished, and keep going. The important part is that you cannot list out all the primes: there's always another one to put on the list, just like trying to list out all the whole numbers. Thus, there must be infinitely many primes.
I hope you found this logic as gorgeous as I did.
It is interesting that, although prime numbers are simple things (by mathematical standards), there are still many unsolved problems about them. Two of the famous such problems are Goldbach's conjecture and the Twin Primes conjecture. Goldbach observed that every even number he tested was the sum of two primes: 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, 10 = 5 + 5, 12 = 7 + 5, 14 = 7 + 7, and so on. Despite trying many millions of millions of numbers (using computers --- it's too boring for anyone to do by hand!), no one has found a single even number that's not the sum of two prime numbers. Goldbach conjectured (in a 1742 letter to a famous mathematician named Leonard Euler) that every even number is the sum of two prime numbers. Despite many attempts over the years, no one has been able to prove or disprove this conjecture.
The Twin Primes conjecture says that there are infinitely many "twin primes." Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), and so on. Again, no one has proved it or disproved it.
Why are these so hard to prove? In some sense, it's because the definition of a prime number is based on multiplication, but these results talk about addition. But this is just an excuse: we really don't know. They are, however, very interesting problems, and just working on them can provide new insights about the structure of numbers.
I hope I have demonstrated to you that mathematics is very much alive, and quite worth pursuing. The beauty we have seen, and have yet to see, is truly staggering. It is yet another wonder of mathematics that so many results that mathematicians study for their beauty turn out to be important and useful in many places in the real world. Gauss called mathematics "the queen of the sciences," and I must say that I think he was right.
