My Math Alphabet: F is for Factoring
Factoring Numbers
Factoring numbers is usually pretty straightforward. If we're trying to find the factors of a number n, we check each prime up to the square root of n*. There are easy ways to check whether 2, 3, and 5 are factors, and about 73% of randomly picked numbers will have at least one of these factors. (I just now figured that out. Challenge #1: How?) But once we've pulled out those factors, or purposely chosen a number that doesn't have any of those factors, it gets just a bit harder; with the help of a calculator, we just check whether n/p is a whole number. Here's a cute bit of math trivia: The first three numbers I'd check with a calculator - 7, 11, and 13 - multiply to make 1001, so for example, 137,137 factors to 7 . 11 . 13 . 137. (137 does not have 2, 3, 5, 7, 11, or 13 as factors, so I know it's prime.)
Although this search is relatively simple, it depends on knowing the list of primes to check (2, 3, 5, 7, 11, 13, 17, 19, ...), and that gets lots harder to produce for really big numbers. I remember reading the Scientific American article in 1978 about the mathematics behind a new form of cryptography, that involved multiplying two very big prime numbers to get a number that would be too big to factor (in a reasonable amount of time). This scheme, now dubbed RSA after the 3 mathematicians who worked it out, is part of what helps keep online information private. Of course, what qualifies as a big enough number changes over time as computers become more powerful. Here's a list of the first ten thousand primes.
Factoring numbers is good practice for improving your number sense, and mathematicians sometimes find that a meditative exercise. If you have a calculator handy, it's pretty quick to check out the factoring of the number in this xkcd comic (the square root of 1453 is 38.118..., so you only have to check the primes up to 37). That's challenge #2.
Some mathematicians are good at arithmetic and would check out possible factors in their heads. I'd rather not myself, but G.H. Hardy and Ramanujan, perhaps like most number theorists, would have liked that sort of thing. A much-quoted story has Hardy saying his taxi number, 1729, was dull, and Ramanujan responding that it was a very interesting number because it's the smallest number expressible as the sum of two cubes in two different ways. Hmm, I just factored 1729 (challenge #3), and I think it's interesting in another way - its prime factors are in an arithmetic sequence.
Factoring Polynomials
We spend a lot of our time in beginning algebra courses on factoring polynomials, and some teachers question whether that's a good topic to spend the time on. One reason for factoring polynomials is to find solutions to equations. But computers and calculators can answer those questions so easily, perhaps this topic become has become archaic. (And besides, most polynomials can't be factored. I remember a lovely post about that, and cannot find it. Can someone point me to it?) Here's a cool picture by Dan Christensen, of all the complex roots of polynomials with integers coefficients of degree 5 or less. (More here.)
In the comments to dy/dan's post Grocery Shrink Ray, one person wrote: (comment #13)
One is Pythagorean triples, which I've posted about before. But here's a mathematical use for factoring I'd never seen before: George Sicherman wanted to find a way to create a pair of dice that would have the same sums as regular dice, with the same probabilities (7 comes up most often, for instance). He used polynomials (and their factors) to find his unusual dice. (Found at Plus Magazine.)
What are some of the other cool mathematical uses of factoring?
_____
*If f is a factor of n, then f . g = n. One of these two numbers must be less than the square root of n, and the other more, unless n = f . f.
Factoring Numbers
Factoring numbers is usually pretty straightforward. If we're trying to find the factors of a number n, we check each prime up to the square root of n*. There are easy ways to check whether 2, 3, and 5 are factors, and about 73% of randomly picked numbers will have at least one of these factors. (I just now figured that out. Challenge #1: How?) But once we've pulled out those factors, or purposely chosen a number that doesn't have any of those factors, it gets just a bit harder; with the help of a calculator, we just check whether n/p is a whole number. Here's a cute bit of math trivia: The first three numbers I'd check with a calculator - 7, 11, and 13 - multiply to make 1001, so for example, 137,137 factors to 7 . 11 . 13 . 137. (137 does not have 2, 3, 5, 7, 11, or 13 as factors, so I know it's prime.)
Although this search is relatively simple, it depends on knowing the list of primes to check (2, 3, 5, 7, 11, 13, 17, 19, ...), and that gets lots harder to produce for really big numbers. I remember reading the Scientific American article in 1978 about the mathematics behind a new form of cryptography, that involved multiplying two very big prime numbers to get a number that would be too big to factor (in a reasonable amount of time). This scheme, now dubbed RSA after the 3 mathematicians who worked it out, is part of what helps keep online information private. Of course, what qualifies as a big enough number changes over time as computers become more powerful. Here's a list of the first ten thousand primes.
Some mathematicians are good at arithmetic and would check out possible factors in their heads. I'd rather not myself, but G.H. Hardy and Ramanujan, perhaps like most number theorists, would have liked that sort of thing. A much-quoted story has Hardy saying his taxi number, 1729, was dull, and Ramanujan responding that it was a very interesting number because it's the smallest number expressible as the sum of two cubes in two different ways. Hmm, I just factored 1729 (challenge #3), and I think it's interesting in another way - its prime factors are in an arithmetic sequence.
Factoring Polynomials
We spend a lot of our time in beginning algebra courses on factoring polynomials, and some teachers question whether that's a good topic to spend the time on. One reason for factoring polynomials is to find solutions to equations. But computers and calculators can answer those questions so easily, perhaps this topic become has become archaic. (And besides, most polynomials can't be factored. I remember a lovely post about that, and cannot find it. Can someone point me to it?) Here's a cool picture by Dan Christensen, of all the complex roots of polynomials with integers coefficients of degree 5 or less. (More here.)In the comments to dy/dan's post Grocery Shrink Ray, one person wrote: (comment #13)
Is teaching factoring the best use of time in the classroom? Is it the best topic we can teach students, at this point in their mathematical learning? WCYDWT can�t answer those questions, but it can raise them and get us thinking. In the case of factoring, I think these questions are fair ones. It�s true that I�m asking some pretty leading questions here; as you might be able to guess, my answer would probably be �no�. In my view, factoring is not terribly critical in real life, and this happens to be correlated to the fact that it�s hard to come up with WCYDWT problems (or real-world problems) to motivate learning factoring.I replied:
Factoring may not have real-life applications, but it leads into cool deeper math topics. I see it as very important in algebra.So, what are those cool deeper math topics?
One is Pythagorean triples, which I've posted about before. But here's a mathematical use for factoring I'd never seen before: George Sicherman wanted to find a way to create a pair of dice that would have the same sums as regular dice, with the same probabilities (7 comes up most often, for instance). He used polynomials (and their factors) to find his unusual dice. (Found at Plus Magazine.)
What are some of the other cool mathematical uses of factoring?
_____
*If f is a factor of n, then f . g = n. One of these two numbers must be less than the square root of n, and the other more, unless n = f . f.
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