Fundamental Theorem of ArithmeticDate: 07/08/97 at 18:44:42 From: bill Subject: Fundamental Theorem of Arithmetic What is the big deal? I have excelled in math for many years, all the while ignorant of what I recently learned to be the "Fundamental Theorem of Arithmetic". Date: 07/13/97 at 02:06:48 From: Doctor Chita Subject: Re: Fundamental Theorem of Arithmetic Dear Bill, It's a philosophical thing, like enjoying a Mozart symphony without knowing anything about composition, or a poem without knowing anything about iambic pentameter, or art without knowing anything about Romanticism. Sure, you can combine numbers without quoting the fundamental theorem of arithmetic, solve equations without quoting the fundamental theorem of algebra, differentiate and integrate functions without quoting the fundamental theorem of calculus. But isn't learning all about the search for truth, beauty, and consistency? The fundamental theorem of arithmetic is at the center of number theory, and simply, but elegantly, says that all composite numbers are products of smaller prime numbers, unique except for order. For example, 12 = 3*2*2, where 2 and 3 are prime numbers. The prime numbers, themselves, are unique, starting with 2. (1 is not considered a prime, since prime numbers have exactly two divisors; themselves and 1. The number 1 has only 1 divisor: itself.) Like atoms in chemistry, prime and composite numbers are the building blocks of arithmetic. For a nice, non-technical discussion of number theory and its intriguing mysteries (for example, is there a largest prime number?), check out Ivars Peterson's book, _The Mathematical Tourist_. For me, the fundamental theorem of arithmetic is one of the few things I can depend on. I know I sleep better at night knowing it. -Doctor Chita, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 07/13/97 at 12:25:40 From: Anonymous Subject: Re: Fundamental Theorem of Arithmetic Dr. Chita, Your answer to my inquiry about the fundamental theorem of arithmetic, although elegant, just didn't do it for me. I understood what the theorem states, and perhaps now you have given me an appreciation for its simplicity, but I am looking for something more. What would you say to a student who asks, "Why did we just spend 20 minutes stating/ writing down/discussing this theorem? So what if every number can be expressed as a product of primes?" I would like to be able to explain/show the student where this theorem fits into the big picture: what it leads to, what important concepts use it as a basis, why we would still be in the 18th century without it. Thanks for your original response. I would also appreciate any further insight which might calm my frustrated engineer brain. Bill Sanford Date: 07/14/97 at 23:11:17 From: Doctor Chita Subject: Re: Fundamental Theorem of Arithmetic Dear Bill, Now that you've centered the original question in a context, I can more easily appreciate your frustration. I know how students have a way of asking questions that can deflate many flights of fancy. I am not sure that students understand the need that mathematicians have for defining their world as one that starts from a few basic premises. For example, in Euclidean plane geometry, there are just a few defined terms (e.g., point, line, plane, space) and a minimum number of postulates (axioms). For example, "A straight line can be drawn from any point to any point." and "All right angles are equal to one another." From this set of geometric building blocks, all else is derived in the form of theorems. In the case of number theory, the subject is of course numbers - the natural numbers, that is: {1, 2, 3, . . . }. Number theorists want to know as much about them as possible. The specialness of the fundamental theorem lies in the fact that there there is an infinite set of prime numbers and that unique combinations of them generate an infinite set of composite numbers. The fact that there is no largest prime suggests that there is no largest composite number. Again, this is a sort of metaphysical observation. As for uses; modern cryptography relies on the fundamental theorem of arithmetic. By combining primes and forming very large composite numbers, code makers can create a key that makes a cipher virtually unbreakable. So, knowing how to construct numbers has a very practial use in our world today. If you will forgive another suggestion, you and your students might want to read a little book called _Cryptology_ by Albrecht Beutelspacher, published by the MAA. And of course, there's the book _Enigma_ about Turing (the father of artificial intelligence) and how he cracked the German's code during WW II. Finally, arguments about the beauty of mathematics may fall on deaf ears when the audience consists of kids who have somehow picked up the idea that mathematics is all about solving arithmetic problems. With such a narrow view, they cannot yet appreciate that mathematics is a way of thinking, made up of many kinds of objects (numbers, equations, figures, etc.), and each conforming to a logical flow of postulates, definitions, and theorems. Encourage your students to read more about math and mathematicians. Have them research Euclid's proof that there is no largest prime. What is a Mersenne prime? What is the current "largest" prime, discovered by a computer just a short while ago? And what is the product of this prime and a smaller prime? Maybe if you could get them to think more about the big ideas of numbers, they will gradually appreciate what makes me sleep more soundly at night. Thanks for getting back. I don't know if I've helped you any further. I hope so. Good luck! -Doctor Chita, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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