swimmermom3 Posted May 19, 2014 Share Posted May 19, 2014 :blushing: :blushing: :blushing: :blushing: :blushing: :blushing: I have "heard" you all talk about "number theory" for years and it's usually in reference to AoPS, which led me to believe it was something for the mathematically gifted and not mere mortals. So this morning, I am doing my own work in the AoPS Intermediate Algebra book and on p. 81 in the concept box, it says, "Using a little number theory can help factor quadratics." What was being discussed was something I have done intuitively, but I was intrigued by the explanation. What exactly is number theory (in terms slow mortals can understand) and should I be studying it this summer as we move into precalculus and calculus? Are we talking about "playing" with numbers? For anyone that has read Cheaper by the Dozen and remembers the kids performing amazing mental math tricks, were they employing number theory? I am beginning to suspect that the "counting" part of AoPS's Counting and Probability books does not simply mean "1, 2, 3, 4...." :tongue_smilie: You have no idea how long it has taken me to get up the nerve to post this. It's a bit like walking around the airport with toilet paper stuck to your shoe. My ignorance is showing. Quote Link to comment Share on other sites More sharing options...
quark Posted May 19, 2014 Share Posted May 19, 2014 There is a good explanation here. And more of where that came from here (a number theory text for non math majors). :001_smile: I'll wait for the more mathy folk to explain more (because although I have a number theory and counting nut at home, I have no idea how to explain it well! :ph34r: ) ETA: More books and links if you are interested (apart from AoPS) Number Theory: A Lively Introduction with Proofs, Applications and Stories The first two volumes by James Tanton (and others too I'm sure but I'm mentioning the first two because they are the only two I've peeked into) Some free math circle materials (scroll down to the pdf links) I have some links in my siggy as well. Quote Link to comment Share on other sites More sharing options...
JanetC Posted May 19, 2014 Share Posted May 19, 2014 Number theory is puzzle solving using mathematical relationships and either integers or natural numbers. For quadratic equation factory, it's being able to answer the question, "are there two numbers with a sum of X and product of Y?" I had really hoped to do the AOPS number theory book with my kids, but I don't think we'll ever find the time now. They did, however, get a taste of it in elementary ages with CSMP and activities like this one: http://ceure.buffalostate.edu/~csmp/CSMPProgram/Storybooks/StrangeCountry.pdf Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted May 19, 2014 Share Posted May 19, 2014 Don't be afraid to ask good questions, Lisa!! There is nothing magical or only-for-the-gifted in those parts of mathematics. American schools have long over-emphasized algebra & geometry in their race to calculus, and have neglected poor old discrete math, which includes the topics of number theory and counting & probability. Richard R has a great little essay on AoPS about the importance of discrete math in the world today. At its simplest, number theory is the study of the integers: how they factor, prime & composite numbers, modular arithmetic (think of clock arithmetic: 2:00 pm is the same as14:00 military time) working in different bases like base 2, etc. There are lots of pretty theorems that are surprisingly deep. Here's an example of one called Fermat's Theorem: Pick a prime number p, and take any other number a (as long as "a" isn't a multiple of p). Then a^(p-1) is always 1 more than a multiple of p. This sort of result is beautiful and easy enough to understand, but deep & difficult to prove. By the way, number theory is a great field if you want to learn how to write good proofs. You can learn why the square root of 2 cannot be a rational number, or how to find all possible integers solutions for the Pythagorean Theorem a^2 + b^2 = c^2. We all know that 3^2 + 4^2 =5^2, but are there others? (no fractions allowed!) This is included in the study of Diophantine equations, a fancy name for equations that require integer solutions only. Number theory has applications in cryptography (internet encryption is important to anyone buying stuff online, and relies on number theory), computing (all those 0's and 1's that computers understand are in base 2), and fun things like magic squares. In counting and probability, you can learn to solve problems like: If there are 25 teams and they all play each other two times, how many games need to be scheduled? If I take the SAT where wrong answers have 1/4 pt deducted, should I guess an answer if I can eliminate one of the choices? How many ways can I deal a five-card run from a standard card deck? What is the probability of getting such a hand in a random deal? As for practical applications of C&P, I just finished an edX MOOC this week on probability, and we had a professional gambler in our class. Guess he felt that it was worthwhile! A favorite little book of mine on number theory written for the general public is From Zero to Infinity by Constance Reid. That's the one I started my kids on when they expressed an interest. Ds loved Probability without Tears when he was young (he spent $1 on it on the Barnes & Noble bargain rack & read it in bed every night for a month!) Do you have an Alcumus account on AoPS? You can play around in the number theory and counting & prob sections there, too, to get an idea if you'd enjoy either topic. Quote Link to comment Share on other sites More sharing options...
Julie of KY Posted May 19, 2014 Share Posted May 19, 2014 :blushing: :blushing: :blushing: :blushing: :blushing: :blushing: What exactly is number theory (in terms slow mortals can understand) and should I be studying it this summer as we move into precalculus and calculus? Are we talking about "playing" with numbers? For anyone that has read Cheaper by the Dozen and remembers the kids performing amazing mental math tricks, were they employing number theory? I think number theory was explained pretty well by previous posters. No, you don't need number theory to be able to do precalculus or calculus, and I don't think knowing more number theory would make it particularly easier. I like Kathy's suggestion about playing around with number theory on Alcumus to see if you like it. For the tricks preformed by the kids in Cheaper by the Dozen, I think you would be better looking at Secrets of Mental Math by Arthur Benjamin http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?ie=UTF8&qid=1400533474&sr=8-1&keywords=mental+math+arthur+benjamin Personally, I like counting and probability better than number theory, but I'm coming around as my oldest likes to turn so many algebra problems into number theory (usually using modular arithmetic) and then he has to explain to me how he got the answer so quickly and easily. Quote Link to comment Share on other sites More sharing options...
swimmermom3 Posted May 20, 2014 Author Share Posted May 20, 2014 There is a good explanation here. And more of where that came from here (a number theory text for non math majors). :001_smile: I'll wait for the more mathy folk to explain more (because although I have a number theory and counting nut at home, I have no idea how to explain it well! :ph34r: ) ETA: More books and links if you are interested (apart from AoPS) Number Theory: A Lively Introduction with Proofs, Applications and Stories The first two volumes by James Tanton (and others too I'm sure but I'm mentioning the first two because they are the only two I've peeked into) Some free math circle materials (scroll down to the pdf links) I have some links in my siggy as well. Quark, thank you. These are really helpful resources and the first two links give me a place to start. I don't recognize 1 (modulo 4), 3 (modulo 4), triangular numbers (which make sense when I look at squares), and Fibonacci I only know from here and something amazing to do with sunflowers. Do you have Number Theory: A Lively Introduction with Proofs, Applications and Stories? The reviews are glowing and it sounds like something I might really enjoy. That price tag is a bugger, though. Number theory is puzzle solving using mathematical relationships and either integers or natural numbers. For quadratic equation factory, it's being able to answer the question, "are there two numbers with a sum of X and product of Y?" I had really hoped to do the AOPS number theory book with my kids, but I don't think we'll ever find the time now. They did, however, get a taste of it in elementary ages with CSMP and activities like this one: http://ceure.buffalostate.edu/~csmp/CSMPProgram/Storybooks/StrangeCountry.pdf I think we are running out of time for the number theory book too, but I would like to learn more over the summer and see if I can incorporate it into our regular lessons. Is the material in your link representative of what kids involved with math competitions would be working on? I love puzzles, but always think of them in terms of language like logic problems. Monster sudoku puzzles are really neither words nor math and I find them comforting when I need to slow my brain down and/or distract it. Quote Link to comment Share on other sites More sharing options...
quark Posted May 20, 2014 Share Posted May 20, 2014 Quark, thank you. These are really helpful resources and the first two links give me a place to start. I don't recognize 1 (modulo 4), 3 (modulo 4), triangular numbers (which make sense when I look at squares), and Fibonacci I only know from here and something amazing to do with sunflowers. Do you have Number Theory: A Lively Introduction with Proofs, Applications and Stories? The reviews are glowing and it sounds like something I might really enjoy. That price tag is a bugger, though. You are very welcome! Yes, I have that textbook. :tongue_smilie: Bought a used copy at the height of his number theory craze which has now given way to something else. The pdf links I sent are free though and should help for a while at least! Good luck! Quote Link to comment Share on other sites More sharing options...
EndOfOrdinary Posted May 20, 2014 Share Posted May 20, 2014 Do not feel badly about the questions. I had to ask about verb tenses at homeschool group a year or two ago. People looked at me like I was totally ridiculous. I can go all out about number theory for you, but please do not ask me to use a semicolon correctly! We all have our strengths. No need to feel embarrassed or ashamed. Quote Link to comment Share on other sites More sharing options...
lewelma Posted May 20, 2014 Share Posted May 20, 2014 A lot of my son's number theory questions just look like an algebra question, until you realize that you can only have integer solutions. And somehow that makes it really hard. :001_smile: That is about as technical as I get. Ruth in NZ Quote Link to comment Share on other sites More sharing options...
Colleen in NS Posted May 20, 2014 Share Posted May 20, 2014 You have no idea how long it has taken me to get up the nerve to post this. It's a bit like walking around the airport with toilet paper stuck to your shoe. My ignorance is showing. You have no idea how many times I have felt ridiculous posting "ignorant" questions here! :D But it's these kinds of questions that catch my eye because they help me learn, too. Thanks for being brave! Quote Link to comment Share on other sites More sharing options...
Recommended Posts
Join the conversation
You can post now and register later. If you have an account, sign in now to post with your account.