Rader's NUMBERNUT.COM Chapter 5 : Repeating Decimals 1/81 = 0.012345679 ... (from 0 to 7 (one letter), last is 9. length=9) 1/891 = 0.001122334455667789 ... (from 00 to 77 (two letters), last is 89. length=18) 1/8991 = 0.000111222333444555666777889 ... (from 000 to 777 (three letters), last is 889. length=27) 1/89991 = 0.000011112222333344445555666677778889 ... (from 0000 to 7777 (four letters), last is 8889. length=36) ... 1/81 = 0.012345679 ... Power of n Power of 2 appears. At a first glance, the pattern corrupts at 65, but so the pattern continues eternaly. [general formula] For power of "a" in k-digits, a/(10^k-a) In case of 11, This is, so the expansion becomes Fibonacci number. Multiple Using gereration function. s = kx + 2kx^2 + 3kx^3 + 4kx^4 + ... = k(x + 2x^2 + 3x^3 + 4x^4 ...) So, start from s = x + 2x^2 + 3x^3 + 4x^4 ... sx = x^2 + 2x^3 + 3x^4 + 4x^5 ... ∴ (1-x)s = x + x^2 + x^3 + ... The right side is a geometric series which first term is x and ratio is x, so, = x/(1-x) ∴ s = x/(1-x)^2 Let x replace by 1/x then, s = x/(x-1)^2 For example,
6174 (number) 6174 is known as Kaprekar's constant[1][2][3] after the Indian mathematician D. R. Kaprekar. This number is notable for the following property: Take any four-digit number, using at least two different digits. 9990 – 0999 = 8991 (rather than 999 – 999 = 0) 9831 reaches 6174 after 7 iterations: 8820 – 0288 = 8532 (rather than 882 – 288 = 594) 8774, 8477, 8747, 7748, 7487, 7847, 7784, 4877, 4787, and 4778 reach 6174 after 4 iterations: Note that in each iteration of Kaprekar's routine, the two numbers being subtracted one from the other have the same digit sum and hence the same remainder modulo 9. Sequence of Kaprekar transformations ending in 6174 Sequence of three digit Kaprekar transformations ending in 495 Kaprekar number Bowley, Rover. "6174 is Kaprekar's Constant".
New Pattern Found in Prime Numbers (PhysOrg.com) -- Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But on the other hand, the global distribution of primes reveals a remarkably smooth regularity. In a recent study, Bartolo Luque and Lucas Lacasa of the Universidad Politécnica de Madrid in Spain have discovered a new pattern in primes that has surprisingly gone unnoticed until now. “Mathematicians have studied prime numbers for centuries,” Lacasa told PhysOrg.com. Benford’s law (BL), named after physicist Frank Benford in 1938, describes the distribution of the leading digits of the numbers in a wide variety of data sets and mathematical sequences. Since the late ‘70s, researchers have known that prime numbers themselves, when taken in very large data sets, are not distributed according to Benford’s law. The set of all primes - like the set of all integers - is infinite.
Mathematical Atlas: A gateway to Mathematics Welcome! This is a collection of short articles designed to provide an introduction to the areas of modern mathematics and pointers to further information, as well as answers to some common (or not!) questions. The material is arranged in a hierarchy of disciplines, each with its own index page ("blue pages"). For resources useful in all areas of mathematics try 00: General Mathematics. There is a backlog of articles awaiting editing before they are referenced in the blue pages, but you are welcome to snoop around VIRUS WARNING: The Mathematical Atlas receives but does not send mail using the math-atlas.org domain name. Please bookmark any pages at this site with the URL This URL forces frames; for a frame-free version use
15-Pound, Retro-Tech Flywheel Helps You Pedal Your Bike To Tomorrow The technology of a flywheel is simple and old: Use energy to spin up a wheel very quickly. Later, you can take that spinning energy and use it for something else. But you normally think of flywheels as enormous steel monstrosities spinning in factories. But 22-year-old inventor Maxwell von Stein's new bike employs a small flywheel to boost his speed and take a load off his legs while pedaling: When braking, the biker simply shifts gears and allows the energy to transfer from the back wheel to the flywheel (instead of transferring uselessly to the brake pads). The wheel weighs 15 pounds, so you certainly need the extra help it provides to keep moving. / 10 Pin / 29 Plus / 50 Tweet / 602 Like / 13 Share
- StumbleUpon Hammack Home This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition), and the Amazon reviews. The second edition is identical to the first edition, except some mistakes have been corrected, new exercises have been added, and Chapter 13 has been extended. (The Cantor-Bernstein-Schröeder theorem has been added.) The two editions can be used interchangeably, except for the last few pages of Chapter 13. Order a copy from Amazon or Barnes & Noble for $13.75 or download a pdf for free here. Part I: Fundamentals Part II: How to Prove Conditional Statements Part III: More on Proof Part IV: Relations, Functions and Cardinality Thanks to readers around the world who wrote to report mistakes and typos! Instructors: Click here for my page for VCU's MATH 300, a course based on this book.
How Products Are Made A Non-Mathematical Introduction to Using Neural Networks The goal of this article is to help you understand what a neural network is, and how it is used. Most people, even non-programmers, have heard of neural networks. There are many science fiction overtones associated with them. And like many things, sci-fi writers have created a vast, but somewhat inaccurate, public idea of what a neural network is. Most laypeople think of neural networks as a sort of artificial brain. Neural networks would be used to power robots or carry on intelligent conversations with human beings. Neural networks are one small part of AI. The human brain really should be called a biological neural network (BNN). There are some basic similarities between biological neural networks and artificial neural networks. Like I said, neural networks are designed to accomplish one small task. The task that neural networks accomplish very well is pattern recognition. Figure 1: A Typical Neural Network Neural Network Structure Programming hash tables use keys and values.
Roman Numerals The Romans were active in trade and commerce, and from the time of learning to write they needed a way to indicate numbers. The system they developed lasted many centuries, and still sees some specialized use today. Roman numerals traditionally indicate the order of rulers or ships who share the same name (i.e. Queen Elizabeth II). They are also sometimes still used in the publishing industry for copyright dates, and on cornerstones and gravestones when the owner of a building or the family of the deceased wishes to create an impression of classical dignity. The Roman numbering system also lives on in our languages, which still use Latin word roots to express numerical ideas.
An Introduction to Neural Networks Prof. Leslie Smith Centre for Cognitive and Computational Neuroscience Department of Computing and Mathematics University of Stirling. lss@cs.stir.ac.uk last major update: 25 October 1996: minor update 22 April 1998 and 12 Sept 2001: links updated (they were out of date) 12 Sept 2001; fix to math font (thanks Sietse Brouwer) 2 April 2003 This document is a roughly HTML-ised version of a talk given at the NSYN meeting in Edinburgh, Scotland, on 28 February 1996, then updated a few times in response to comments received. Please email me comments, but remember that this was originally just the slides from an introductory talk! Why would anyone want a `new' sort of computer? What is a neural network? Some algorithms and architectures. Where have they been applied? What new applications are likely? Some useful sources of information. Some comments added Sept 2001 NEW: questions and answers arising from this tutorial Why would anyone want a `new' sort of computer? Good at Not so good at Fast arithmetic
Brain size and evolution - complexity, "behavioral complexity", and brain size From Serendip Organisms have indeed gotten more "complex" over evolutionary time, at least on a broad scale Organisms differ in "behavioral complexity" Organisms differ in brain sizeThere is some relation between "behavioral complexity" and brain size, but humans do not have the largest brains. There is a better relation between "behavioral complexity" and brain size in relation to body size. from Harry J. I'm still more interested in whether there is life on Mars?.