3 Dimensional Fractals & Mandelbulb. Why is a Minute Counted as 60 Seconds and not 100 Seconds. How to construct (draw) the incircle of a triangle with compass and straightedge or ruler. Irrationality of the square root of 2. Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah It was one of the most surprising discoveries of the Pythagorean School of Greek mathematicians that there are irrational numbers. According to Courant and Robbins in "What is Mathematics": This revelation was a scientific event of the highest importance. Quite possibly it marked the origin of what we consider the specifically Greek contribution to rigorous procedure in mathematics. Certainly it has profoundly affected mathematics and philosophy from the time of the Greeks to the present day. Specifically, the Greeks discovered that the diagonal of a square whose sides are 1 unit long has a diagonal whose length cannot be rational.
By the Pythagorean Theorem, the length of the diagonal equals the square root of 2. The following proof is a classic example of a proof by contradiction: We want to show that A is true, so we assume it's not, and come to contradiction. . [16-Aug-1996] What is the next number in the sequence 1, 1, 1, 1? If we let be then we have Therefore the next number might as well be 42. The same trick can be pulled off with any given integer sequence. In fact, it can be any number you like. This is not particularly hard to prove. If we have been given a sequence of numbers and we wish to find the value of , then we are effectively being asked to find the the function () such that and then to evaluate (5). Through them. Which will go through of all of them. And we want to be 42. We have equations with unknowns (being the ).
This is a very pedantic point to make, but you have to be a pedant to be a mathematician, because making unjustified assumptions while answering questions can be catastrophic. Equations - EquationSheet.com. Animated Bézier Curves. Play with the control points to modify the curves! These animations illustrate how a parametric Bézier curve is constructed. The parameter t ranges from 0 to 1. In the simplest case, a first-order Bézier curve, the curve is a straight line between the control points.
For a second-order or quadratic Bézier curve, first we find two intermediate points that are t along the lines between the three control points. Then we perform the same interpolation step again and find another point that is t along the line between those two intermediate points. Written using the D3 visualisation library. Requires a SVG-capable browser e.g. . © Jason Davies | Privacy Policy. Nerd Paradise : Divisibility Rules for Arbitrary Divisors.
It's rather obvious when a number is divisible by 2 or 5, and some of you probably know how to tell if a number is divisible by 3, but it is possible to figure out the division 'rule' for any number. Here are the rules for 2 through 11... The last digit is divisible by 2. The sum of all the digits in the number is divisible by 3. The last 2 digits are divisible by 4. The last digit is 5 or 0. The number is both divisible by 2 and divisible by 3. Cut the number into 2 parts: the last digit and everything else before that. The last 3 digits are divisible by 8 The sum of all the digits in the number is divisible by 9. The last digit is a 0. Break the number into two parts (like you did for the division by 7 rule). Also there is a quick way for determining divisibility by 11 for 3-digit numbers: If the inner digit is larger than the two outer digits, then it is divisible by 11 if the inner digit is the sum of the two outer digits.
Rules for all divisors ending in 1... User Comments: 9 Dividing By 12. Math Fun Facts! Pascal's Triangle. Patterns Within the Triangle Using Pascal's Triangle Heads and Tails Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you the probability of any combination. For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three that will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern "1,3,3,1" in Pascal's Triangle. Example: What is the probability of getting exactly two heads with 4 coin tosses?
There are 1+4+6+4+1 = 16 (or 24=16) possible results, and 6 of them give exactly two heads. Combinations The triangle also shows you how many Combinations of objects are possible. Example: You have 16 pool balls. Answer: go down to the start of row 16 (the top row is 0), and then along 3 places (the first place is 0) and the value there is your answer, 560.
Here is an extract at row 16: Gametheory101.com. Big Numbers. Two-dimensional Geometry and the Golden section. On this page we meet some of the marvellous flat (that is, two dimensional) geometry facts related to the golden section number Phi. A following page turns our attention to the solid world of 3 dimensions. Contents of this Page The icon means there is a Things to do investigation at the end of the section. 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More.. Let's start by showing how to construct the golden section points on any line: first a line phi (0·618..) times as long as the original and then a line Phi (1·618..) times as long. Constructing the internal golden section points: phi If we have a line with end-points A and B, how can we find the point which divides it at the golden section point?
(In fact we can do it with just the compasses, but how to do it without the set-square is left as an exercise for you.) We want to find a point G between A and B so that AG:AB = phi (0·61803...) by which we mean that G is phi of the way along the line. Using only circles. Why couldn’t I have been shown this in maths class? Stuff You MUST Know Cold for AP Calc. Plus.maths.org. History Topics Index. l8zVz.jpg (1536×2048) Folding Paper in Half Twelve Times. Folding Paper in Half 12 Times: The story of an impossible challenge solved at the Historical Society office Alice laughed: "There's no use trying," she said; "one can't believe impossible things. " "I daresay you haven't had much practice," said the Queen. Through the Looking Glass by L. Carroll The long standing challenge was that a single piece of paper, no matter the size, cannot be folded in half more than 7 or 8 times.
The most significant part of Britney's work is actually not the geometric progression of a folding sequence but rather the detailed analysis to find why geometric sequences have practical limits that prevent them from expanding. Her book provides the size of paper needed to fold paper and gold 16 times using different folding techniques. Britney Gallivan has solved the Paper Folding Problem. In April of 2005 Britney's accomplishment was mentioned on the prime time CBS television show Numb3rs. The task was commonly known to be impossible. A few references: Day of the Week. Probabilities in the Game of Monopoly® Probabilities in the Game of Monopoly® Table of Contents I recently saw an article in Scientific American (the April 1996 issue with additional information in the August 1996 and April 1997 issues) that discussed the probabilities of landing on the various squares in the game of Monopoly®.
They used a simplified model of the game without considering the effects of the Chance and Community Chest cards or of the various ways of being sent to jail. I was intrigued enough with this problem that I started working on trying to find the probabilities for landing on the different squares with all of the rules taken into account. I ran into some interesting problems but finally came up with the right answers, which you will find here along with some other useful derived data. Incidentally, I'm not much of a Monopoly® player myself, but I've always enjoyed interesting problems involving probability and statistics, of which this was one. Average Income per Roll from other Squares Back to my homepage. Cryptograms/stats.htm. Weierstrass functions. Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere". Here is an example of one: It is not hard to show that this series converges for all x.
In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr.
How To Analyze Data Using the Average. The average is a simple term with several meanings. The type of average to use depends on whether you’re adding, multiplying, grouping or dividing work among the items in your set. Quick quiz: You drove to work at 30 mph, and drove back at 60 mph. What was your average speed? Hint: It’s not 45 mph, and it doesn’t matter how far your commute is. But what does it mean? Let’s step back a bit: what is the “average” all about? To most of us, it’s “the number in the middle” or a number that is “balanced”. The average is the value that can replace every existing item, and have the same result. One goal of the average is to understand a data set by getting a “representative” sample. The Arithmetic Mean The arithmetic mean is the most common type of average: Let’s say you weigh 150 lbs, and are in an elevator with a 100lb kid and 350lb walrus.
The real question is “If you replaced this merry group with 3 identical people and want the same load in the elevator, what should each clone weigh?” Pros: Cons: Divisibility Rules (Tests) Tanya Khovanova’s Math Blog » Blog Archive » Divisibility by 7 is a Walk on a Graph, by David Wilson. My guest blogger is David Wilson, a fellow fan of sequences. It is a nice exercise to understand how this graph works. When you do, you will discover that you can use this graph to calculate the remainders of numbers modulo 7. Back to David Wilson: I have attached a picture of a graph. Write down a number n. For example, if n = 325, follow 3 black arrows, then 1 white arrow, then 2 black arrows, then 1 white arrow, and finally 5 black arrows.
If you end up back at the white node, n is divisible by 7. Nothing earth-shattering, but I was pleased that the graph was planar. How to Read Mathematics. This article is part of my new book Rediscovering Mathematics, now in paperback! How to Read Mathematics by Shai Simonson and Fernando Gouvea Mathematics is “a language that can neither be read nor understood without initiation.” 1 A reading protocol is a set of strategies that a reader must use in order to benefit fully from reading the text.
Poetry calls for a different set of strategies than fiction, and fiction a different set than non-fiction. Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics. When we read a novel we become absorbed in the plot and characters. Novelists frequently describe characters by involving them in well-chosen anecdotes, rather than by describing them by well-chosen adjectives.
Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. What are the common mistakes people make in trying to read mathematics? Don’t Read Too Fast. Number Names. Here are large number names and their scientific notation equivalents. 1,000,000 = one million = 10^6 1,000,000,000 = one billion = 10^9 1,000,000,000,000 = one trillion = 10^12 1,000,000,000,000,000 = one quadrillion = 10^15 1,000,000,000,000,000,000 = one quintillion = 10^18 1,000,000,000,000,000,000,000 = one sextillion = 10^21 1,000,000,000,000,000,000,000,000 = one septillion = 10^24 1,000,000,000,000,000,000,000,000,000 = one octillion = 10^27 1,000,000,000,000,000,000,000,000,000,000 = one nonillion = 10^30 1,000,000,000,000,000,000,000,000,000,000,000 = one decillion = 10^33 1,000,000,000,000,000,000,000,000,000,000,000,000 = one undecillion = 10^36 1,000,000,000,000,000,000,000,000,000,000,000,000,000 = one duodecillion = 10^39 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000 = one tredecillion = 10^42 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 = one quattuordecillion = 10^45 That would mean one googol is A while back I received an e-mail from Chris Meador.
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". What's Special About This Number? YTMND - 3.141592653589793... The MegaPenny Project | Index Page. Fibonacci Number. The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation with .
As a result of the definition (1), it is conventional to define The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials Fibonacci numbers are implemented in the Wolfram Language as Fibonacci[n]. The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation). The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. The plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). Ends in zeros. And . As. RF Cafe - Mathematical References.
Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. Collected Algorithms of the ACM. The On-Line Encyclopedia of Integer Sequences™ (OEIS™) The 0! Story. The Free-Form Linguistics Revolution in Mathematica. 50 Best Mathematics Blogs. David Eppstein. Math Gems. The Thirty Greatest Mathematicians.