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Pascal's Triangle

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. 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). 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: A Formula for Any Entry in The Triangle Notation: "n choose k" can also be written C(n,k), nCk or even nCk.

gametheory101.com Algebra Help Math Sheet Arithmetic Operations The basic arithmetic operations are addition, subtraction, multiplication, and division. These operators follow an order of operation. Addition Addition is the operation of combining two numbers. Subtraction Subtraction is the inverse of addition. Multiplication Multiplication is the product of two numbers and can be considered as a series of repeat addition. Division Division is the method to determine the quotient of two numbers. Arithmetic Properties The main arithmetic properties are Associative, Commutative, and Distributive. Associative The Associative property is related to grouping rules. Commutative The Commutative property is related the order of operations. Distributive The law of distribution allows operations in some cases to be broken down into parts. Arithmetic Operations Examples Exponent Properties Properties of Radicals Properties of Inequalities Properties of Absolute Value Definition of Complex Numbers Properites of Complex Numbers Definition of Logarithms

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

Elementary Calculus: Example 3: Inscribing a Cylinder Into a Sphere Find the shape of the cylinder of maximum volume which can be inscribed in a given sphere. The shape of a right circular cylinder can be described by the ratio of the radius of its base to its height. This ratio for the inscribed cylinder of maximum volume should be a number which does not depend on the radius of thesphere. For example, we should get the same shape whether the radius of the sphere is given in inches or centimeters. Let r be the radius of the given sphere, x the radius of the base of the cylinder, h its height, and V its volume. First, we draw a sketch of the problem in Figure 3.6.4. Figure 3.6.4 From the sketch we can read off the formulas V = πx2h, x2 + (½h)2 = r2, 0 ≤ x ≤ r. r is a constant. Solution One: Eliminating One Variable Sollution Two: Implicit Differentiation

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. Written using the D3 visualisation library. Requires a SVG-capable browser e.g. © Jason Davies | Privacy Policy. Tau Day | No, really, pi is wrong: The Tau Manifesto by Michael Hartl 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.

Geometry Help Looking for some Geometry Help? Our materials here review the basic terms and concepts in geometry and provide further lessons to help you develop your understanding of geometry and its applications to solving problems in real life. Geometry is about the shape and size of things. It is the study of points, lines, angles, shapes, their relationships, and their properties. Videos have been included in almost all the following topics to help reinforce your understanding. Angles Triangles Polygons Circles Circle Theorems Solid Geometry Geometrical Formulas Coordinate Geometry and Graphs Geometric Construction Geometry Transformation Geometry Proofs (Videos) Triangle Medians and Centroids (2D Proof) Area Circumradius Formula Proof Proof that the diagonals of a rhombus are perpendicular bisectors of each other Geometry Practice Questions Free SAT Practice Questions (with Hints & Solutions) - Geometry OML Search We welcome your feedback, comments and questions about this site or page.

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

Learning Calculus: Overcoming Our Artificial Need for Precision Accepting that numbers can do strange, new things is one of the toughest parts of math: There’s numbers between the numbers we count with? (Yes — decimals)There’s a number for nothing at all? (Sure — zero)The number line is two dimensional? (You bet — imaginary numbers) Calculus is a beautiful subject, but challenges some long-held assumptions: Numbers don’t have to be perfectly accurate? Today’s post introduces a new way to think about accuracy and infinitely small numbers. Counting Numbers vs. Not every number is the same. Our first math problems involve counting: we have 5 apples and remove 3, or buy 3 books at $10 each. We later learn about fractions and decimals, and things get weird: What’s the smallest fraction? It gets worse. We’re hit with a realization: we have limited accuracy for quantities that are measured, not counted. What do I mean? That’s cute, but you didn’t answer my question — what number is it? You may pout, open your calculator and say it’s “18.8495…”. We don’t know!

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. 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. Fine print, your comments, more links, Peter Alfeld, PA1UM [16-Aug-1996]

A Gentle Introduction To Learning Calculus I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education. Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). Calculus is similarly enlightening. They are. Unfortunately, calculus can epitomize what’s wrong with math education. It really shouldn’t be this way. Math, art, and ideas I’ve learned something from school: Math isn’t the hard part of math; motivation is. Teachers focused more on publishing/perishing than teachingSelf-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject”Textbooks and curriculums more concerned with profits and test results than insight Poetry is similar. Feisty, are we?

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. 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. The price is $16.00 including shipping.

GEM1518K - Mathematics in Art & Architecture - Project Submission “For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.” - M.C. Escher Contents Page 1 Introduction The Art of Alhambra Our Area of Focus Our Aim Page 2 The Principals behind Tessellations - Translation - Rotation - Reflection - Glide Reflection Page 3 Mathematics in Escher's Art - Translation - Rotation - Glide Reflection - Combination Page 4 Our Original Tessellations - The Catch (Translation) - Under the Sea (Rotation) - The Herd (Glide Reflection) - Dumbo & Butterfly (Combination) Page 5 Possible links with Architecture Conclusion References Mathematics in Escher's Art In this page we attempt to try to use simple mathematical terms to explain a few of Escher's pieces. One of the basic principals behind his tessellations is the use of what we call an "addition and subtraction" method within the grid. Translation One of Escher's first explorations into the tessllations is that of the use of translation. Rotation Glide Reflection Combination

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