Calculus. Free e-books and e-lectures on Algorithms, Math, and Cryptography. Mathematics for Computer Science | Electrical Engineering and Computer Science. Math21b, Fall 2003, Linear Algebra and Differential Equations. Calculus - Exampleproblems. Math Help - SolveMyMath. The momentum representation. How to Calculate a Square Root by Hand: 21 steps (with pictures) Edit Article CalculatorUsing Prime FactorizationFinding Square Roots Manually Edited by NatK, Maluniu, Luís Miguel Armendáriz, Webster and 44 others In the days before calculators, students and professors alike had to calculate square roots by hand.
Several different methods have evolved for tackling this daunting process, some giving a rough approximation, others giving an exact value. To learn how to find a number's square root using only simple operations, see Step 1 below to get started. Ad Steps Method 1 of 2: Using Prime Factorization 1Divide your number into perfect square factors. Method 2 of 2: Finding Square Roots Manually Using a Long Division Algorithm 1Separate your number's digits into pairs. 9To continue to calculate digits, drop a pair of zeros on the left, and repeat steps 4, 5 and 6.
Understanding the Process 1Consider the number you are calculating the square root of as the area S of a square. 11To calculate the next digit C, repeat the process. Tips. Free Maths Video Lecture courses. Touch Mathematics | Trigonometry.
Mathematics reference: Rules for differentiation. An Interactive Guide To The Fourier Transform. The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations: Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor: What does the Fourier Transform do? Given a smoothie, it finds the recipe.How? Run the smoothie through filters to extract each ingredient.Why? Here's the "math English" version of the above: The Fourier Transform takes a time-based pattern, measures every possible cycle, and returns the overall "cycle recipe" (the strength, offset, & rotation speed for every cycle that was found).
Time for the equations? If all goes well, we'll have an aha! This isn't a force-march through the equations, it's the casual stroll I wish I had. From Smoothie to Recipe A math transformation is a change of perspective. The Fourier Transform changes our perspective from consumer to producer, turning What did I see? In other words: given a smoothie, let's find the recipe. Why? Pauls Online Math Notes. Mathematics Applied to Physics/Engineering. Why Does e^(pi i) + 1 = 0? This page is just a collection of a couple of answers on the LiveJournal Mathematics Community in a thread about eπi + 1 = 0.
Soon, I will whip them into a more coherent form. In collegiate calculus, you probably learned about something called Taylor series. You can use Taylor series to make polynomial approximations of infinitely-differentiable functions. If you take the Taylor series out infinitely, you actually have the function. The general form for a Taylor series is this: ∑ f(n)(x0) * (x - x0)n / n! Where the sum goes from n=0 to n=infinity and f(n)(x0) means the n-th derivative of f(x) evaluated at x0. Also, note that f(0)(x) is just f(x). Fortunately, we know all of the derivatives of ex, sin x, and cos x. All of the derivatives of ex are equal to ex. Ex = ∑ xn / n! The derivatives of sin x are a bit more tricky. f(0)(x) = sin x f(1)(x) = cos x f(2)(x) = -sin x f(3)(x) = -cos x f(4)(x) = sin x And, from there the pattern repeats... f(k+4)(x) = f(k). sin x = ∑ (-1)n x2n+1/ (2n+1)!
i0 = 1. Graduate Program in Mathematics, Study guides for Comp Exams, University of Illinois at Urbana-Champaign. Demystifying the Natural Logarithm (ln) After understanding the exponential function, our next target is the natural logarithm. Given how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of e^x, a strange enough exponent already. But there’s a fresh, intuitive explanation: The natural log gives you the time needed to reach a certain level of growth. Suppose you have an investment in gummy bears (who doesn’t?) With an interest rate of 100% per year, growing continuously.
If you want 10x growth, assuming continuous compounding, you’d wait only ln(10) or 2.302 years. E and the Natural Log are twins: e^x is the amount of continuous growth after a certain amount of time.Natural Log (ln) is the amount of time needed to reach a certain level of continuous growth Not too bad, right? E is About Growth The number e is about continuous growth.
We can take any combination of rate and time (50% for 4 years) and convert the rate to 100% for convenience (giving us 100% for 2 years). Main Page - Exampleproblems. Untitled. Two by Two. Essentially the technique converts the first factor into binary, multiplies each of its constituents by the second factor, and sums the results. Imagine that each line is associated with a power of 2: the first line with 20, the second with 21,and so on. The business in the first column, halving the first factor successively and crossing out those lines with even numbers, effectively reduces the first factor to its binary constituents — here, the lines that remain are those associated with 20, 25, and 26, and, sure enough, 20 + 25 + 26 = 97. Now we need to multiply each of those constituents by the second factor, 23. In other words, we want to find: That’s what’s accomplished in the second column.
So if we add those values, we’ll get the product of the original two numbers, which is what we sought: 23 + 736 + 1472 = 2231. Here’s essentially what we’ve done, from the top: It works with any pair of numbers. The Thirty Greatest Mathematicians. Click for a discussion of certain omissions. Please send me e-mail if you believe there's a major flaw in my rankings (or an error in any of the biographies). Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different. I'm sure I've overlooked great mathematicians who obviously belong on this list. Please e-mail and tell me! Following are the top mathematicians in chronological (birth-year) order. Earliest mathematicians Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic.
Early Vedic mathematicians The greatest mathematics before the Golden Age of Greece was in India's early Vedic (Hindu) civilization. Top Thales of Miletus (ca 624 - 546 BC) Greek domain Apastambha (ca 630-560 BC) India Pythagoras of Samos (ca 578-505 BC) Greek domain Panini (of Shalatula) (ca 520-460 BC) Gandhara (India) Tiberius(?) Geocentrism vs. Color Theory 101. Three-Cornered Things | Weekly wanderings through higher math—from not so high a vantage point. Calculus. Intuitive Understanding Of Euler’s Formula. Euler's identity seems baffling: It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive? Not according to 1800s mathematician Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.
Argh, this attitude makes my blood boil! Euler's formula describes two equivalent ways to move in a circle. That's it? Starting at the number 1, see multiplication as a transformation that changes the number (1 * e^(i*pi))Regular exponential growth continuously increases 1 by some rate; imaginary exponential growth continuously rotates a numberGrowing for "pi" units of time means going pi radians around a circleTherefore, e^(i*pi) means starting at 1 and rotating pi (halfway around a circle) to get to -1 That's the high-level view -- let's dive into the details. The statement Hrm. Geometry. The word geometry is Greek for geos - meaning earth and metron - meaning measure. Geometry was extremely important to ancient societies and was used for surveying, astronomy, navigation, and building. Geometry, as we know it is actually known as Euclidean geometry which was written well over 2000 years ago in Ancient Greece by Euclid, Pythagoras, Thales, Plato and Aristotle just to mention a few.
The most fascinating and accurate geometry text was written by Euclid, and was called Elements. Euclid's text has been used for over 2000 years! Geometry is the study of angles and triangles, perimeter, area and volume. It differs from algebra in that one develops a logical structure where mathematical relationships are proved and applied. Terms (Undefined) Point Points show position.
Terms (Defined) Line Segment A line segment is a straight line segment which is part of the straight line between two points. Calculus Derivatives Cheat Sheet. What are the best rules, formulas and tricks in math? - math formulas rules. 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. 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?
Numbers aren’t all the same size (i.e. 1 times some number)? 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? We don’t know! 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.
Tau Day | No, really, pi is wrong: The Tau Manifesto by Michael Hartl. 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. 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. If more than two numbers are added this can be called summing. 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 Logarithms.
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).
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 Yes, it works!
Calculus and Differential Equations. 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). It all fits together. 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. Reflections on Relativity. Mathematics Department - Mathematics Web Sites. Maths Online Courses. Ask Dr. Math - College Level Archive.
English names of the first 10000 powers of 10 - American System without dashes. Mathematics Department : Mathematics Courses. The Math Guy. Visualizing Basic Algebra. How to Become a Pure Mathematician (or Statistician) Free Mathematics Books Download Free Mathematics Ebooks Online Mathematics tutorials.
Analysis index. Undergraduate contest problems and solutions. Abstract Algebra - Free Harvard Courses. Foundations of Mathematics. S Introduction to Complex Systems. Dave's short course in trigonometry. Piety within Progression. Tutorials related to Kinematic Mechanisms.