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An Interactive Guide To The Fourier Transform

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. What does the Fourier Transform do? 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? So... given a smoothie, how do we find the recipe? Well, imagine you had a few filters lying around: We can reverse-engineer the recipe by filtering each ingredient. Filters must be independent. See The World As Cycles Oh!

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. 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

Probability Theory — A Primer | Math ∩ Programming - FrontMotion Firefox It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables. And so it’s about time we got to the bottom of probability theory. We should make a quick disclaimer before we get into the thick of things: this primer is not meant to connect probability theory to the real world. So let us begin with probability spaces and random variables. Finite Probability Spaces We begin by defining probability as a set with an associated function. Definition: A finite set equipped with a function is a probability space if the function

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. ∑ 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). Since sin 0 = 0 and cos 0 = 1, we can then make a Taylor series for sin x: sin x = ∑ (-1)n x2n+1/ (2n+1)! And, in a similar manner, we can make a Taylor series for cos x: cos x = ∑ (-1)n x2n/ (2n)! Now, comes the fun part. i0 = 1 i1 = i i2 = -1 i3 = -i ik+4 = ik

Don Cross - personal website - math, science, software, electronics - FrontMotion Firefox 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. 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). Intuitively, e^x means: Natural Log is About Time Now what does this inverse or opposite stuff mean?

Lijst van grote getallen - Wikipedia - FrontMotion Firefox De termen zijn volgens de lange schaalverdeling. Als de waarde van de termen volgens de korte schaalverdeling gevonden moet worden, kan vanaf biljoen de macht van 10 berekend worden door de helft van de exponent volgens de lange schaal te nemen en er 3 bij op te tellen. Bijvoorbeeld: 1 biljoen volgens de lange schaal is (zie tabel) 1012, volgens de korte schaal 109 (9 = 12÷2 + 3).

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. 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.

Spirals What is a spiral? A spiral is a curve in the plane or in the space, which runs around a centre in a special way. Different spirals follow. Most of them are produced by formulas. Spirals by Polar Equations top Archimedean Spiral topYou can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. You get formulas analogic to the circle equations. SpiralThe radius r(t) and the angle t are proportional for the simpliest spiral, the spiral of Archimedes. If you connect both spirals by a straight (red) or a bowed curve, a double spiral develops. Equiangular Spiral (Logarithmic Spiral, Bernoulli's Spiral) top More Spirals topIf you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. I chose equations for the different spiral formulas suitable for plotting. Clothoide (Cornu Spiral) top Spirals Made of Arcs topHalf circle spirals Spirals Made of Line Segments top German D.H.O. top

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.