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

Inceptionism: Going Deeper into Neural Networks Posted by Alexander Mordvintsev, Software Engineer, Christopher Olah, Software Engineering Intern and Mike Tyka, Software EngineerUpdate - 13/07/2015Images in this blog post are licensed by Google Inc. under a Creative Commons Attribution 4.0 International License. However, images based on places by MIT Computer Science and AI Laboratory require additional permissions from MIT for use.Artificial Neural Networks have spurred remarkable recent progress in image classification and speech recognition. But even though these are very useful tools based on well-known mathematical methods, we actually understand surprisingly little of why certain models work and others don’t. So let’s take a look at some simple techniques for peeking inside these networks. So here’s one surprise: neural networks that were trained to discriminate between different kinds of images have quite a bit of the information needed to generate images too. Why is this important?

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

How to Study and Learn (Part One) All thinking occurs within, and across, disciplines and domains of knowledge and experience, yet few students learn how to think well within those domains. Despite having taken many classes, few are able to think biologically, chemically, geographically, sociologically, anthropologically, historically, artistically, ethically, or philosophically. Students study literature, but do not think in a literary way as a result. They study poetry, but do not think poetically. To study well and learn any subject is to learn how to think with discipline within that subject. To become a skilled learner is to become a self-directed, self-disciplined, self-monitored, and self-corrective thinker, who has given assent to rigorous standards of thought and mindful command of their use. Because we recognize the fact that students generally lack the intellectual skills and discipline to learn independently and deeply, we have designed a Thinker's Guide for Students on How to Study and Learn.

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?

The Art of Learning Summary - Deconstructing Excellence The Art of Learning Summary Josh Waitzkin’s story is a fascinating one, culminating in a book that surpasses any other writing in its insight into how a world champion is made. Everyone in the chess world knew the name Josh Waitzkin by the time he earned the Chess Master designation at the age of twelve, somewhere in the middle of his eight national championship titles. Notoriety in the chess world then morphed into pop culture fame five years later with the movie Searching for Bobby Fischer, which was based on Waitzkin’s life. In seeking an escape from the inner turmoil caused by his child celebrity status, Josh stumbled upon the Tao Te Ching, and was drawn by the Buddhist and Taoist philosophies of inner tranquility. The journey from king of the chess nerds to martial arts legend is astounding in itself, but the real story here is that Josh subsequently accomplished what few have done. Part I: The Foundation Chapter 1: Innocent Moves & Chapter 2: Losing to Win Dr. Part II: My Second Art

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.

geometric double-meanings Jen-chung Chuan Department of Mathematics National Tsing Hua University Hsinchu, Taiwan 300 jcchuan@math.nthu.edu.tw Introduction Why is a geometric figure important? A geometric figure clarifies a theorem, motivates a proof, stimulates the thinking process, sums up a lengthy animation, provides a counterexample to a wild conjecture, or just plainly announces the existence of a significant piece of mathematics. Why is a geometric figure interesting? "Given" and "To Construct" Switched A geometric construction problem has three parts: "Given", "To Construct", and the construction itself. By switching the "Given" and the "To Construct" parts, we see that the picture may have these two interpretations: This tiny example shows that dynamic geometry is at least twice as interesting as the traditional one. The picture carries two messages: Evolute and Involute Involute is the path of a point of a string tautly unwound from the curve. The figure may be interpreted in two ways: Peaucellier Cell

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.

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