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
Immersive Linear Algebra
immersivemath immersive linear algebra by J. Ström, K. The world's first linear algebra book with fully interactive figures. Learn More Check us out on Twitter and Facebook Table of Contents Preface A few words about this book. Chapter 1: Introduction How to navigate, notation, and a recap of some math that we think you already know. Chapter 2: Vectors The concept of a vector is introduced, and we learn how to add and subtract vectors, and more. Chapter 3: The Dot Product A powerful tool that takes two vectors and produces a scalar. Chapter 4: The Vector Product In three-dimensional spaces you can produce a vector from two other vectors using this tool. Chapter 5: Gaussian Elimination A way to solve systems of linear equations. Chapter 6: The Matrix Enter the matrix. Chapter 7: Determinants A fundamental property of square matrices. Chapter 8: Rank Discover the behaviour of matrices. Chapter 9: Linear Mappings Learn to harness the power of linearity... Chapter 10: Eigenvalues and Eigenvectors
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
Math ∩ Programming
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).
Differential Equations Explained
$\cos$PLAY You're probably used to equations like $$(t-.5)(t-1)= 0,$$ where 'solving' means finding an unknown number. A differential equation (DE), by contrast, is a fact about the derivative of an unknown function, and 'solving' one means finding a function that fits. To visualize derivatives, we can draw a right triangle whose hypoteneuse is tangent to a function. If the triangle's width is $1$, then its height is the derivative. With that one weird trick, the plots to the right show how the derivative of $\sin(t)$ is $\cos(t)$. That's a pretty basic DE, though. Consider a cart rolling to a stop. The solution is a function $v(t)$ giving velocity at time $t$. It turns out the exponential function, $e^{-kt}$, has the properties $$ \begin{align} \frac{d}{dt}e^{-kt}=-ke^{-kt} && e^{-k\cdot 0}=1. To make the solution more intuitive, here you'll solve the cart's DE manually by picking a series of $\left( t, v \right)$ points. The first cart below obeys the $v(t)$ function you designed.
