**Root Into Europe** is an ITV comedy-drama based on the character from
William Donaldson's book The Henry Root Letters. Henry Root, a fish
dealer who disapproves of the impending European Union, declares himself
England's 'European regulator' in a letter to the British Prime
Minister, then John Major. He takes his wife Muriel (Pat Heywood) on a
tour of Europe to represent English values to mainland Europe. His
adventures are captured on a camcorder by his wife to be sent to the BBC
upon his return for a future documentary, which one expects will never
be made. The episodes bring him to France, Spain, Italy, Germany,
Belgium and The Netherlands.

Type: Scripted

Languages: English

Status: Ended

Runtime: 60 minutes

Premier: 1992-05-17

## Root Into Europe - Square root - Netflix

In mathematics, a square root of a number a is a number y such that y2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y⋅y) is a. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 • 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two square roots: √a, which is positive, and −√a, which is negative. Together, these two roots are denoted as ± √a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation “the square root” is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2. Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of “squaring” of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

## Root Into Europe - Computation - Netflix

the computation of the square root of a positive number can be reduced to that of a number in the range [1,4). This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used. The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers. Another useful method for calculating the square root is the shifting nth root algorithm, applied for n = 2. The name of the square root function varies from programming language to programming language, with sqrt (often pronounced “squirt” ) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python.

as it allows one to adjust the estimate x by some amount c and measure the square of the adjustment in terms of the original estimate and its square. Furthermore, (x + c)2 ≈ x2 + 2xc when c is close to 0, because the tangent line to the graph of x2 + 2xc + c2 at c=0, as a function of c alone, is y = 2xc + x2. Thus, small adjustments to x can be planned out by setting 2xc to a, or c=a/(2x). The most common iterative method of square root calculation by hand is known as the “Babylonian method” or “Heron's method” after the first-century Greek philosopher Heron of Alexandria, who first described it. The method uses the same iterative scheme as the Newton–Raphson method yields when applied to the function y = f(x) = x2 − a, using the fact that its slope at any point is dy/dx = f'(x) = 2x, but predates it by many centuries. The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if x is an overestimate to the square root of a nonnegative real number a then a/x will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted below), and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find x: Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision. Replace x by the average (x + a/x) / 2 between x and a/x. Repeat from step 2, using this average as the new value of x. That is, if an arbitrary guess for √a is x0, and xn + 1 = (xn + a/xn) / 2, then each xn is an approximation of √a which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If a = 0, the convergence is only linear. Using the identity

where ln and log10 are the natural and base-10 logarithms. By trial-and-error, one can square an estimate for √a and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it's prudent to use the identity

a = 2 − n 4 n a , {\displaystyle {\sqrt {a}}=2^{-n}{\sqrt {4^{n}a}},}

## Root Into Europe - References - Netflix

- http://cygnus-x.net/geekstuff/programs/sqrt/21/100000.txt
- http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt2.5mil
- https://books.google.com/books?id=KywWBAAAQBAJ
- https://web.archive.org/web/20160901081936/https://books.google.com/books?id=YKZqY8PCNo0C
- https://web.archive.org/web/20160901091516/https://books.google.com/books?id=g3AlWip4R38C&pg=PA92
- http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt8.1mil
- http://cygnus-x.net/geekstuff/programs/sqrt/13/200000.txt
- https://www.netflixtvshows.com
- http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt2.2mil
- https://books.google.com/books?id=YKZqY8PCNo0C&pg=PA78
- http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00570001&seq=12&frames=0&view=50
- http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII14.html
- https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1268
- https://books.google.com/books?id=g3AlWip4R38C&pg=PA92