Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain written as Z; this integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic; the Gaussian integers are the set Z =, where i 2 = − 1. In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, a subring of the field of complex numbers, it is thus an integral domain. When considered within the complex plane, the Gaussian integers constitute the 2-dimensional integer lattice; the conjugate of a Gaussian integer a + bi is the Gaussian integer a – bi. The norm of a Gaussian integer is its product with its conjugate. N = = a 2 + b 2; the norm of a Gaussian integer is thus the square of its absolute value as a complex number.

The norm of a Gaussian integer is a nonnegative integer, a sum of two squares. Thus a norm cannot be of the form 4k + 3, with k integer; the norm is multiplicative, that is, one has N = N N, for every pair of Gaussian integers z, w. This can be shown directly, or by using the multiplicative property of the modulus of complex numbers; the units of the ring of Gaussian integers are the Gaussian integers with norm 1, that is, 1, –1, i and –i. Gaussian integers have a Euclidean division similar to that of integers and polynomials; this makes the Gaussian integers a Euclidean domain, implies that Gaussian integers share with integers and polynomials many important properties such as the existence of a Euclidean algorithm for computing greatest common divisors, Bézout's identity, the principal ideal property, Euclid's lemma, the unique factorization theorem, the Chinese remainder theorem, all of which can be proved using only Euclidean division. A Euclidean division algorithm takes, in the ring of Gaussian integers, a dividend a and divisor b ≠ 0, produces a quotient q and remainder r such that a = b q + r and N < N.

In fact, one may make the remainder smaller: a = b q + r and N ≤ N 2. With this better inequality, the quotient and the remainder are not unique, but one may refine the choice to ensure uniqueness. To prove this, one may consider the complex number quotient x + iy = a/b. There are unique integers m and n such that –1/2 < x – m ≤ 1/2 and –1/2 < y – n ≤ 1/2, thus N ≤ 1/2. Taking q = m + in, one has a = b q + r, with r = b, N ≤ N 2; the choice of x – m and y – n in a semi-open interval is required for uniqueness. This definition of Euclidean division may be interpreted geometrically in the complex plane, by remarking that the distance from a complex number ξ to the closest Gaussian integer is at most √2/2. Since the ring G of Gaussian integers is a Euclidean domain, G is a principal ideal domain, which means that every ideal of G is principal. Explicitly, an ideal I is a subset of a ring R such that every sum of elements of I and every product of an element of I by an element of R belong to I. An ideal is principal, if it consists of all multiples of a single element g, that is, it has the form.

In this case, one says that g is a generator of the ideal. Every ideal I in the ring of the Gaussian integers is principal, because, if one chooses in I an nonzero element g of minimal norm, for every element x of I, the remainder of Euclidean division of x by g belongs to I and has a norm, smaller than that of g; that is, one has x = qg, where q is the quotient. For any g, the ide

Sack of Baturyn

Sack of Baturyn, sometimes referred to as the Slaughtering in Baturyn, was a part of series of punishing raids conducted by the Russian Imperial Army against Mazepa and Cossack state. On 2 November 1708, upon the sack of Baturyn, its entire civil population was exterminated, while the "Hetman Residence" was obliterated. Before the storm of Baturyn, Menshikov had at his disposal twenty regiments of dragoons, numbering fifteen to twenty thousand troopers. Baturyn at that time was a fortified city reinforced with a high number of artillery, he decided to use his diplomatic skills to convince the defenders to surrender and sent down Andrei Markovich with a message, but the Baturyn defenders refused and opened fire onto the Menshikov's positions. The appointed colonel Ivan Nis of Pryluky Regiment and an interpreter Stefan Zertis were arrested by Serdyuk Guards as saboteurs and attached to artillery guns. Trying to save himself from being executed for desertion, Nis was able to send to Menshikov one of his officers, who pointed to a secret entrance to the fortress.

The next morning on 2 November at six o'clock the Russian forces penetrated into the city and, after some two hours of resistance from cossacks, were victorious. After being left defenceless, the whole civil population of the city was tortured to death. Losses: Cossack rebels, 15 thousand including civilians. Imperial forces, 3 thousand soldiers. Pavlenko, S. "Perishing of Baturyn on 2 November 1708". "Ukrainska vydavnycha spilka". Kiev, 2007. Pavlenko, S. "Ivan Mazepa". "Alternatyvy". Kiev, 2003. Tairova-Yakovleva, T. "Mazepa". "Molodaya gvardiya". Moscow, 2007; the Western Europe about Mazepa by the director of the Scientific-Research Institute of Cossackdom at NANU Institute of History Pavlenko, S. Baturyn tragedy of 1708: thoughts and facts. All-Ukrainian daily newspaper "Day" #210, 1 December 2007. Was Baturyn doomed? Petr I. Order to Zaporizhian Host Petr I. To Prince Menshikov

The Gruffalo's Child (film)

The Gruffalo's Child is a 2011 British short computer animated TV film based on the picture book written by Julia Donaldson and illustrated by Axel Scheffler. A sequel to The Gruffalo, the film was shown on Christmas Day 2011 in the United Kingdom two years after the debut of the first film. Directed by Johannes Weiland and Uwe Heidschotter, the film was produced by Michael Rose and Martin Pope of Magic Light Pictures, London, in association with Studio Soi in Ludwigsburg, Germany. In June 2013, the film was given the Award for Best TV Special at the 8th Festival of European Animated Feature Films and TV Specials In a snowy wood, the squirrel mother's daughter gets her brother; the squirrel boy asks, "Whose footprints are these?". The girl squirrel tells him, "The Gruffalo", so the squirrel boy tries to tell his mother that the gruffalo was there. However, the Mother Squirrel doesn't believe them until she tells her two children a story of how the Gruffalo escaped from the mouse; the story begins with the Gruffalo's daughter attempting to follow a hedgehog, because she just wants to explore the outside world.

Her father, does not want her to do that. He tells her about the time when he met the mouse, he describes the mouse as a monster, the daughter imagines the mouse to be just as her father depicts it. One snowy night, the Gruffalo's daughter decides to explore the outside world, she begins to follow some footprints. On her journey through the deep dark wood, she meets first the snake and the owl and the fox, realizing that the monster doesn't exist, she begins to cry, wishes to be back with her father when she runs into the mouse. The mouse tells her that the monster is real, he makes a scary shadow on the branch of a hazel tree; the Gruffalo's daughter runs away to her father's cave, but the shadow-like mouse begins to follow her. In the cave of the Gruffalo, she is now comfortable at her father's side. Helena Bonham Carter as Mother Squirrel Shirley Henderson as The Gruffalo's Child Robbie Coltrane as The Gruffalo James Corden as Mouse Tom Wilkinson as Fox John Hurt as Owl Rob Brydon as Snake Sam Lewis as First Little Squirrel Phoebe Givron-Taylor as Second Little Squirrel Magic Light Pictures The Gruffalo's Child on IMDb The Gruffalo's Child at The Big Cartoon DataBase