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Page "Algebraic number" ¶ 6
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Gaussian and integers
* Z, the ring of Gaussian integers.
The original algorithm was described only for natural numbers and geometric lengths ( real numbers ), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable.
This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers.
If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.
An example of such a domain is the Gaussian integers Z, that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers.
* Z: the ring of Gaussian integers
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers.
The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z.
The Gaussian integers are a special case of the quadratic integers.
Gaussian integers as lattice point s in the complex plane
Formally, Gaussian integers are the set
Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.
The Gaussian integers form a principal ideal domain with units 1, − 1, i, and − i. If x is a Gaussian integer, the four numbers x, ix, − x, and − ix are called the associates of x.
As for every principal ideal domain, the Gaussian integers form also a unique factorization domain.
The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q ( i ) consisting of the complex numbers whose real and imaginary part are both rational.
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity ( 1832 ) ( see ).
Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e. the Gaussian integers ) than they are as statements about ordinary whole numbers ( i. e. the integers ).

Gaussian and those
Often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence.
The error-rates quoted here are those in additive white Gaussian noise ( AWGN ).
The Gaussian functions are thus those functions whose logarithm is a quadratic function.
Gaussian functions are among those functions that are elementary but lack elementary antiderivatives ; the integral of the Gaussian function is the error function.
The atomic orbitals used are typically those of hydrogen-like atoms since these are known analytically i. e. Slater-type orbitals but other choices are possible like Gaussian functions from standard basis sets.
The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions ( which give a displaced Gaussian ) has led many to expand them in terms of Gaussians.
The name originates from Pople's use of Gaussian orbitals to speed up calculations compared to those using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree-Fock calculations.
Since those prime factors are Gaussian primes, this means that 133 is a Blum integer.
Since those prime factors are Gaussian primes, this means that 141 is a Blum integer.

Gaussian and complex
The solution, in the form of a Gaussian function, represents the complex amplitude of the beam's electric field.
Information about the spot size and radius of curvature of a Gaussian beam can be encoded in the complex beam parameter,:
In terms of the complex beam parameter, a Gaussian field with one transverse dimension is proportional to
It is easy to see graphically that every complex number is within units of a Gaussian integer.
Put another way, every complex number ( and hence every Gaussian integer ) has a maximal distance of
Although an arbitrarily curved space is very complex to describe, the curvature of a space which is locally isotropic and homogeneous is described by a single Gaussian curvature, as for a surface ; mathematically these are strong conditions, but they correspond to reasonable physical assumptions ( all points and all directions are indistinguishable ).
As described above, a Rayleigh fading channel itself can be modelled by generating the real and imaginary parts of a complex number according to independent normal Gaussian variables.
There is a direct method of eliminating multiple ( or repeated ) roots from polynomials with exact coefficients ( integers, rational numbers, Gaussian integers or rational complex numbers ).
* Bogert, Bruce P .; Ossanna, Joseph F., " The heuristics of cepstrum analysis of a stationary complex echoed Gaussian signal in stationary Gaussian noise ", IEEE Transactions on Information Theory, v. 12, issue 3, July 19 1966, pp. 373-380
In mathematics, the Morlet wavelet ( or Gabor wavelet ) is a wavelet composed of a complex exponential ( carrier ) multiplied by a Gaussian window ( envelope ).
where is the optical phase error introduced by atmospheric turbulence, R ( k ) is a 2 dimensional square array of independent random complex numbers which have a Gaussian
In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers.
The E < sub > 6 </ sub > lattice, E < sub > 8 </ sub > lattice and Coxeter – Todd lattice also have constructions as complex lattices, over either the Eisenstein or Gaussian integers.
Here " whole real numbers " are ordinary integers and " whole complex numbers " are Gaussian integers ; one should similarly interpret " real and complex prime numbers ".

Gaussian and numbers
For large numbers the Poisson distribution approaches a normal distribution, typically making shot noise in actual observations indistinguishable from true Gaussian noise except when the elementary events ( photons, electrons, etc.
The positive integer Gaussian primes are the prime numbers congruent to 3 modulo 4,.
One should not refer to only these numbers as " the Gaussian primes ", which term refers to all the Gaussian primes, many of which do not lie in Z.
Prime numbers of the form are also Gaussian primes.
The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases ( such as for the Gaussian integers Z ), it may also fail, as in the case of Z

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