Gaussian integer information

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition...

A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x +...

generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic...

Gaussian integers. The sum, difference, or product of Gaussian integers is also a Gaussian integer: ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i...

of a prime Gaussian integer if the hypotenuse is prime. If the Gaussian integer is not prime then it is the product of two Gaussian integers p and q with...

root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice...

polynomial p(x) with integer coefficients and where anxn is the highest-degree term of p(x). Integral element Gaussian integer Eisenstein integer Root of unity...

the square roots of rational integers, such as √2, and the complex number i = √−1, which generates the Gaussian integers. Another common example is the...

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ⁡ ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}...

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of...

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued...

are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z}...

/twoː/, /twuː/, and finally /tuː/. An integer is determined to be even if it is divisible by 2. For integers written in a numeral system based on an...

optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function;...

and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or C {\displaystyle...

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}...

without using quartic reciprocity. For an odd Gaussian prime π {\displaystyle \pi } and a Gaussian integer α {\displaystyle \alpha } relatively prime to...

integers. Define f (n) = |n|, the absolute value of n. Z[ i ], the ring of Gaussian integers. Define f (a + bi) = a2 + b2, the norm of the Gaussian integer...

such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of...

qualified as quadratic integers. Gaussian integers, complex numbers a + bi for which both a and b are integers, are also quadratic integers. This is because...

partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only...

Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers...

second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said that a third monograph...

In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs...

algorithm Gaussian brackets – described on WolframMathWorld Gaussian's modular arithmetic Gaussian integer, usually written as Z[i] Gaussian prime Gaussian logarithms...

= Q and L = Q(i), so OK is simply Z, and OL = Z[i] is the ring of Gaussian integers. Although this case is far from representative — after all, Z[i] has...

P; this is a matrix formulation of Gaussian elimination. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the...

of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers. If we regard the ring of Gaussian integers, we...

§ Blackman window. The Fourier transform of a Gaussian is also a Gaussian. Since the support of a Gaussian function extends to infinity, it must either...