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...

of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer. The norm of a product of Gaussian integers is the...

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

number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic...

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})}...

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

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...

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

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

twā would thus be /twɑː/, /twɔː/, /twoː/, /twuː/, and finally /tuː/. An integer is determined to be even if it is divisible by two. When written in base...

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...

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

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}}}...

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...

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

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...

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...

= 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...

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

the field of Gaussian rationals is neither ordered nor complete (as a metric space). The Gaussian integers Z[i] form the ring of integers of Q(i). The...

functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods { ( 1 + i ) ϖ , ( 1 − i ) ϖ } , {\displaystyle...

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...

§ 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...