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Gaussian integer
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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, usually written as or
![{\displaystyle \mathbf {Z} [i]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a617cf5867f951fefb72f3ab7278e0f6f1eedd)
![{\displaystyle \mathbb {Z} [i].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03900897cdf515bbbc52879377653a871b9efc06)
Gaussian integers share many properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic.
Gaussian integers are algebraic integers and form the simplest ring of quadratic integers.
Gaussian integers are named after the German mathematician Carl Friedrich Gauss.
The Gaussian integers are the set
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, which is 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 square lattice.
The conjugate of a Gaussian integer a + bi is the Gaussian integer a − bi.
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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, usually written as or
Gaussian integers share many properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total order that respects arithmetic.
Gaussian integers are algebraic integers and form the simplest ring of quadratic integers.
Gaussian integers are named after the German mathematician Carl Friedrich Gauss.
The Gaussian integers are the set
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, which is 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 square lattice.
The conjugate of a Gaussian integer a + bi is the Gaussian integer a − bi.
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