Gaussian integer

Gaussian integer

English

Noun

Gaussian integer (plural Gaussian integers)

1. (algebra) Any complex number of the form a + bi, where a and b are integers.
   • 1997, Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, page 44,

     We could say that a Gaussian integer is larger than another if its norm is larger, that is, if its distance from the origin is larger. That's all right, although it has the peculiarity that ${\\displaystyle 5}$ is larger than ${\\displaystyle -7}$ as an integer, but smaller than ${\\displaystyle -7}$ as a Gaussian integer! In a way, this notion of larger makes more sense: shouldn't ${\\displaystyle -7}$ be considered larger than ${\\displaystyle 5}$?

   • 2000, André Weilert, Asymptotically fast GCD Computation in ${\\displaystyle \\mathbb {Z} [i]}$, Wieb Bosma (editor), Algorithmic Number Theory: 4th International Symposium, ANTS-IV, Proceedings, Springer, LNCS 1838, page 595,

     We present an asymptotically fast algorithm for the computation of the greatest common divisor (GCD) of two Gaussian integers.

   • 2008, Timothy Gowers, June Barrow-Green, Imre Leader (editors), The Princeton Companion to Mathematics, Princeton University Press, page 319,

     For example, in the ring of Gaussian integers, ${\\displaystyle R_{-1}}$, we have the factorizations

     ${\\displaystyle 2=(1+i)(1-i)}$,

     ${\\displaystyle 5=(1+2i)(1-2i)}$,

     ${\\displaystyle 13=(2+3i)(2-3i)}$,

     ${\\displaystyle 17=(1+4i)(1-4i)}$,

     ${\\displaystyle 29=(2+5i)(2-5i)}$,

     ${\\displaystyle \\vdots }$

     where all the Gaussian integer factors in brackets above are irreducible elements of the ring of Gaussian integers.

Hypernyms

• complex number
• quadratic integer
  • algebraic integer
• Gaussian rational number, Gaussian rational

Hyponyms

• Gaussian prime
• integer

See also

• Eisenstein integer
• Blum integer