Gaussian Integer
High School
Definition
A complex number (x + yi) whose real and imaginary parts (both x and y) are both integers. Gaussian integers were originally introduced by Carl Friedrich Gauss.
Worked examples
\(3 + 2i\), \(-5 + 0i = -5\), \(0 - 4i\)
Each has integer real and imaginary parts, so all three are Gaussian integers.
\(\frac{1}{2} + 3i\) is not a Gaussian integer; \(4 + \sqrt{2}i\) is not either.
Both the real and imaginary parts must be integers—no fractions or irrationals allowed.
Common mistakes
- \(2.5 + 3i\) is a Gaussian integer because 3 is an integer → Not a Gaussian integer—both parts must be integers The real part 2.5 is not an integer, so the whole number fails the requirement.
- Only numbers like \(a + bi\) with \(b \ne 0\) are Gaussian integers → Ordinary integers like \(5 = 5 + 0i\) are also Gaussian integers Gaussian integers include all regular integers as the special case when the imaginary part is zero.
- \(i\) itself is not a Gaussian integer → \(i = 0 + 1i\) is a Gaussian integer Both 0 and 1 are integers, so \(i\) qualifies.
Where you'll use it next
Gaussian integers appear in number theory when you study unique factorization in the complex plane, solve Diophantine equations, and explore algebraic number theory and ring structures in abstract algebra.
Found in 1 StudyPug lesson
Unlocking the Power of Complex Numbers and Complex Planes
12th Grade12thGrade 12 Math
Dive into the fascinating world of complex numbers and planes. Master essential concepts, operations, and real-world applications in engineering, physics, and advanced mathematics.
See also
iAbsolute Value of a Complex NumberCartesian FormNatural NumbersFactor of an IntegerDe Moivre's Theorem
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026