the Gaussian integers

Gaussian integers are complex numbers where both the real and imaginary parts are whole numbers. They are expressed in the form \( a + bi \), where \( a \) and \( b \) are integers, and \( i \) represents the imaginary unit, which satisfies \( i^2 = -1 \). This set extends the traditional integers by including imaginary components, allowing for new mathematical concepts. Gaussian integers play a role in number theory, algebra, and various applications in computer science and cryptography, enabling complex calculations while maintaining a structure similar to regular integers.