Gaussian Integer

Gaussian Integer

A complex number of the form a + bi for which both a and b are integers. For example, 2 + 3i, 8 – 7i, –5, and 12i are all Gaussian integers.

Key Formula

z = a + bi, \quad a, b \in \mathbb{Z}

Where:

$z$ = The Gaussian integer
$a$ = The real part, which must be an ordinary integer
$b$ = The imaginary part, which must be an ordinary integer
$i$ = The imaginary unit, where $i^2 = -1$

Worked Example

Problem: Determine whether the product of the Gaussian integers 2 + 3i and 1 − i is itself a Gaussian integer.

Step 1: Multiply using the distributive property (FOIL).

(2 + 3i)(1 - i) = 2(1) + 2(-i) + 3i(1) + 3i(-i)

Step 2: Simplify each term, remembering that

i^2 = -1

= 2 - 2i + 3i - 3i^2 = 2 - 2i + 3i - 3(-1) = 2 + i + 3

Step 3: Combine the real parts and the imaginary parts.

= 5 + i

Answer: The product is

5 + i

. Since both 5 and 1 are integers, the result is indeed a Gaussian integer.

Why It Matters

Gaussian integers extend familiar ideas from ordinary integer arithmetic—like primes, divisibility, and factoring—into the complex plane. For instance, the ordinary prime 5 factors as

(2 + i)(2 - i)

in the Gaussian integers, revealing structure invisible in the real numbers alone. This concept plays a key role in number theory, including proofs about which integers can be written as a sum of two squares.

Common Mistakes

Mistake: Assuming any complex number is a Gaussian integer.

Correction: Both the real and imaginary parts must be integers. For example,

1.5 + 2i

is not a Gaussian integer because 1.5 is not an integer.

Related Terms

Complex Numbers — The broader set that contains all Gaussian integers
Integers — The components a and b must be integers
Imaginary Numbers — Numbers involving the imaginary unit i
Real Numbers — Every ordinary integer is a Gaussian integer with b = 0