alreich/gaussian_integers

Gaussian Integers and Gaussian Rational Numbers

Gaussian Integers and Gaussian Rational Numbers

This module defines two classes, Zi and Qi, the Gaussian integers and Gaussian rational numbers, respectively.

Mathematically, the integers are denoted by $\mathbb{Z}$, the rational numbers by $\mathbb{Q}$, and the complex numbers by $\mathbb{C}$.

$\mathbb{C} \equiv \lbrace a + bi: a, b \in \mathbb{R} \rbrace$ where $\mathbb{R}$ is the set of real numbers and $i^2 = -1$.

The Gaussian integers are denoted by $\mathbb{Z}[i] \equiv \lbrace n + mi: n, m \in \mathbb{Z} \rbrace \subset \mathbb{C}$,

and the Gaussian rationals are denoted by $\mathbb{Q}[i] \equiv \lbrace r + si: r, s \in \mathbb{Q} \rbrace \subset \mathbb{C}$.

NOTE:

• Python uses $j$ instead of $i$ to represent complex numbers
• Although, both Zi and Qi are subclasses of numbers.Complex, and $\mathbb{Z}[i] \subset \mathbb{Q}[i] \subset \mathbb{C}$, the class Zi is not a subclass of the class Qi.

For more information, see the two Jupyter notebooks in the notebooks directory.

For a quick look, see the examples following the plot of Gaussian primes, below.

Quick Look

The examples below are from "The Gaussian Integers" by Keith Conrad

>>> from gaussians import Zi, Qi

The Division Theorem

>>> alpha = Zi(27, -23)
>>> beta = Zi(8, 1)

>>> quot, rem = Zi.modified_divmod(alpha, beta)

>>> print(f"{beta * quot + rem} = {beta} * {quot} + {rem}")

(27-23j) = (8+1j) * (3-3j) + -2j

The Euclidean Algorithm

>>> alpha = Zi(11, 3)
>>> beta = Zi(1, 8)

>>> gcd = Zi.gcd(alpha, beta, verbose=True)  # Prints intermediate results

>>> print(f"\ngcd({alpha}, {beta}) -> {gcd}")

   (11+3j) = (1+8j) * (1-1j) + (2-4j)
   (1+8j) = (2-4j) * (-2+1j) + (1-2j)
   (2-4j) = (1-2j) * (2+0j) + 0j

gcd((11+3j), (1+8j)) -> (1-2j)

Bezout's Theorem

>>> a, x, y = Zi.xgcd(alpha, beta)

>>> print(f"a = {a}")
>>> print(f"{alpha * x  + beta * y} = {alpha} * {x} + {beta} * {y}")

a = (1-2j)
(1-2j) = (11+3j) * (2-1j) + (1+8j) * 3j

True Division

If $\alpha, \beta \in \mathbb{Z}[i]$, then ${\large \frac{\alpha}{\beta}} \in \mathbb{Q}[i]$

>>> a = Zi(4, 5)
>>> b = Zi(1, -2)
>>> a / b

Qi('-6/5', '13/5')

>>> print(f"{a} / {b} -> {a / b}")

(4+5j) / (1-2j) -> (-6/5+13/5j)