NAME
Sidef::Types::Number::Gauss - Gaussian integer arithmetic in Sidef
DESCRIPTION
This class implements Gaussian integers, which are complex numbers of the form a + bi where both a and b are integers. Gaussian integers form a unique factorization domain and are useful in number theory, particularly in studying prime factorizations and solving Diophantine equations.
A Gaussian integer can be created using the Gauss(a, b) constructor, which represents the complex number a + bi.
SYNOPSIS
var a = Gauss(17, 19) #=> 17+19i
var b = Gauss(43, 97) #=> 43+97i
say a+b #=> 60+116i
say a-b #=> -26-78i
say a*b #=> -1112+2466i
say a/b #=> 99/433-32i/433
say a.norm #=> 650 (17² + 19²)
say a.conj #=> 17-19i
var g = Gauss(120, 84)
say g.factor #=> [Gauss(-1,1), Gauss(1,1), Gauss(1,1), Gauss(1,1), Gauss(2,1), Gauss(3,0)]
var p = Gauss(3, 2)
say p.is_prime #=> trueINHERITS
Inherits methods from:
* Sidef::Types::Number::NumberMETHODS
!=
a != bReturns true if two Gaussian integers are not equal (their real or imaginary parts differ).
Aliases: ne
%
a % bReturns the remainder when dividing Gaussian integer a by b, using the Euclidean division algorithm for Gaussian integers.
Aliases: mod
&
a & bPerforms bitwise AND operation on the real and imaginary parts separately.
Aliases: and
*
a * bReturns the product of two Gaussian integers. If a = (p+qi) and b = (r+si), then a*b = (pr-qs) + (ps+qr)i.
Aliases: mul
**
a ** bReturns the Gaussian integer a raised to the power b.
Aliases: pow
+
a + bReturns the sum of two Gaussian integers by adding their real and imaginary parts separately.
Aliases: add
++
a++Increments the Gaussian integer by 1 (adds 1 to the real part).
Aliases: inc
-
a - bReturns the difference of two Gaussian integers by subtracting their real and imaginary parts separately.
Aliases: sub
--
a--Decrements the Gaussian integer by 1 (subtracts 1 from the real part).
Aliases: dec
/
a / bReturns the quotient when dividing Gaussian integer a by b. The result may have rational real and imaginary parts.
Aliases: ÷, div
<
a < bCompares two Gaussian integers by their norms. Returns true if the norm of a is less than the norm of b.
Aliases: lt
<<
a << bPerforms left bitwise shift on both real and imaginary parts by b positions.
Aliases: lsft, shift_left
<=>
a <=> bCompares two Gaussian integers by their norms. Returns -1 if norm(a) < norm(b), 0 if equal, and 1 if norm(a) > norm(b).
Aliases: cmp
==
a == bReturns true if two Gaussian integers are equal (both real and imaginary parts match).
Aliases: eq
>
a > bCompares two Gaussian integers by their norms. Returns true if the norm of a is greater than the norm of b.
Aliases: gt
>>
a >> bPerforms right bitwise shift on both real and imaginary parts by b positions.
Aliases: rsft, shift_right
^
a ^ bPerforms bitwise XOR operation on the real and imaginary parts separately.
Aliases: xor
|
a | bPerforms bitwise OR operation on the real and imaginary parts separately.
Aliases: or
≤
a ≤ bReturns true if the norm of a is less than or equal to the norm of b.
Aliases: <=, le
≥
a ≥ bReturns true if the norm of a is greater than or equal to the norm of b.
Aliases: >=, ge
a
self.aReturns the real part of the Gaussian integer.
Aliases: re, real
abs
x.absReturns the absolute value (magnitude) of the Gaussian integer, which is sqrt(a² + b²) where the Gaussian integer is a+bi.
b
self.bReturns the imaginary part of the Gaussian integer.
Aliases: im, imag
ceil
x.ceilReturns a new Gaussian integer with the ceiling function applied to both the real and imaginary parts.
conj
x.conjReturns the complex conjugate of the Gaussian integer. If x = a+bi, returns a-bi.
divisors
n.divisorsReturns an array of all Gaussian integer divisors of n.
dump
x.dumpReturns a string representation of the Gaussian integer suitable for debugging, showing its internal structure.
eval
x.eval(v)Evaluates the Gaussian integer in some context, replacing symbolic variables with the value v.
factor
z.factorReturns an array of Gaussian prime factors of z. Each factor is itself a Gaussian integer.
Aliases: factors
factor_exp
z.factor_expReturns an array of pairs [prime, exponent] representing the prime factorization of the Gaussian integer z.
float
x.floatConverts the Gaussian integer to a complex number with floating-point real and imaginary parts.
floor
x.floorReturns a new Gaussian integer with the floor function applied to both the real and imaginary parts.
gcd
n.gcd(k)Returns the greatest common divisor of two Gaussian integers using the Euclidean algorithm.
gcd_norm
n.gcd_norm(k)Returns the norm (squared magnitude) of the greatest common divisor of two Gaussian integers.
i
x.iReturns the imaginary unit multiplied by the Gaussian integer, effectively rotating it 90 degrees in the complex plane.
iabs
x.iabsReturns the squared absolute value (norm) of the Gaussian integer, which is a² + b² for a+bi.
inv
x.invReturns the multiplicative inverse of the Gaussian integer, equal to conj(x)/norm(x).
invmod
x.invmod(m)Returns the modular multiplicative inverse of x modulo m in the Gaussian integers, if it exists.
is_coprime
n.is_coprime(k)Returns true if Gaussian integers n and k are coprime (their GCD is a unit: ±1 or ±i).
is_div
x.is_div(y)Returns true if Gaussian integer x is divisible by y (i.e., x/y is also a Gaussian integer).
Aliases: is_divisible
is_imag
x.is_imagReturns true if the Gaussian integer is purely imaginary (real part is zero).
is_mone
x.is_moneReturns true if the Gaussian integer equals -1.
is_one
x.is_oneReturns true if the Gaussian integer equals 1.
is_prime
x.is_primeReturns true if the Gaussian integer is a Gaussian prime. A Gaussian integer is prime if it cannot be factored into non-unit Gaussian integers.
is_real
x.is_realReturns true if the Gaussian integer is purely real (imaginary part is zero).
is_zero
x.is_zeroReturns true if the Gaussian integer equals 0+0i.
lift
x.liftConverts the Gaussian integer to a standard complex number representation.
neg
x.negReturns the negation of the Gaussian integer (multiplies by -1).
new
self.new(a, b)
Gauss(a, b)Creates a new Gaussian integer with real part a and imaginary part b.
Aliases: call
norm
x.normReturns the norm of the Gaussian integer, which is a² + b² for x = a+bi. This is the squared magnitude.
parts
self.partsReturns an array containing the real and imaginary parts: [a, b] for a+bi.
powmod
x.powmod(n, m)Returns x raised to the power n modulo m in the Gaussian integers.
pretty
x.prettyReturns a human-readable string representation of the Gaussian integer in the form "a+bi" or "a-bi".
Aliases: stringify
reals
self.realsReturns an array containing both the real and imaginary parts as real numbers.
round
x.round(r)Rounds both the real and imaginary parts to r decimal places, returning a new Gaussian integer.
sgn
x.sgnReturns the sign of the Gaussian integer as a unit Gaussian integer (one of 1, -1, i, -i) with the same argument.
sqr
x.sqrReturns the square of the Gaussian integer (x * x).
to_c
self.to_cConverts the Gaussian integer to a complex number.
Aliases: to_n
to_s
x.to_sConverts the Gaussian integer to a string representation.
MATHEMATICAL BACKGROUND
Gaussian integers form a unique factorization domain, meaning every Gaussian integer can be uniquely factored into Gaussian primes (up to units and order). The units in the Gaussian integers are {1, -1, i, -i}.
A Gaussian integer a+bi is a Gaussian prime if and only if:
a = 0 and |b| is a prime number of the form 4k+3, or
b = 0 and |a| is a prime number of the form 4k+3, or
Both a and b are nonzero and a²+b² is a rational prime
The norm function N(a+bi) = a²+b² is multiplicative: N(zw) = N(z)N(w).
SEE ALSO
AUTHOR
Daniel "Trizen" Șuteu
LICENSE
This library is free software; you can redistribute it and/or modify it under the same terms as Sidef itself.