Gaussian Integers, Part 1
November 4, 2014
Gaussian integers are complex numbers of the form a + b i where both a and b are integers. They obey the normal laws of algebra for addition, subtraction and multiplication. Division works, too, but is a little bit complicated:
Addition: (a + b i) + (x + y i) = (a + x) + (b + y) i.
Subtraction: add the negative: (a + b i) − (x + y i) = (a − x) + (b − y) i.
Multiplication: cross-multiply, with i 2 = −1: (a + b i) × (x + y i) = (a x − b y) + (a y + b x) i.
Quotient: multiply by the conjugate of the divisor, then round: (a + b i) ÷ (x + y i) = ⌊(a x + b y) / n⌉ + ⌊(b x − a y) / n⌉ i, where n = x 2 + y 2.
Remainder: compute the quotient, then subtract quotient times divisor from dividend.
Your task is to write a small library of functions for operating on Gaussian integers. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.
Programming Praxis 
Haskell: http://codepad.org/1VZIecvt
Question1: is the quotient not like this (ax+by) +(ay-bx)i ?
Question2: is it ok to use floor instead of round?
Then in Scala:
class GaussianInteger(val a: Int, val b: Int) {
def addition(that: GaussianInteger): GaussianInteger = new GaussianInteger(a + that.a, b + that.b)
def subtraction(that: GaussianInteger): GaussianInteger = new GaussianInteger(a – that.a, b – that.b)
def crossMultiply(that: GaussianInteger): GaussianInteger = new GaussianInteger(a * that.a – b * that.b, a * that.b + b * that.a)
def quotient(that: GaussianInteger): GaussianInteger = {
val n = that.a * that.a + that.b * that.b
new GaussianInteger((a * that.a – b * that.b) / n, (b * that.a – a * that.b) / n)
}
def remainder(that: GaussianInteger): GaussianInteger = subtraction(quotient(that))
override def toString: String = a + ” + ” + b + “i”
}
The quotient is ((ax+by) + (bx-ay)i) / n; the code was right, I fixed the description. Using floor instead of round doesn’t work, because the norm of the result must be less than half the norm of the divisor, by convention; see the discussion of the Division Theorem in the paper linked from the solution page.
Using Scheme’s complex numbers, I get addition and multiplication for free. I find that Gambit-C does Gaussian rationals exactly, and for it the norm is automatically real. In Guile I had to take the real part explicitly. (I needed the norm to be real in order to test that the remainder is smaller than the divisor.)
(define (zi-conjugate a) (make-rectangular (real-part a) (- (imag-part a))))
(define (zi-norm a) (real-part (* a (zi-conjugate a)))) ;imag-part is zero
(define (zi-quotient a b)
(let ((q (/ (* a (zi-conjugate b)) (zi-norm b))))
(make-rectangular (round (real-part q)) (round (imag-part q)))))
(define (zi-remainder a b) (- a (* b (zi-quotient a b))))
Printing the quotient and remainder from the paper:
(write (cons (zi-quotient 27-23i 8+i) (zi-remainder 27-23i 8+i))) (newline)
The Gambit-C interpreter (gsi) prints this, where the absence of decimal points indicates that the components are exact integers (with a zero real component omitted):
(3-3i . -2i)
Guile prints this, where the presence of the decimal points indicates that the computation may have used inexact methods (floating point, I’m sure):
(3.0-3.0i . 0.0-2.0i)
Floating point might fail (rounding, overflowing), but my limited tests, with all components small, were fine even in Guile.