Question 6 of 9: Object-Oriented Design — A C++ Complex Number Class

Question 6: Object-Oriented Design — A C++ Complex Number Class (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. No numeric data; a design requirement for a class that can be declared both with and without initial real/imaginary values, gives read/write access to each part, and supports $+,-,\times$ plus printing.

Find. A Complex class, split into Complex.h (declaration) and Complex.cc (implementation), whose arithmetic operators implement the standard complex number formulas.

Approach. Store the real and imaginary parts as two double members; provide a single constructor whose two default arguments $(0,0)$ make it serve as both the no-argument declaration Complex c; and the initialised one Complex c(3,4); — which is exactly the "with and without initialization" requirement — then accessor/mutator pairs for each part, overload +, - and * as member functions returning a new Complex by value, and overload << as a free function so cout << c reads naturally.

1. Write Complex.h.

   // Complex.h
   #ifndef COMPLEX_H
   #define COMPLEX_H
   #include <iostream>
   
   class Complex {
   public:
       Complex(double re = 0.0, double im = 0.0);
   
       double getReal() const;
       double getImag() const;
       void   setReal(double re);
       void   setImag(double im);
   
       Complex operator+(const Complex &other) const;
       Complex operator-(const Complex &other) const;
       Complex operator*(const Complex &other) const;
   
       friend std::ostream &operator<<(std::ostream &os, const Complex &c);
   
   private:
       double real, imag;
   };
   
   #endif

2. Write Complex.cc, implementing the arithmetic formulas directly. $$ (a+bi)+(c+di) = (a{+}c) + (b{+}d)i, \qquad (a+bi)-(c+di) = (a{-}c) + (b{-}d)i $$ $$ (a+bi)\times(c+di) = (ac{-}bd) + (ad{+}bc)i $$

   // Complex.cc
   #include "Complex.h"
   
   Complex::Complex(double re, double im) : real(re), imag(im) {}
   
   double Complex::getReal() const { return real; }
   double Complex::getImag() const { return imag; }
   void   Complex::setReal(double re) { real = re; }
   void   Complex::setImag(double im) { imag = im; }
   
   Complex Complex::operator+(const Complex &o) const
   {
       return Complex(real + o.real, imag + o.imag);
   }
   
   Complex Complex::operator-(const Complex &o) const
   {
       return Complex(real - o.real, imag - o.imag);
   }
   
   Complex Complex::operator*(const Complex &o) const
   {
       return Complex(real * o.real - imag * o.imag,
                       real * o.imag + imag * o.real);
   }
   
   std::ostream &operator<<(std::ostream &os, const Complex &c)
   {
       os << c.real;
       if (c.imag >= 0) os << " + " << c.imag << "i";
       else             os << " - " << -c.imag << "i";
       return os;
   }

3. Trace the arithmetic on $c_1=3+4i$, $c_2=1-2i$. $$c_1+c_2 = (3{+}1)+(4{+}(-2))i = 4+2i, \qquad c_1-c_2 = (3{-}1)+(4{-}(-2))i = 2+6i$$ $$c_1\times c_2 = (3\cdot1 - 4\cdot(-2)) + (3\cdot(-2) + 4\cdot1)i = 11-2i$$ $$\boxed{c_1+c_2=4+2i,\quad c_1-c_2=2+6i,\quad c_1\times c_2=11-2i}$$