% Two complex numbers are added by adding their
% real components and their imaginary components
% separately.
%
% If
%
%     w = a + bj
%
% and
%
%     z = c + dj,
%
% then
%
%     w + z = (a + c) + (b + d)j.
%
% Complex-number addition is commutative, just
% like addition of real numbers:
%
%     w + z = z + w.

w = 5.3 + 3.2j
z = 3.2 - 4.7j

fprintf( '\nCalculations with ''w'' and ''z'':\n' )

w + z
z + w

% We can see how this addition is performed by
% separately adding the real components and the
% imaginary components.

real( z ) + real( w )
imag( z ) + imag( w )

fprintf( 'Re(z) + Re(w) = %f\n', ...
         real( z ) + real( w ) )

fprintf( 'Im(z) + Im(w) = %f\n', ...
         imag( z ) + imag( w ) )

% The imaginary component is multiplied by j
% when reconstructing the complex number.

fprintf( 'j(Im(z) + Im(w)) = %fj\n\n', ...
         imag( z ) + imag( w ) )

% Putting the real and imaginary components
% back together gives the sum of the two
% complex numbers.

fprintf( '(Re(w) + Re(z)) + j(Im(w) + Im(z)) = %f%+fj\n\n', ...
         real( z ) + real( w ), ...
         imag( z ) + imag( w ) )

% Thus, adding complex numbers is equivalent to
% separately adding their real and imaginary
% components:
%
%     Re(w + z) = Re(w) + Re(z)
%
% and
%
%     Im(w + z) = Im(w) + Im(z).


% Now consider the following five complex
% numbers.

w1 = -0.217 + 8.326j
w2 = -9.653 + 3.757j
w3 = -1.592 + 5.909j
w4 =  5.608 - 4.460j
w5 =  0.928 + 0.350j

% Based on the results below, what do you think
% is the actual sum of these five complex
% numbers?
%
% Mathematically, addition is associative:
%
%     (a + b) + c = a + (b + c).
%
% Consequently, changing the order in which
% additions are performed should not change
% the result.
%
% On a computer, however, real numbers are
% usually represented using floating-point
% numbers. Most decimal fractions cannot be
% represented exactly using the finite binary
% representation used by the computer.
%
% Consequently, small rounding errors can be
% introduced during each arithmetic operation.
% Changing the order of the operations may
% therefore change these small rounding errors.
%
% Thus, floating-point addition is not always
% exactly associative, even though mathematical
% addition is associative.
%
% Fortunately, these errors are generally very
% small for calculations such as these.

fprintf( '\nDifferent ways of calculating w1 + ... + w5:\n' )

% Display additional digits so that any small
% differences caused by floating-point
% arithmetic are visible.

format long

w1 + w2 + w3 + w4 + w5

(w1 + w2) + (w3 + (w4 + w5))

w1 + (w2 + ((w3 + w4) + w5))

(w1 + w2) + (w3 + w4) + w5