% We have already seen addition, subtraction,
% multiplication, conjugation, and
% multiplicative inverses of complex numbers.
%
% We can now define division of complex numbers.
%
% Suppose that w and z are complex numbers and
%
%     z ~= 0.
%
% Division by z means multiplication by the
% multiplicative inverse of z:
%
%     w/z = w*(1/z).
%
% Thus, complex division follows directly from
% our definition of a multiplicative inverse.


w = 2.3 + 4.1j;
z = 1.7 - 2.4j;

fprintf('\nw:\n')
w

fprintf('\nz:\n')
z

fprintf('\nw/z:\n')
w/z

fprintf('\nw*(1/z):\n')
w*(1/z)


% Apart from possible floating-point rounding,
% these last two values are the same.


% We can derive a formula for division in terms
% of the real and imaginary components.
%
% Suppose
%
%     w = a + bj
%
% and
%
%     z = c + dj.
%
% We want to calculate
%
%     a + bj
%     ------.
%     c + dj
%
% Recall that the multiplicative inverse of z is
%
%        1       c - dj
%     ------ = ---------.
%     c + dj     c^2+d^2
%
% Therefore,
%
%     a + bj
%     ------
%     c + dj
%
%                  c - dj
%       = (a + bj) --------
%                  c^2+d^2
%
%         (a + bj)(c - dj)
%       = -----------------.
%              c^2+d^2
%
% Multiplying out the numerator gives
%
%     (a + bj)(c - dj)
%
%         = ac + bd + (bc - ad)j.
%
% Therefore,
%
%     a + bj
%     ------
%     c + dj
%
%       ac + bd     bc - ad
%       -------  +  ------- j.
%       c^2+d^2     c^2+d^2


% Let us verify this formula.

a = real(w);
b = imag(w);
c = real(z);
d = imag(z);

quotient_formula = ...
    (a*c + b*d)/(c^2 + d^2) ...
    + 1j*(b*c - a*d)/(c^2 + d^2);

fprintf('\nw/z calculated directly:\n')
w/z

fprintf('\nw/z calculated from the components:\n')
quotient_formula


% Another way to derive the same result is to
% multiply the numerator and denominator by the
% complex conjugate of the denominator:
%
%     w     w*conj(z)
%     -  =  ---------.
%     z     z*conj(z)
%
% Recall that
%
%                         2
%     z*conj(z) = |z|.
%
% Thus,
%
%     w     w*conj(z)
%     -  =  ----------.
%     z          2
%              |z|
%
% This is often the most useful general formula
% for complex division.


fprintf('\nw/z:\n')
w/z

fprintf('\nw*conj(z)/abs(z)^2:\n')
w*conj(z)/abs(z)^2


% Notice that the denominator
%
%      2
%     |z|
%
% is a positive real number whenever z ~= 0.
%
% Multiplying by the conjugate has therefore
% removed the complex number from the
% denominator.


% Division has many useful properties.
%
% First, dividing a number by itself gives 1,
% provided the number is nonzero:
%
%     z/z = 1.


fprintf('\nz/z:\n')
z/z


% Dividing by 1 leaves a number unchanged:
%
%     z/1 = z.

fprintf('\nz/1:\n')
z/1


% Dividing zero by any nonzero number gives
% zero:
%
%     0/z = 0.

fprintf('\n0/z:\n')
0/z


% Division by zero, however, is not defined.
%
% Thus, expressions such as
%
%     w/0
%
% do not represent a complex number.


% Division is not commutative.
%
% In general,
%
%     w/z ~= z/w.


fprintf('\nw/z:\n')
w/z

fprintf('\nz/w:\n')
z/w


% Division is also not associative.
%
% In general,
%
%     (w/z)/v ~= w/(z/v).

v = -0.8 + 1.3j;

fprintf('\n(w/z)/v:\n')
(w/z)/v

fprintf('\nw/(z/v):\n')
w/(z/v)


% In fact,
%
%     (w/z)/v = w/(z*v),
%
% while
%
%     w/(z/v) = w*v/z.


fprintf('\n(w/z)/v and w/(z*v):\n')
(w/z)/v
w/(z*v)

fprintf('\nw/(z/v) and w*v/z:\n')
w/(z/v)
w*v/z


% Conjugation works particularly nicely with
% division.
%
% The conjugate of a quotient is the quotient
% of the conjugates:
%
%           / w \     conj(w)
%     conj( | - | ) = -------.
%           \ z /     conj(z)
%
% provided z ~= 0.


fprintf('\nConjugate of w/z:\n')
conj(w/z)

fprintf('\nconj(w)/conj(z):\n')
conj(w)/conj(z)


% This follows from the corresponding
% properties for multiplication and
% multiplicative inverses.


% Multiplicative inverses also work naturally
% with division.
%
% Recall that we represent the multiplicative
% inverse of a scalar z by
%
%     1/z.
%
% The inverse of a quotient is
%
%       1
%     -----
%      w/z
%
%       z
%     = -.
%       w
%
% Thus,
%
%       1
%     -----
%      w/z
%
%       1/w
%     = -----.
%       1/z
%
% In other words,
%
%     inverse(w/z)
%
%         = inverse(w)/inverse(z).
%
% Notice that we do not use MATLAB's
% 'inv(...)' function here. The function
% 'inv(...)' is primarily intended for matrix
% inverses. For scalar complex numbers, '1/z'
% expresses the multiplicative inverse directly.


fprintf('\nInverse of w/z:\n')
1/(w/z)

fprintf('\nz/w:\n')
z/w

fprintf('\n(1/w)/(1/z):\n')
(1/w)/(1/z)


% Magnitudes also behave very simply under
% division.
%
% The magnitude of a quotient is the quotient
% of the magnitudes:
%
%     |w/z| = |w|/|z|.


fprintf('\nabs(w/z):\n')
abs(w/z)

fprintf('\nabs(w)/abs(z):\n')
abs(w)/abs(z)


% This follows immediately from
%
%     w/z = w*(1/z)
%
% together with
%
%     |w*z| = |w|*|z|
%
% and
%
%     |1/z| = 1/|z|.


% Arguments also have a simple relationship.
%
% Multiplication adds arguments, while division
% subtracts them:
%
%     arg(w/z)
%
%         = arg(w) - arg(z)    modulo 2*pi.
%
% Because MATLAB's angle(...) returns a
% principal argument, the two numerical values
% may differ by a multiple of 2*pi.


fprintf('\nangle(w/z):\n')
angle(w/z)

fprintf('\nangle(w) - angle(z):\n')
angle(w) - angle(z)


% The same relationship can be expressed in
% degrees:
%
%     arg(w/z)
%
%         = arg(w) - arg(z)    modulo 360 deg.


fprintf('\nangle(w/z) in degrees:\n')
rad2deg(angle(w/z))

fprintf('\nangle(w) - angle(z) in degrees:\n')
rad2deg(angle(w)) - rad2deg(angle(z))


% Geometrically, division therefore has two
% effects.
%
% If
%
%     w = |w| exp(j*alpha)
%
% and
%
%     z = |z| exp(j*beta),
%
% then
%
%     w     |w|
%     -  =  --- exp(j*(alpha - beta)).
%     z     |z|
%
% Thus, dividing by z:
%
%   1. divides the magnitude by |z|, and
%
%   2. subtracts the argument of z.
%
% Equivalently, dividing by z scales by
%
%     1/|z|
%
% and rotates by
%
%     -arg(z).


% There are also several useful algebraic
% identities involving division.
%
% Provided all denominators are nonzero,
%
%     (w*z)/z = w.

fprintf('\n(w*z)/z:\n')
(w*z)/z

fprintf('\nw:\n')
w


% Multiplication can be distributed between
% numerator and denominator:
%
%     (w*v)/(z*v) = w/z
%
% provided v ~= 0.

fprintf('\n(w*v)/(z*v):\n')
(w*v)/(z*v)

fprintf('\nw/z:\n')
w/z


% Division distributes over addition and
% subtraction in the numerator:
%
%     (w + v)/z = w/z + v/z
%
% and
%
%     (w - v)/z = w/z - v/z.


fprintf('\n(w + v)/z:\n')
(w + v)/z

fprintf('\nw/z + v/z:\n')
w/z + v/z

fprintf('\n(w - v)/z:\n')
(w - v)/z

fprintf('\nw/z - v/z:\n')
w/z - v/z


% Be careful: division does NOT distribute over
% addition in the denominator.
%
% In general,
%
%       w       w     w
%     ----- ~= --- + ---.
%     z + v     z     v


fprintf('\nw/(z + v):\n')
w/(z + v)

fprintf('\nw/z + w/v:\n')
w/z + w/v


% Quotients can also be multiplied:
%
%     w   u     w*u
%     - * -  =  ---.
%     z   v     z*v

u = 3.2 - 0.6j;

fprintf('\n(w/z)*(u/v):\n')
(w/z)*(u/v)

fprintf('\n(w*u)/(z*v):\n')
(w*u)/(z*v)


% Dividing one quotient by another gives
%
%     w/z     w*v
%     ---  =  ---.
%     u/v     z*u
%
% This is the familiar rule of multiplying by
% the reciprocal.

fprintf('\n(w/z)/(u/v):\n')
(w/z)/(u/v)

fprintf('\n(w*v)/(z*u):\n')
(w*v)/(z*u)


% Integer powers also behave naturally with
% division:
%
%             n
%     (w/z)^n
%
%          n
%         w
%       = --.
%          n
%         z
%
% This holds for integer n whenever the
% expressions involved are defined.

n = 4;

fprintf('\n(w/z)^n:\n')
(w/z)^n

fprintf('\nw^n/z^n:\n')
w^n/z^n


% We can summarize the most important
% properties of complex division as follows.
%
% Division by z is multiplication by its
% multiplicative inverse:
%
%     w/z = w*(1/z).
%
% The conjugate of a quotient is
%
%                  conj(w)
%     conj(w/z) = ---------.
%                  conj(z)
%
% The inverse of a quotient is
%
%       1       z       1/w
%     -----  =  -  =  -----.
%      w/z      w       1/z
%
% The magnitude of a quotient is
%
%               |w|
%     |w/z| = -----.
%               |z|
%
% The argument of a quotient is
%
%     arg(w/z)
%
%         = arg(w) - arg(z)    modulo 2*pi.
%
% Thus, geometrically, division divides
% magnitudes and subtracts arguments.