# Subtraction can be defined in terms of addition
# and the additive inverse.
#
# If
#
#     w = a + bj
#
# and
#
#     z = c + dj,
#
# then
#
#     w - z = w + (-z)
#
#           = (a + bj) + (-c - dj)
#
#           = (a - c) + (b - d)j.
#
# Thus, complex numbers are subtracted by
# subtracting their real components and their
# imaginary components separately.

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

print( "\nCalculations with subtraction:" )

print( w - z )

# We can obtain the same result by adding the
# additive inverse of z to w.

print( w + (-z) )

# Addition is commutative, so we can also add
# the two numbers in the opposite order.

print( (-z) + w )

# Thus,
#
#     w - z = w + (-z) = (-z) + w.

print( "\nEquivalent ways of calculating w - z:" )

print( w - z )
print( w + (-z) )
print( (-z) + w )

# We can also see how subtraction is performed
# by separately subtracting the real and
# imaginary components.

print( w.real - z.real )
print( w.imag - z.imag )

print( "Re(w) - Re(z) =",
       w.real - z.real )

print( "Im(w) - Im(z) =",
       w.imag - z.imag )

# Putting these two components back together
# gives the difference of the two complex
# numbers.

print( "(Re(w) - Re(z)) + j(Im(w) - Im(z)) =",
       (w.real - z.real)
       + 1j*(w.imag - z.imag) )

# Thus,
#
#     Re(w - z) = Re(w) - Re(z)
#
# and
#
#     Im(w - z) = Im(w) - Im(z).


# Unlike addition, subtraction is not
# commutative. In general,
#
#     w - z != z - w.
#
# Instead, reversing the order gives the
# additive inverse:
#
#     w - z = -(z - w).

print( "\nReversing the order of subtraction:" )

print( w - z )
print( z - w )
print( -(z - w) )

# Confirm that
#
#     w - z = -(z - w).

print( (w - z) == -(z - w) )


# Subtracting a number from itself gives zero.

print( "\nSubtracting a number from itself:" )

print( w - w )
print( z - z )

# Subtracting zero does not change a number.

print( "\nSubtracting zero:" )

print( w - 0 )
print( z - 0 )

# Subtracting a number from zero gives its
# additive inverse.

print( "\nSubtracting a number from zero:" )

print( 0 - w )
print( -w )

print( 0 - z )
print( -z )

# Thus,
#
#     w - 0 = w
#
# while
#
#     0 - w = -w.


# Subtracting an additive inverse is equivalent
# to addition:
#
#     w - (-z) = w + z.

print( "\nSubtracting an additive inverse:" )

print( w - (-z) )
print( w + z )

print( (w - (-z)) == (w + z) )


# Parentheses are important when more than two
# numbers are being subtracted. Subtraction is
# not associative. In general,
#
#     (w - z) - v != w - (z - v).

v = -1.8 + 2.1j

print( "\nSubtraction is not associative:" )

print( (w - z) - v )
print( w - (z - v) )

# The first expression can be written as
#
#     (w - z) - v
#       = w + (-z) + (-v)
#       = w - z - v.
#
# The second expression is
#
#     w - (z - v)
#       = w - z + v.
#
# Consequently, these expressions generally
# have different values.

print( ((w - z) - v) == (w - (z - v)) )