Complex Conjugate Proof

Posted on January 9, 2014 by Miles Steele

We will set out to prove the equation (a - b)* = a* - b* where * means complex conjugate.

As a refresher, a complex conjugate of a complex number is the number which is the same but for a negated imaginary component. For example, the complex conjugate of 4 + 3i is 4 - 3i.

We will do this by exploring through programming. First, we will tell python what a complex number is by creating a class to store the real and imaginary components and teach it how to manipulate the numbers in such ways as addition and subtraction by using methods on the complex number class.

Python actually has support for complex numbers built in. This is fantastic, but we are not going to use its built in support. The reason is two-fold. First, it is fun to see the inner workings of the complex number class. Secondly, writing our own will make it easier to debug, understand, and ensure the validity of the funny business we will be doing later, which will involve shoveling things that are not quite numbers through the complex number class machinery.

You can grab a copy of the python source for this at any point, or clone the github repository.

class ComplexNumber(object):
  """
  Class for manipulating complex numbers.

  This example is limited to things which might be useful for the
  exercise and thus does not include multiplication or division.
  """

  def __init__(self, real, imag):
    """ Complex numbers have a real and imaginary part. """
    self.real = real
    self.imag = imag

  def conj(self):
    """ Complex conjugate of the number. """
    return ComplexNumber(self.real, -self.imag)

  def __add__(left, right):
    """ Addition operator: x + y """
    return ComplexNumber(left.real + right.real, left.imag + right.imag)

  def __sub__(left, right):
    """ Subtraction operator: x - y """
    return left + (-right)

  def __pos__(self):
    """ Unary plus operator: (+x) """
    return self

  def __neg__(self):
    """ Unary minus operator: -x """
    return ComplexNumber(-self.real, -self.imag)

  def __eq__(left, right):
    """ Equality testing: x == y """
    return (left.real == right.real) and (left.imag == right.imag)

  def __ne__(left, right):
    """ Inequality testing: x != y """
    return not left == right

  def __repr__(self):
    """
    This tells python how to represent ComplexNumber's as strings.
    This way, when we print a ComplexNumber, it shows something
    nice like "8 + 3i"
    """
    return "{} + {}i".format(self.real, self.imag)

Numerical Tests

Now that we have our complex number class, let’s test it out on a few examples.

print "\nComplex number examples:"
x = ComplexNumber(8, 5)
y = ComplexNumber(3, 2)
print x                        # -> 8 + 5i
print y                        # -> 3 + 2i
print -x                       # -> -8 + -5i
print x + y                    # -> 11 + 7i
print x - y                    # -> 5 + 3i
print -y + x                   # -> 5 + 3i
print x == y                   # -> False
print x == ComplexNumber(8, 5) # -> True
print x.conj()                 # -> 8 + -5i

Looks good so far. The printing for negative imaginary components is a little funky, but it’s readable enough.

Now let’s try the equation in question with a few examples. Remember, the equation we’re testing is (a - b)* = a* - b*.

print "\nEquation example:"
a = ComplexNumber(8, 5)
b = ComplexNumber(3, 2)
# left side of the equation
print (a - b).conj()                        # -> 5 + -3i
# right side of the equation
print a.conj() - b.conj()                   # -> 5 + -3i
print (a - b).conj() == a.conj() - b.conj() # -> True

Great, it looks like the equation holds for those values. How about a few more?

print "\nMore equation examples:"
az = [ComplexNumber(8, 5),  ComplexNumber(-2, 3),
      ComplexNumber(-9, -3), ComplexNumber(1027, -304) ]
bz = [ComplexNumber(9, -2), ComplexNumber(8, 4),
      ComplexNumber(6, 9),   ComplexNumber(0, 0) ]
for a,b in zip(az, bz):
  print (a - b).conj() == a.conj() - b.conj() # -> True every time!

But this does not prove the equation true, you say. Well, what if we try for 10000 different randomly generated examples?

print "\nRandomly generated examples:"
from random import randint
for _ in xrange(10000):
  a = ComplexNumber(randint(-1000, 1000), randint(-1000, 1000))
  b = ComplexNumber(randint(-1000, 1000), randint(-1000, 1000))
  truth = (a - b).conj() == a.conj() - b.conj() # -> ... True
  if not truth:
    print "It was false!"
print "\nIf nothing to the contrary is printed above,"\
      + "then all the examples checked out."

Those all worked! At least for me. You can give it a shot if you want. The equation seems to hold up. It seems very unlikely for there to be holes in the coverage of this equation. Is that good enough?
No?
Ok. Well let’s try to show it more generally then.

A General Approach

In order to prove the equation generally, we will have to stop plugging in actual complex numbers. Every time we plug in an actual complex number, we doom ourselves to a loss of generality. But does that mean that all of our hard work in creating the machinery of the ComplexNumber class will go to waste? Certainly not.

We will now create a class to represent something that is not quite a number. It will behave a lot like a number, but without every having an actual value. We will call such nebulous artifacts “variables” for now. I’m not sure that is quite the right word, you can decide for yourself.

We will put Variables inside ComplexNumbers as real and imaginary components. So Variables will need to be able to do everything that a number does.

class Variable(object):
  """
  Variables are like numbers, but they have no value.
  Variables must be able to do everything that numbers can do.

  Each Variable we create will have its own unique identity.
  It will be distinct from every other Variable that exists.
  However, we must make this so WITHOUT assigning a value to the variable.
  """

  def __init__(self):
    """
    Variables have an empty constructor because they do not
    have any value to remember.
    """
    pass

  def __add__(left, right):
    """ The sum of two variables is an object representing just that. """
    return VariableSum(left, right)

  def __sub__(left, right):
    """
    The difference of two variables is just the sum where the right one is negated.
    (a - b) = (a + (-b))
    """
    return VariableSum(left, -right)

  def __pos__(self):
    """ +v is the same as v """
    return self

  def __neg__(self):
    """ -v is the negated version of the variable v """
    return NegatedVariable(self)

  def __eq__(left, right):
    """
    How should we determine whether two variables are equal?

    Well first off, the simple case, if both the left and right sides
    of our comparison are the same variable then they should be equal.
    However, if the left and right are different variables, they could,
    in some world, have the same value. So it wouldn't be quite fair
    to return False for such a comparison. For this sort of ambiguous
    answer, we will just throw an exception and hope this never happens
    in our proof.

    Another complication when we run across a comparison between a variable
    and the negated form of some variable. These could have the same value,
    but we can't be sure, even if they are the positive and negated form of
    the same variable. So, here we will throw an exception as well.
    """
    if isinstance(right, NegatedVariable):
      # The right variable is negated, but the left is not.
      # This is an ambiguous comparison.
      # The values of the variables could be equal, but there is no way
      # for us to answer definitively with a True or False
      raise RuntimeError("Ambiguous Equality")
    else:
      # Both variables are non-negated.
      # The "is" comparison tests whether the left and right sides
      # are the same instance of the variable class.
      return left is right

  def __ne__(left, right):
    return not (left == right)

Whew, that’s pretty weird. We seem to have referenced a bunch of classes in the methods of Variable which do not exist yet. Let’s go ahead and create those classes.

class VariableSum(object):
  """ A VariableSum represents the result of adding two Variables """

  def __init__(self, left, right):
    """ A variable sum stores the left and right elements in the addition """
    self.left = left
    self.right = right

  def __neg__(self):
    """
    To negate a sum like -(x + y),
    just negate it's parts  (-x + -y)
    """
    return VariableSum(-self.left, -self.right)

  def __eq__(left, right):
    """ Check whether a sum of variables is the same as another sum of variables. """
    # Yes, this line is confusing.
    if left.left == right.left and left.right == right.right:
      return True
    else:
      # There are other ways that sums could be equal.
      # For example, (a + b) = (b + a)
      # We are not going to implement every equality comparison, so instead
      # we will throw a NotImplementedError to indicate the if you wanted
      # to use a comparison that currently throws an error in your proof,
      # then you might consider writing more code here.
      raise NotImplementedError("Comparison not fully implemented.")

  def __neq__(left, right):
    return not (left == right)

class NegatedVariable(Variable):
  """
  Notice that Variable appears inside instead of object.
  This denotes that a NegatedVariable is really a kind of Variable.
  Technically, NegatedVariable inherits all of the methods of
  from the Variable class. So NegatedVariables know how to do all
  the same tricks, like participating in addition and subtraction.

  The exception is negation of a NegatedVariable.
  NegatedVariable has its own special definition
  of negation, which you will see below.
  """

  def __init__(self, variable):
    """ A NegatedVariable must remember what it is the negation of. """
    self.variable = variable

  def __neg__(self):
    """
    A twice negated variable is just the original variable.
    -(-v) = v
    """
    return self.variable

  def __eq__(left, right):
    if isinstance(right, NegatedVariable):
      # Both variables are negated.
      return -left == -right
    else:
      # The left variable is negated, but the right is not.
      # This is an ambiguous comparison.
      # The values of the variables could be equal, but there is no way
      # for us to answer definitively with a True or False
      raise RuntimeError("Ambiguous Equality")

  def __ne__(left, right):
    return not (left == right)

Final Test

Now that all of the pieces are in place we should be able to test whether (a - b)* really does equal a* - b*.

We will compose the complex numbers a and b from two variables each. One for the real component, and another for the imaginary component.

print "\nGeneral evaluation:"
a = ComplexNumber(Variable(), Variable())
b = ComplexNumber(Variable(), Variable())
print (a - b).conj() == a.conj() - b.conj()  # -> True

It’s true! We have shown that (a - b)* = a* - b*. But don’t take my word for it, you can run this program yourself.

Here is the python source.

That’s assuming there were no bugs in the program. Did you see any bugs? Fix them or play with the code on github.