Complex Conjugates and Properties of Complex Numbers

The content in today's blog is taken from Bruce E. Meserve's Fundamental Concepts of Algebra.

The presentation builds on a previous blog I did on the definitions needed to build the complex numbers from the real numbers.

Definition 1: Complex Conjugate

For any complex number a+bi (see Definition 6, here), the complex conjugate is the form a-bi.


In other words, for the complex number (a,b), its complex conjugate is (a,-b). The complex conjugate of (a,-b) is likewise (a,b).


Definition 2: Norm of complex number: n(z)

The norm of a complex number which is represented as n(z) is the product of a complex number with its conjugate.

This definition is demonstrated in the following lemma:

Lemma 1: For any complex number (a,b), its norm is a real number: (a² + b²,0)

Proof:

(1) Let (a,b) be any complex number.

(2) It's complex conjugate is (a,-b) [see Definition 1 above]

(3) Using the definition for multiplication of complex numbers (see Definition 3, here) and the definition for norms (see Definition 2 above):

It's norm is (a,b)*(a,-b) = (a*a - (b*-b),a*(-b) + (b*a)) = (a² + b²,0)

(4) We can see that this is a real number (see Theorem 15, here)

QED


Definition 3: Absolute value of a complex number (the modulus)

The absolute value of a complex number is the nonnegative square root of the norm so that:

if z = (a,b), then abs(z) = √a² + b²


Definition 4: Division of complex numbers

(a,b) / (c,d) = (p,q) if and only if (a,b) = (c,d)*(p,q)


That this definition is well-defined is established in the next theorem.

Theorem 2: For the set of complex numbers, division by an nonzero number is well-defined.

Proof:

(1) Let (a,b), (c,d) be two complex numbers.

(2) Using the Definition of Multiplication (see Definition 3, here):

(c,d)*(p,q) = (cp - dq,cq + dp)

(3) So, if (a,b) = (c,d)*(p,q), then we have (see Definition 4, here):

a = cp - dq b = cq + dp

(4) Solving the first equation in terms of p and the second equation in terms of q gives us:

p = (a + dq)/c q = (b - dp)/c

(5) Combining the equations for p, we have:



(6) Now, rearranging the equation gives us:



And solving for p gives us:




which results in:




(7) Combining the equations for q, we have:



(8) Now, rearranging the equation gives us:



And solving for q gives us:



which results in:



(9) By assumption (c,d) is not nonzero so c² + d² is nonzero

(8) Since a,b,c,d are real numbers and the operations on real numbers are well-defined, it follows that p,q are well-defined and therefore division of (a,b) by (c,d) is well-defined.

QED


Theorem 3: The norm of a product is equal to the product of the norms.

Proof:

(1) Using Lemma 1 above, the norm for (a,b) is (a² +b²,0)

(2) Likewise, the norm for (c,d) is (c² + d²,0)

(3) Using the definition of multiplication for complex numbers (see Definition 3, here),

(a,b)*(c,d) = (a*c -b*d,a*d + b*c)

(4) Norm(a*c - b*d,a*d + b*c) is ([ac-bd]² + [ad+bc]²,0)

(5) Finally:

(ac - bd)² + (ad + bc)² = a²c² - 2abcd + b²d² + a²d² + 2abcd + b²c² =

= a²c² + b²d² + a²d² + b²c² = (a² + b²)(c² + d²)

which shows that:

Norm[(a,b)*(c,d)] = Norm(a,b)*Norm(c,d)

QED


Theorem 4:

The absolute value of a sum of complex numbers is less than or equal to the sum of the absolute values:

abs(z₁ + z₂) ≤ abs(z₁) + abs(z₂)

Proof:

(1) Assume that abs(z₁ + z₂) is greater than abs(z₁) + abs(z₂)

(2) Let:

z₁ = a + bi

z₂ = c + di

(3) Using the definition of absolute values for complex numbers (see Definition 3 above):

√(a + c)² + (b + d)² is greater than √a² + b² + √c² + d²

(4) Squaring both sides gives us:

(a + c)² + (b+d)² is greater than a² + b² + 2√(a² + b²)(c² + d²) + c² + d²

So that:

a² + c² + b² + d² + 2ac + 2bd is greater than a² + b² + c² + d² + 2√(a² + b²)(c² + d²)

So that:

ac + bd is greater than √(a² + b²)(c² + d²)

(5) Squaring both sides gives us:

a²c² + b²d² + 2abcd is greater than a²c² + b²d² + a²d² + b²c²

So that:

0 is greater than a²d² + b²c² +-2abcd

So that:

0 is greater than (bc - ad)²

(6) But this is impossible since (bc -ad)² ≥ 0.

(7) So, we reject our assumption in step #1.

QED


References

• Bruce E. Meserve, Fundamental Concepts of Algebra, Dover Books, 1981.