Conjugate. The conjugate of z=x+iy is z=x−iy — obtained by flipping the sign of the imaginary part, equivalently by reflecting z across the real axis in the Argand plane. A key fact: the product of a complex number with its own conjugate is always a non-negative real number, zz=(x+iy)(x−iy)=x2+y2.
Ten conjugate properties (each provable directly from the definition, several proved in the text):
z1+z2=z1+z2
z1−z2=z1−z2
z1z2=z1z2
(z2z1)=z2z1,z2=0
Re(z)=2z+z
Im(z)=2iz−z
zn=(z)n, n an integer
z is real⟺z=z
z is purely imaginary⟺z=−z
z=z
Proof idea (property 1): writing z1=x1+iy1,z2=x2+iy2, z1+z2=(x1+x2)−i(y1+y2)=(x1−iy1)+(x2−iy2)=z1+z2. Proof idea (property 9): z=−z⟺x+iy=−(x−iy)=−x+iy⟺2x=0⟺x=0, i.e. z is purely imaginary.
The conjugate is the standard tool for dividing by a complex number: multiplying numerator and denominator by the conjugate of the denominator makes the denominator real (exactly like rationalising a surd).
Modulus. The modulus of z=x+iy, written ∣z∣, is ∣z∣=x2+y2 — the distance from z to the origin in the Argand plane, generalising the real-number absolute value. Note zz=∣z∣2.
Both properties follow by writing z=x+iy, z=x−iy (Definition 2.3), and comparing the two sides directly — exactly as the book verifies the analogous 'purely imaginary' property.
Step 1. (i) Set up. Let z=x+iy with x,y∈R; then by definition z=x−iy.
Step 2. (i) Prove the forward direction. If z is real then y=0, so z=x and z=x−i(0)=x=z. Hence z=z.
Step 3. (i) Prove the converse. Suppose z=z. Then x+iy=x−iy⇒2iy=0⇒y=0 (since 2i=0). So z=x is real.
Step 4. (i) Conclude. Combining both directions, z is real ⟺z=z.
Step 5. (ii) Compute z+z.z+z=(x+iy)+(x−iy)=2x. Since Re(z)=x, dividing by 2 gives Re(z)=2z+z. …