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.
Use conjugate property (7), zn=(z)n: if w=z then zn−wn=zn−zn is purely imaginary, and zn+wn=zn+zn is real — exactly the book's Example 2.8 technique. …
Both parts reduce to the pattern w−w (purely imaginary) or w+w (real), using conjugate property (7) zn=(z)n — the same technique as the book's worked Example 2.8.
Step 1. (i) Identify the conjugate pair. Let z=2+i3. Then z=2−i3, so the given expression is z10−(z)10.
Step 2. (i) Use property (7). Since z10=(z)10, we can replace (z)10 by z10. So the expression equals z10−z10.
Step 3. (i) Show a difference w−w is always purely imaginary. For any complex w, let d=w−w. Then d=w−w=w−w=−(w−w)=−d, so d=−d; by property (9) this means d is purely imaginary.
Step 4. (i) Conclude. Taking w=z10, d=z10−z10=(2+i3)10−(2−i3)10 is purely imaginary.
Step 5. (ii) Simplify the first fraction. Rationalise 9+i19−7i by the conjugate 9−i: