Exercise 1.2 — Question 1
For two complex numbers z1=a+bi and z2=c+di where a,b,c,d∈R:
z1+z2=(a+c)+(b+d)i
z1−z2=(a−c)+(b−d)i
Using FOIL and i2=−1:
z1⋅z2=(ac−bd)+(ad+bc)i
Multiply numerator and denominator by the conjugate of the denominator:
z2z1=(c+di)(c−di)(a+bi)(c−di)=c2+d2(ac+bd)+(bc−ad)i
If z=a+bi, then:
iz=i(a+bi)=ai+bi2=ai−b=−b+ai
Therefore:
- Re(iz)=−b=−Im(z)
- Im(iz)=a=Re(z)
Multiplying a complex number by i rotates it by 90° counterclockwise in the Argand plane.
| Type | Condition | Conjugate Condition |
|---|
| Purely Real | b=0 | z=zˉ |
| Purely Imaginary | a=0 | z=−zˉ |
Example 1: If z1=3+2i and z2=1−4i, find z1+z2, z1−z2, and z1⋅z2.
z1+z2=(3+1)+(2−4)i=4−2i
z1−z2=(3−1)+(2+4)i=2+6i
z1⋅z2=(3)(1)−(2)(−4)+[(3)(−4)+(2)(1)]i=3+8+(−12+2)i=11−10i
Example 2: Find Re(iz) and Im(iz) for z=5−3i.
iz=i(5−3i)=5i−3i2=3+5i
Re(iz)=3=−(−3)=−Im(z)✓
Im(iz)=5=Re(z)✓