Exercise 1.1 — Question 4
The imaginary unit i is defined as i=−1, so i2=−1.
The powers of i follow a cyclic pattern with period 4:
| Power | Value |
|---|
| i0 | 1 |
| i1 | i |
| i2 | −1 |
| i3 | −i |
| i4 | 1 |
| i5 | i |
| ⋮ | ⋮ |
Since i4=1, the cycle repeats every 4 steps.
To evaluate in for any positive integer n:
- Divide n by 4 and find the remainder r (where r∈{0,1,2,3}).
- Then in=ir.
in=i4q+r=(i4)q⋅ir=1q⋅ir=ir
Example 1: Evaluate i23
23=4×5+3⟹r=3⟹i23=i3=−i
Example 2: Evaluate i100
100=4×25+0⟹r=0⟹i100=i0=1
Example 3: Evaluate i4k+2 for any non-negative integer k
i4k+2=(i4)k⋅i2=1⋅(−1)=−1
Example 4: Evaluate i−3
i−3=i31=−i1=−i1⋅ii=−i2i=1i=i
For z1=a+bi and z2=c+di:
Addition:
z1+z2=(a+c)+(b+d)i
Subtraction:
z1−z2=(a−c)+(b−d)i
Multiplication:
z1⋅z2=(ac−bd)+(ad+bc)i
(using i2=−1)
Division (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