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Operations on Complex Numbers
Mathematics 4
November 29, 2011
Mathematics 4 () Operations on Complex Numbers November 29, 2011 1 / 18
Review of Multiplication of Complex Numbers
Find the product of 4 + 4i and −2− 3i
1. Multiply Algebraically
(4 + 4i)(−2− 3i) = −8− 12i− 8i− 12i2
= −8− 20i+ 12
= 4− 20i
Mathematics 4 () Operations on Complex Numbers November 29, 2011 2 / 18
Review of Multiplication of Complex Numbers
Find the product of 4 + 4i and −2− 3i
2. Multiply in their polar forms
(4 + 4i)(−2− 3i) = (4√2 cis 45o) · (
√13 cis 236.31o)
= (4√2 ·√13) cis(45 + 236.31)o
= 4√26 cis 281.31o
= 4− 20i
Mathematics 4 () Operations on Complex Numbers November 29, 2011 3 / 18
Review of Multiplication of Complex Numbers
Rule for Multiplication of Complex Numbers in Polar Form
Given:
z1 = r1 cisα z2 = r2 cisβ
z1 · z2 = (r1 · r2) cis(α+ β)
Mathematics 4 () Operations on Complex Numbers November 29, 2011 4 / 18
Review of Multiplication of Complex Numbers
Rule for Multiplication of Complex Numbers in Polar Form
Given:
z1 = r1 cisα z2 = r2 cisβ
z1 · z2 = (r1 · r2) cis(α+ β)
Mathematics 4 () Operations on Complex Numbers November 29, 2011 4 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 =
1z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1
z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 =
r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θ
z2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 =
(r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 = (r cis θ) · (r cis θ)
= r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θ
z3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 =
(r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ)
= r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
Given: z = r cis θ
z0 = 1z1 = r cis θz2 = (r cis θ) · (r cis θ) = r2 cis 2θz3 = (r2 cis 2θ) · (r cis θ) = r3 cis 3θ
Mathematics 4 () Operations on Complex Numbers November 29, 2011 5 / 18
Raising Complex Numbers to a Power
De Moivre’s Theorem
(r cis θ)n = rn cis(n · θ)
Mathematics 4 () Operations on Complex Numbers November 29, 2011 6 / 18
De Moivre’s Theorem
Example 1: Find (√2 cis 20o)10
(√2 cis 20o)10 = (
√2)10 cis(10 · 20)o
(√2 cis 20o)10 = 32 cis 200o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 7 / 18
De Moivre’s Theorem
Example 1: Find (√2 cis 20o)10
(√2 cis 20o)10 = (
√2)10 cis(10 · 20)o
(√2 cis 20o)10 = 32 cis 200o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 7 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)0 = 1 cis 0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)1 =
√2 cis 20o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)2 = 2 cis 400
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)3 = 2
√2 cis 60o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)4 = 4 cis 80o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)5 = 4
√2 cis 100o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)6 = 8 cis 120o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)7 = 8
√2 cis 140o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)8 = 16 cis 160o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)9 = 16
√2 cis 180o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s TheoremExample 1: Find (
√2 cis 20o)10
(√2 cis 20o)10 = 32 cis 200o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 8 / 18
De Moivre’s Theorem
Example 2: Find (3 cis 120o)5
(3 cis 120o)5 = 35 cis(5 · 120)o
(3 cis 120o)5 = 243 cis 600o = 243 cis 240o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 9 / 18
De Moivre’s Theorem
Example 2: Find (3 cis 120o)5
(3 cis 120o)5 = 35 cis(5 · 120)o(3 cis 120o)5 = 243 cis 600o = 243 cis 240o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 9 / 18
De Moivre’s TheoremExample 2: Find (3 cis 120o)5
(3 cis 120o)0 = 1 cis 0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
De Moivre’s TheoremExample 2: Find (3 cis 120o)5
(3 cis 120o)1 = 3 cis 120o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
De Moivre’s TheoremExample 2: Find (3 cis 120o)5
(3 cis 120o)2 = 9 cis 240o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
De Moivre’s TheoremExample 2: Find (3 cis 120o)5
(3 cis 120o)3 = 27 cis 360o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
De Moivre’s TheoremExample 2: Find (3 cis 120o)5
(3 cis 120o)4 = 81 cis 480o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
De Moivre’s TheoremExample 2: Find (3 cis 120o)5
(3 cis 120o)5 = 243 cis 600o = 243 cis 240o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 10 / 18
De Moivre’s Theorem
Example 3: Find (1− i)8
(1− i)8 = (√2 cis 315o)8
(√2 cis 315o)8 = 16 cis 2520o = 16 cis 0 = 16
Mathematics 4 () Operations on Complex Numbers November 29, 2011 11 / 18
De Moivre’s Theorem
Example 3: Find (1− i)8
(1− i)8 = (√2 cis 315o)8
(√2 cis 315o)8 = 16 cis 2520o = 16 cis 0 = 16
Mathematics 4 () Operations on Complex Numbers November 29, 2011 11 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)0 = (√2 cis(−45)o)0 = 1 cis 0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)1 = (√2 cis(−45)o)1 =
√2 cis(−45)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)2 = (√2 cis(−45)o)2 = 2 cis(−90)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)3 = (√2 cis(−45)o)3 = 2
√2 cis(−135)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)4 = (√2 cis(−45)o)4 = 4 cis(−180)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)5 = (√2 cis(−45)o)5 = 4
√2 cis(−225)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)6 = (√2 cis(−45)o)6 = 8 cis(−270)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)7 = (√2 cis(−45)o)7 = 8
√2 cis(−315)o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)8 = (√2 cis(−45)o)8 = 16 cis(−360)o = 16 cis 0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s TheoremExample 3: Find (1− i)8
(1− i)8 = (√2 cis(−45)o)8 = 16 cis(−360)o = 16 cis 0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 12 / 18
De Moivre’s Theorem
De Moivre’s Theorem can be used to find the nth of a complex number:
Find the three cube roots of −2− i2√3.
We wish to find values of r and θ such that:
(r cis θ)3 = −2− i2√3
Using De Moivre’s Theorem and expressing the complex number in polarform:
r3 cis 3θ = 4 cis 240o
Therefore:
r3 = 4 and 3θ = 240o + k · 360o, k ∈ Zr = 3√4 and θ = 80o + k · 120o, k ∈ Z
Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18
De Moivre’s Theorem
De Moivre’s Theorem can be used to find the nth of a complex number:
Find the three cube roots of −2− i2√3.
We wish to find values of r and θ such that:
(r cis θ)3 = −2− i2√3
Using De Moivre’s Theorem and expressing the complex number in polarform:
r3 cis 3θ = 4 cis 240o
Therefore:
r3 = 4 and 3θ = 240o + k · 360o, k ∈ Z
r = 3√4 and θ = 80o + k · 120o, k ∈ Z
Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18
De Moivre’s Theorem
De Moivre’s Theorem can be used to find the nth of a complex number:
Find the three cube roots of −2− i2√3.
We wish to find values of r and θ such that:
(r cis θ)3 = −2− i2√3
Using De Moivre’s Theorem and expressing the complex number in polarform:
r3 cis 3θ = 4 cis 240o
Therefore:
r3 = 4 and 3θ = 240o + k · 360o, k ∈ Zr = 3√4 and θ = 80o + k · 120o, k ∈ Z
Mathematics 4 () Operations on Complex Numbers November 29, 2011 13 / 18
Finding the nth roots of complex numbers
For any complex number r cis θ and n ∈ Z+:
The nth roots of r cis θ is given by:
n√r cis θk
θk =θ + k360o
n, k = 0, 1, 2, ...(n− 1)
Mathematics 4 () Operations on Complex Numbers November 29, 2011 14 / 18
Finding the nth roots of complex numbers
Example 1: Find the fourth roots of 16 cis 120o
r4 cis 4θ = 16 cis 120o
r4 = 16 and 4θ = 120o + k360o
r = 2 and θ = 30o + k90o
2 cis 30o
2 cis 120o
2 cis 210o
2 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
Finding the nth roots of complex numbers
Example 1: Find the fourth roots of 16 cis 120o
r4 cis 4θ = 16 cis 120o
r4 = 16 and 4θ = 120o + k360o
r = 2 and θ = 30o + k90o
2 cis 30o
2 cis 120o
2 cis 210o
2 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
Finding the nth roots of complex numbers
Example 1: Find the fourth roots of 16 cis 120o
r4 cis 4θ = 16 cis 120o
r4 = 16 and 4θ = 120o + k360o
r = 2 and θ = 30o + k90o
2 cis 30o
2 cis 120o
2 cis 210o
2 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
Finding the nth roots of complex numbers
Example 1: Find the fourth roots of 16 cis 120o
r4 cis 4θ = 16 cis 120o
r4 = 16 and 4θ = 120o + k360o
r = 2 and θ = 30o + k90o
2 cis 30o
2 cis 120o
2 cis 210o
2 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
Finding the nth roots of complex numbers
Example 1: Find the fourth roots of 16 cis 120o
r4 cis 4θ = 16 cis 120o
r4 = 16 and 4θ = 120o + k360o
r = 2 and θ = 30o + k90o
2 cis 30o
2 cis 120o
2 cis 210o
2 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 15 / 18
Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o
(2 cis 30)0
(2 cis 120)0
(2 cis 210)0
(2 cis 300)0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o
(2 cis 30)1
(2 cis 120)1
(2 cis 210)1
(2 cis 300)1
Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o
(2 cis 30)2
(2 cis 120)2
(2 cis 210)2
(2 cis 300)2
Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o
(2 cis 30)3
(2 cis 120)3
(2 cis 210)3
(2 cis 300)3
Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o
(2 cis 30)4
(2 cis 120)4
(2 cis 210)4
(2 cis 300)4
Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
Finding the nth roots of complex numbersExample 1: Find the fourth roots of 16 cis 120o
(2 cis 30)4
(2 cis 120)4
(2 cis 210)4
(2 cis 300)4
Mathematics 4 () Operations on Complex Numbers November 29, 2011 16 / 18
Finding the nth roots of complex numbers
Example 2: Find the cube roots of −8
r3 cis 3θ = −8 = 8 cis 180o
r3 = 8 and 3θ = 180o + k360o
r = 2 and θ = 60o + k120o
2 cis 60o
2 cis 180o = −22 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
Finding the nth roots of complex numbers
Example 2: Find the cube roots of −8r3 cis 3θ = −8 = 8 cis 180o
r3 = 8 and 3θ = 180o + k360o
r = 2 and θ = 60o + k120o
2 cis 60o
2 cis 180o = −22 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
Finding the nth roots of complex numbers
Example 2: Find the cube roots of −8r3 cis 3θ = −8 = 8 cis 180o
r3 = 8 and 3θ = 180o + k360o
r = 2 and θ = 60o + k120o
2 cis 60o
2 cis 180o = −22 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
Finding the nth roots of complex numbers
Example 2: Find the cube roots of −8r3 cis 3θ = −8 = 8 cis 180o
r3 = 8 and 3θ = 180o + k360o
r = 2 and θ = 60o + k120o
2 cis 60o
2 cis 180o = −22 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
Finding the nth roots of complex numbers
Example 2: Find the cube roots of −8r3 cis 3θ = −8 = 8 cis 180o
r3 = 8 and 3θ = 180o + k360o
r = 2 and θ = 60o + k120o
2 cis 60o
2 cis 180o = −22 cis 300o
Mathematics 4 () Operations on Complex Numbers November 29, 2011 17 / 18
Finding the nth roots of complex numbersExample 2: Find the cube roots of −8
(2 cis 60)0 (2 cis 180)0 (2 cis 300)0
Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18
Finding the nth roots of complex numbersExample 2: Find the cube roots of −8
(2 cis 60)1 (2 cis 180)1 (2 cis 300)1
Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18
Finding the nth roots of complex numbersExample 2: Find the cube roots of −8
(2 cis 60)2 (2 cis 180)2 (2 cis 300)2
Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18
Finding the nth roots of complex numbersExample 2: Find the cube roots of −8
(2 cis 60)3 (2 cis 180)3 (2 cis 300)3
Mathematics 4 () Operations on Complex Numbers November 29, 2011 18 / 18