Download Powers and Roots of Complex Numbers

POWERS AND
ROOTS OF
COMPLEX
NUMBERS
DR. SHILDNECK
MULTIPLYING COMPLEX NUMBERS
Remember how to multiply from yesterday… It follows
Given 𝑧1 = 𝑟1 𝑐𝑖𝑠𝜃1 = 𝑟1 (𝑐𝑜𝑠𝜃1 + 𝑖 𝑠𝑖𝑛𝜃1 ),
𝑧1
𝑛
𝑧1 2 = 𝑟1 𝑐𝑖𝑠𝜃1 𝑟1 𝑐𝑖𝑠𝜃1 = 𝑟1 𝑟1 𝑐𝑖𝑠 𝜃1 + 𝜃1 = 𝑟1 2 𝑐𝑖𝑠(2𝜃1 )
𝑧1 3 = 𝑟1 2 𝑐𝑖𝑠(2𝜃1 ) 𝑟1 𝑐𝑖𝑠𝜃1 = 𝑟1 2 𝑟1 𝑐𝑖𝑠 2𝜃1 + 𝜃1 = 𝑟1 3 𝑐𝑖𝑠(3𝜃1 )
𝑧1 4 = 𝑟1 3 𝑐𝑖𝑠(3𝜃1 ) 𝑟1 𝑐𝑖𝑠𝜃1 = 𝑟1 3 𝑟1 𝑐𝑖𝑠 3𝜃1 + 𝜃1 = 𝑟1 4 𝑐𝑖𝑠(4𝜃1 )
⋮
= 𝑟1 𝑛−1 𝑐𝑖𝑠((𝑛 − 1)𝜃1 ) 𝑟1 𝑐𝑖𝑠𝜃1 = 𝑟1 𝑛−1 𝑟1 𝑐𝑖𝑠 (𝑛 − 1)𝜃1 +𝜃1 = 𝑟1 𝑛 𝑐𝑖𝑠(𝑛𝜃1 )
DEMOIVRE’S THEOREM
If the polar form of a complex number is 𝑧 = 𝑟𝑐𝑖𝑠𝜃 = 𝑟 (𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃),
𝒛𝒏 = 𝒓𝒏 𝒄𝒊𝒔(𝒏𝜽) = 𝒓𝒏 (𝐜𝐨𝐬 𝒏𝜽 + 𝒊 𝐬𝐢𝐧 𝒏𝜽)
EXAMPLE
[Example 1] Find 4 + 4 3𝑖
6
and express the answer in rectangular form.
ROOTS OF COMPLEX NUMBERS
Remember from the Fundamental Theorem of Algebra that polynomials of degree n
have exactly n zeros (roots) in the complex number system.
This means that x4 = 256, which we can re-write as x4 – 256 = 0, has exactly FOUR
roots.
This indicates that the number 256 has exactly four 4th roots in the complex numbers
(4, -4, 4i, -4i).
In general, all nonzero complex numbers have exactly n distinct nth roots.
That is, they have two square roots, three cube roots, four fourth roots, etc.
DEMOIVRE’S THEOREM (FOR ROOTS)
For a positive integer p, the complex number 𝑧 = 𝑟 (𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃), has exactly p distinct pth
roots. They can be found by
𝟏
𝒓𝒑
𝜽 + 𝟐𝒏𝝅
𝜽 + 𝟐𝒏𝝅
𝐜𝐨𝐬
+ 𝒊 𝐬𝐢𝐧
𝒑
𝒑
Where n = 0, 1, 2, …, (p-1).
EXAMPLE
[Example 2] Find the cube roots of 3 + 4𝑖 . (Round answers to the nearest hundredth if
necessary)
EXAMPLE
[Example 3] Find the fourth roots of 4 − 4𝑖 . (Round answers to the nearest hundredth if
necessary)
ROOTS OF COMPLEX NUMBERS
Observations about the roots of complex numbers
The roots of complex numbers all have the same modulus (which can be
thought of as the radius of a circle).
When plotting the roots, you will notice that the roots are equally spaced
around that circle.
ROOTS OF UNITY (SPECIAL CASE)
Finding the pth roots of 1
When written in polar form, one is written as r = 1.
Thus, the modulus is 1, which means the pth roots of 1 lie on the unit circle.
Like all nonzero complex numbers, 1 has p distinct pth roots in the complex
number system.
EXAMPLE
[Example 4] Find the complex fourth roots of 1.
ASSIGNMENT
Handout # 53-75 odd