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Lecture 22

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Mathematics

MTH 108

Lun- Yi Tsai

Fall

Description

MTH 108
Precalculus Mathematics II
8.3 The Complex Plane and DeMoivre’s Theorem Notes
L. Sterling
Abstract
Provide a generalization to each of the key terms listed in this section.
Complex Numbers
Any numbers that are in the form of
z = a + bi
where the following occur:
▯ a and b are real numbers
▯ z is the standard form
– a is the real part
– b is the imaginary part
▯ i is the imaginary unit
– So this notes that 2
i = i ▯ i = ▯1
Complex Numbers Examples
▯ 2+3i
– Real Part: 2
– Imaginary Part: 3
▯ 5-6i
– Real Part: 5
– Imaginary Part: -6
▯ -3+(-4)i = 3-4i
– Real Part: -3
– Imaginary Part: -4
▯ -(5)-(-2)i = -5+2i
– Real Part: -5
– Imaginary Part: 2
1 Speciﬁc Complex Numbers Examples
▯ a+0i = a
– Real Part: a
– Imaginary Part: 0
▯ In this case, this help note that the real numbers are just subsets of the complex
numbers.
▯ 0+bi = bi
– Real Part: 0
– Imaginary Part: b
▯ In this case, the complex number of bi can be classiﬁed as a pure imaginary number.
Compositions of Complex Numbers
Equality of Complex Numbers
▯
a + bi = c + di
– If and only if a = c and b = d
Sum of Complex Numbers
▯
(a + bi) + (c + di) = (a + c) + (b + d)i
Diﬀerence of Complex Numbers
▯
(a ▯ bi) + (c ▯ di) = (a ▯ c) + (b ▯ d)i
Product of Complex Numbers
▯
(a + bi) ▯ (c + di) = (ac ▯ bd) + (ad + bc)i
Quotient of Complex Numbers
a + bi
c + di
(a + bi)(c ▯ di)
=
(c + di)(c ▯ di)
= (ac ▯ b(▯d)) + (a(▯d) + bc)i
c + d2
ac + bd ▯ adi + bci
= c2+ d2
ac + bd ▯ adi + bci
= 2 2
c + d
ac + bd ▯ad + bc
= 2 2+ 2 2 i
c + d c + d
2 Conjugates
Complex Number and Conjugate
▯ The conjugate of any real number is just the real number.
▯ If z = a + bi is the complex number, then its conjugate, which can be denoted with z, and
deﬁned with the following:
z
= a + bi
= a ▯ bi
▯ The product of a complex number, which is z = a+bi, and its conjugate, which is z = a▯bi
is actually a nonnegative real number, which makes the following:
(a + bi)(a ▯ bi)
= a(a) + a(▯bi) + bi(a) + bi(▯bi)
2 2 2
= a ▯ abi + abi ▯ b i
2 ▯ 2▯
= a + (1 ▯ 1)abi ▯ b (▯1)
2 2
= a + (01)abi + b
2 2
= a + b
Complex Number’s Product and Conjugates
▯ The conjugate’s conjugate of a complex number is just the original complex number, which
I mean by the following:
(z)
= (a ▯ bi)
= a + bi
= z
Therefore : (z) = z
▯ The conjugate of the sum of two complex numbers equals the sum of their conjugates, which
I mean by the following:
(z + q)
= z + q
▯ The conjugate of the product of two complex numbers equals the sum of their conjugates,
which I mean by the following:
(z ▯ q)

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