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Lecture 22

# MTH 108 Lecture 22: 8.3 The Complex Plane and DeMoivre's Theorem Notes Premium

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School
Department
Mathematics
Course
MTH 108
Professor
Lun- Yi Tsai
Semester
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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