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Complex Numbers and Solving Equations Honors Algebra II with Trig. Ms. Lee Essential Stuff Essential Question • How do you simplify expressions involving complex numbers? Essential Vocabulary • Imaginary Number • Complex Number • Complex Conjugate Imaginary Numbers • Not all quadratic equations have real number solutions, so to overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit 𝐢 Imaginary Unit, 𝒊 𝒊= −𝟏 𝟐 𝒊 = −𝟏 • These properties allow us to simply the square root of a negative number. Complex Numbers • A complex number is a number that contains a real and an imaginary part. • The standard form: Real 𝒂 + 𝒃𝒊 Imaginary Add/Subtract/Multiply Add and Subtract 1. 2. Combine Like Terms Write in Standard Form Multiply 1. 2. 3. FOIL Replace 𝒊𝟐 with −𝟏 Combine Like Terms/Standard Form Dividing with Complex Numbers Divide • Multiply the numerator and denominator by the complex conjugate • Complex conjugates are two complex numbers of the form: 𝒂 + 𝒃𝒊 and 𝒂 − 𝒃𝒊 Complex Conjugates The product of complex conjugates will always be a real number. Homework • Homework 5.6: Complex Numbers Homework: • Page 35 Problems 4, 6, 8, 14 • Pages 45-48 Problems 16 – 32 even, 51, 53, 54, 56, 74
