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Published byEgbert Dickerson Modified over 9 years ago
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1 What you will learn Lots of vocabulary! A new type of number! How to add, subtract and multiply this new type of number How to graph this new type of number How to simplify expressions involving this new type of number
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Objective: 5.4 Complex Numbers 2 Lions, Tigers and Complex Numbers…oh my! What is: Mathematicians created an “imaginary unit” to help us do math with negative square roots. i = i is called the imaginary unit.
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Objective: 5.4 Complex Numbers 3 Simplifying with “i” Example: How do we simplify What is i 2 ? Any guesses for 5i 2 ?
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Objective: 5.4 Complex Numbers 4 You Try! Simplify: 1. 2. 3. 12i 2
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Objective: 5.4 Complex Numbers 5 Solving a Quadratic Equation Solve 3x 2 + 10 = -26
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Objective: 5.4 Complex Numbers 6 You Try! Solve 2x 2 + 26 = -10
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Objective: 5.4 Complex Numbers 7 More Vocabulary We are adding to our “types” of numbers. We had whole, integer, rational, irrational and real. We are adding “complex” numbers. A complex number is of the form a + bi where a and b are real numbers. This is called standard form. “a” is the real part. “bi” is the imaginary part. If you have both an “a” term and a “bi” term, the number is an imaginary number. If you have only a “bi” term, the number is a pure imaginary number.
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Objective: 5.4 Complex Numbers 8 “Types” of Numbers – Expanded! Whole 1, 2, 3 Integer -3, -2, 0 Rational ¾, ½, -1/2 Real Numbers Irrational Imaginary Numbers 2+3i, 5-5i Pure Imaginary Numbers -4i, 6i Complex Numbers
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Objective: 5.4 Complex Numbers 9 Graphing Complex Numbers We graph complex numbers in the “complex plane”. Real axis Imaginary axis
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Objective: 5.4 Complex Numbers 10 Graphing continued… Plot the complex numbers in the complex plane. 2 – 3i-3 + 2i4i Real axis Imaginary axis 2i 4i -2i -4i 24 -4
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Objective: 5.4 Complex Numbers 11 Adding and Subtracting Complex Numbers Like adding and subtracting “like terms” 1. ( 4 – i) + (3 + 2i) 2. (7 – 5i) – (1 – 5i) 3. 6 – ( -2 + 9i) + (-8 + 4i)
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Objective: 5.4 Complex Numbers 12 You Try! Write the expression as a complex number in standard form. 1. ( -1 + 2i ) + (3 + 3i) 2. (2 – 3i) – (3 – 7i) 3. 2i – (3 + i) + (2 – 3i)
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Objective: 5.4 Complex Numbers 13 Multiplying Complex Numbers You will use the distributive property or FOIL. Write the expression as a complex number in standard form: 1. 5i(-2 + i) 2. (7 – 4i)(-1 + 2i)
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Objective: 5.4 Complex Numbers 14 You Try Write the expression as a complex number in standard form: 1. -3i(3 + 2i) 2. (3 + 2i)(5 - 2i)
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Objective: 5.4 Complex Numbers 15 Complex Conjugates What happens when we multiply: (6 + 3i)(6 – 3i) (6 + 3i)(6 – 3i) are called complex conjugates.
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Objective: 5.4 Complex Numbers 16 Simplifying “Division” Problems We can use complex conjugates to simplify complex numbers of the form:
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Objective: 5.4 Complex Numbers 17 You Try Write the quotient in standard form.
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Objective: 5.4 Complex Numbers 18 Homework page 277, 18-34 even, 38-42 even, 48-52 even, 56-60 even, 80-85 all
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