Aim #3.2: How do we perform operations on complex numbers?

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Aim #3.2: How do we perform operations on complex numbers? What is a complex number? Well, let’s start with what is an imaginary number? Recall previously, we said that we cannot take the square root of a negative number? √-9 = can’t do because there is not an identical number times itself that makes -9. Well, if only, we could get rid of that pesky negative, we could actually take the square root. Now, with imaginary numbers, we actually can!

What are imaginary numbers? √-1 = i and, also very important, (√-1)2 = -1 So, I can re-write √-9 as √9 ∙√-1. I know √9 is 3, -3 and I know that √-1 = i. So, my final answer would be √-9 = ±3i. Try some: √-25 = √-64 = √-49x2 =

What are complex numbers? Now, on to complex numbers.  These are numbers that are made up of a combination of real and imaginary numbers. The format is: a + bi, where “a” and “b” are real numbers and “i” is the imaginary part (√1). EX: -2 + 3i 4 + 7i 4 – i -6 – 9i

Aim #3.7: How do we perform operations on complex numbers? Add/Subtract complex numbers: Just like terms with variables and radical expressions, complex numbers addition and subtraction are just combining like terms. You combine the real parts together and the imaginary parts together, remembering that the exponents on variables NEVER change. EX: a)(-3 + 4i) + (5 – i) b)(4 – 3i) – (-7 – 2i) c)(-6 + 15i) – (3 – 22i)

Aim #3.7: How do we perform operations on complex numbers? Multiplying complex numbers: Because these are binomials, we multiply just like we would multiply any other binomials. You can use the distributive property, foil, box method, multiplication method, etc. Just remember, that exponents DO change when we multiply! Oh, and don’t forget to check i2 !!! Remember answer should be in STANDARD FORM!

EX: a) (3 + 2i)(-2 + 5i) d) -4i(6 + 7i) b) (-1 + 2i)(-3 – 7i) e) (5 – 2i)2 c) (8 – 3i)(2 – i)

p. 108 #5-12, 21 – 29, 37-44

Summary: Answer in complete sentences. Explain the importance of imaginary numbers. What are the two parts that make up imaginary numbers? Give an example and label each part. Explain how to add or subtract complex numbers.