Math 126CX Handouts 1 - naaa. aaaa aaaaa aaaaaaa aaaaaac cad sd dd ccddf f fffds asdfvd PDF

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CHAPTER 1 MATH 126 NOTE 6 1.6 Complex Numbers 1. Complex Numbers Idea: What is the solution of 𝑥 2 = −4? 𝑥 2 = −4 has no real solution. If we try to solve this equation, we get 𝑥 = ±√−4. But this is impossible, since the square of any real number is positive. [For example, (–2)2 = 4, a positive number.] Thus, negative numbers don’t have real square roots. √−1, √−3, √−4, 𝑎𝑛𝑑 √−9 are not real numbers, they are called imaginary numbers. The imaginary number √−1 is defined by 𝑖 (imaginary unit). Because 𝑖 represents the square root of –1, it follows that 𝑖 2 = −1. For any square root of negative number √−𝑝

Example: Express each number in terms of 𝑖 √−7, √−16,

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Complex N Numbers umbers A complex number is any number that can be written in the standard form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers and 𝑖 = √−1 . In the complex number 𝑎 + 𝑏𝑖, 𝑎 is called the real part, and 𝑏 is called the imaginary part.

Real part

Imaginary part

Important: For the domain problem, we don’t consider complex numbers. For the expression √𝐴, the domain is all 𝑥 values such that 𝐴 ≥ 0. 2. Addition and Subtraction

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Example: Add or subtract and simplify each of the following. (4 + 5𝑖) − (6 − 3𝑖)

3. Multiplication Idea 1: What is (𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) Use the FIOL method. Idea 2: What is 𝑖 2 ? 𝑖 2 = −1 Example: Multiply and simplify each of the following (1 + 2𝑖)(3 − 7𝑖)

4. Conjugates and Division The conjugate of a complex number 𝑎 + 𝑏𝑖 is Product of conjugates (𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) =

Division:

𝑎+𝑏𝑖 𝑐+𝑑𝑖

Idea: Use conjugate to make the denominator real number (cancel 𝑖)

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Example: Compute and simplify each of the following 2 − 5𝑖 1 − 6𝑖

5. Complex Solutions of Quadratic Equations Example: Solve 4𝑥 2 − 24𝑥 + 37 = 0

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