5.9: Imaginary + Complex Numbers -Defining i -Simplifying negative radicands -Powers of i -Solving equations -Complex numbers -Operations with complex #s

Imaginary units Consider the equation 2x = 0 You end up with x 2 = -1 There is no real number that, when squared, equals -1 We define the imaginary unit, i, to be the square root of -1… and i 2 = -1 We can then simplify negative radicands (if the index is even) in terms of i

Example 9-1a Simplify. Answer:

Example 9-1b Simplify. Answer:

Example 9-1c Simplify. a. b. Answer:

Combining terms with negative radicands Recall that by definition, i equals the square root of negative 1 and i 2 = -1 When combining two or more terms with negative radicands, always rewrite each radical in terms of i first!!!

Example 9-2a Answer: = 6 Simplify.

Example 9-2b Answer: Simplify.

Example 9-2c Answer: –15 Answer: Simplify. a. b.

Higher powers of i i raised to ANY power equals either 1, -1, i or –i For this reason, your answer should NEVER contain i raised to a power To simplify, rewrite as i 2 raised to a power, or as i * (i 2 raised to a power) Ex: i 14 = (i 2 ) 7 = (-1) 7 = -1 Ex. i 29 = i * i 28 = i*(i 2 ) 14 = i* (-1) 14 = i*1 = i

Example 9-3a Simplify Multiplying powers Power of a Power Answer:

Example 9-3b Answer: i Simplify.

Solving equations with squared term Isolate the squared term/expression first Then take the square root of each side! REMEMBER when you take the root yourself, stick the ± in front Then simplify the radical, using i if necessary

Example 9-4a Solve Answer: Original equation Subtract 20 from each side. Divide each side by 5. Take the square root of each side.

Example 9-4b Solve Answer:

Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers That is, a complex number contains two parts, a real part (a) and an imaginary part (bi) Examples: 4 + 5i, 7 – 2i Also: 4 (can be written as 4 + 0i) Also: -3i (can be written as 0 – 3i)

Equality of complex numbers Two complex numbers a + bi and c + di are equal iff a = c and b = d If confused, set the coefficients of the I term equal to each other and solve for the variable Then you can set the “real” parts equal and solve

Find the values of x and y that make the equation true. Example 9-5a Set the real parts equal to each other and the imaginary parts equal to each other. Real parts Divide each side by 2. Imaginary parts Answer:

Find the values of x and y that make the equation true. Example 9-5b Answer:

Operations with complex #s Adding/subtracting – just add/subtract the “real” components and the imaginary components Multiplying – distribute or use FOIL.. Just remember that i 2 = -1 Rationalizing (may need to use the COMPLEX CONJUGATE)

Simplify. Example 9-6a Answer: Commutative and Associative Properties

Simplify. Example 9-6b Commutative and Associative Properties Answer:

Simplify. a. b. Example 9-6c Answer:

Application Complex #s are used with electricity.. Except they use j instead of i (the letter i is used elsewhere) E = I * Z, where E is the voltage, I is the current, and Z is the impedance Not that important to know.. Just an example of multiplying complex #s

Answer: The voltage isvolts. Example 9-7a Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formula Find the voltage in a circuit with current j amps and impedance 3 – 6 j ohms. Electricity formula FOIL Multiply. Add.

Example 9-7b Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I Z. Find the voltage in a circuit with current 1 – 3 j amps and impedance j ohms. Answer: 9 – 7 j

and are conjugates. Example 9-8a Multiply. Answer: Standard form Simplify.

Example 9-8b Simplify. Multiply. Answer: Standard form Multiply by

Simplify. a. b. Example 9-8c Answer: