Complex Numbers MATH 018 Combined Algebra S. Rook

2 Overview Section 10.7 in the textbook: –Introduction to imaginary numbers –Multiply and divide square roots with negative numbers –Addition and subtraction of complex numbers –Multiplication of complex numbers –Division of complex numbers –Powers of i

3 Introduction to Imaginary Numbers

44 Thus far we have discussed numbers exclusively in the real number system Consider –Does not exist in the real number system –However it DOES exist in the complex number system Consider if we use the product rule to rewrite as is commonly encountered in the complex number system and is called the imaginary unit or i –Rewriting is called “poking out the i”

55 Introduction to Imaginary Numbers (Continued) Imaginary unit: –Thus, = ? –Any number with an i is called an imaginary number –Also by definition:

66 Introduction to Imaginary Numbers (Continued) In summary: –A negative under an even root index (e.g. ): Does NOT exist in the real number system DOES exist in the complex number system –A negative under an odd root index (e.g. ) ONLY exists in the real number system

Introduction to Imaginary Numbers (Example) Ex 1: Simplify in either the real or complex number system: a) b) c) 7

8 Multiply and Divide Square Roots with Negative Numbers

99 First step is to ALWAYS “poke out the i” WRONG Product/Quotient rule for even roots ONLY applies to positive numbers under the radical! CORRECT After “poking out the i” use the product or quotient rule on the REAL roots –After checking whether the REAL roots can be simplified of course –Only acceptable to have i in the final answer – i 2 can be simplified to -1

Multiply and Divide Square Roots with Negative Numbers (Example) Ex 2: Simplify in the complex number system: a) b) c) 10

11 Adding and Subtracting Complex Numbers

12 Definition of Complex Numbers We have discussed the complex number system in relation to the imaginary unit i Complex Number: a number written in the standard format a + bi where: a and b are real numbers a is the real part bi is the imaginary part Special cases: –If b = 0, the number is real (Chapters 1 – 10.6) –If a = 0, the number is imaginary (10.7) –What are some examples of each?

13 Adding and Subtracting Complex Numbers Recall the definition of like terms: –e.g. (3a + 2b) + (5a – b) = ? What are the two components of a complex number? Think of these as “two different terms” and treat adding complex numbers as adding like terms –e.g. (3 + 2i) + (5 – i) = ? How would we simplify (6a – 8b) – (2a – 7b)? Thus, how would we simplify (6 – 8i) – (2 – 7i)? Write the result in standard form

Addition and Subtraction of Complex Numbers (Example) Ex 3: Simplify and write in a + bi format: a)c) b)d) 14

15 Multiplication of Complex Numbers

16 Multiplication of Complex Numbers Consider multiplying 3i · 2i : –Treat in two steps: multiply the real numbers and then multiply the imaginary units (if they exist) –Remember that it is only acceptable to leave i in the final answer –What did we say about i 2 ? To multiply complex numbers in general: –Use the distributive property or FOIL

Multiplication of Complex Numbers (Example) Ex 4: Simplify and write in a + bi format: a) b) c) 17

18 Division of Complex Numbers

19 Complex Conjugate Consider multiplying (3 + i)(3 – i) –What do you notice? Complex conjugate: the same complex number with real part a and imaginary part bi except with the opposite sign –Very similar to conjugates when we discussed rationalizing in section 10.5 –What would be the complex conjugate of (2 – i)?

20 Division of Complex Numbers Goal is to write the quotient of complex numbers in the format a + bi Consider writing in standard form –How many terms are in the denominator? How would we write in standard form? Finally, consider writing in standard form –How many terms were in the denominator for the previous examples? –What happens when we employ the complex conjugate? Goal is to get a single real number in the denominator

Division of Complex Numbers (Example) Ex 5: Divide and then write in a + bi format: a)c) b) 21

22 Powers of i

23 Powers of i So far we have discussed two powers of i: i i 2 = -1 We can use these to obtain subsequent powers of i: i 3 = -i i 4 = 1 We can continue with subsequent powers until we notice a pattern

24 Powers of i (Continued) Thus, there are only 4 distinct powers of i i, i 2 = -1, i 3 = -i, i 4 = 1 How can we use this to derive ANY power of i? –e.g. Is i 10 equivalent to i, -1, -i, or 1? –There are 4 powers of i in a cycle –How can we determine how many complete cycles we must go through to arrive at 10? –Because we are dividing by 4, what are the possible values for the remainder? –What does each possible value for the remainder mean in terms of the next cycle?

Powers of i (Example) Ex 6: Simplify the power of i to either i, -1, -i, or 1: a) b) c) 25

26 Summary After studying these slides, you should know how to do the following: –Understand the concept of imaginary numbers –Multiply and divide square roots with imaginary numbers –Add, subtract, multiply, and divide complex numbers Understand the form a + bi –Calculate any power of i Additional Practice –See the list of suggested problems for 10.7 Next lesson –Solving Quadratic Equations by Completing the Square (Section 11.1)