Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 161.3 Complex Numbers Chapter 1 Section 3: Complex NumbersIn this section, we will… Express complex numbers in standard form Add, subtract, multiply and divide complex numbers Solve quadratic equations in the complex number system Solve higher-order polynomials in the complex number system Use the discriminant to determine the nature of the complex solutions of a given quadratic equation1.3 Complex Numbers: Express Complex Numbers in Standard FormIn the real number system:To fix this, we introduce the imaginary unit, i where property of i:example: Perform the indicated operation.1- =1 i- =21i =-9-5-28-For any positive real number : b b i b- =Imaginary numbers1.3 Complex Numbers: Express Complex Numbers in Standard FormA complex number is any number of the form: a + bi where a and b are real numbers and examples:1 .i- =real partimaginary part3 45 51020 2iiii+- -++Complex NumbersReal #s Imaginary #sEquality of complex numbers: and a bi c di a c b d+ = + � = =I have an imaginary friend!1-1.3 Complex Numbers: Add, Subtract, Multiply and Divide Complex NumbersAdding and Subtracting Complex Numbers:Examples: Perform the operation and write your answer in standard form.( ) ( ) ( ) ( )( ) ( ) ( ) ( )a bi c di a c b d ia bi c di a c b d i+ + + = + + ++ - + = - + -In other words, combine the real # parts and combine the imaginary parts.( ) ( )8 4 2 2i i- + - -1.3 Complex Numbers: Add, Subtract, Multiply and Divide Complex NumbersMultiplying Complex Numbers:Examples: Perform the operation and write your answer in standard form.( ) ( ) ( ) ( )a bi c di ac bd ad bc i+ + = - + +Just multiply and remember that3 ( 3 4 )i i- +21i =-(5 3 )(2 )i i+ -x1.3 Complex Numbers: Add, Subtract, Multiply and Divide Complex NumbersExamples: Perform the operation and write your answer in standard form.(4 3 )(3 4)i i+ -Caution…Imaginary numbers are not like real numbers! Some properties of real numbers do not apply to imaginary numbers…like the product rule for radicals.the wrong way (using the product rule)the correct wayTo multiply imaginary #s, they should first be expressed in terms of i.9 4- - 9 4- -1.3 Complex Numbers: Add, Subtract, Multiply and Divide Complex NumbersRationalizing the Denominator:Examples: Perform the operation and write your answer in standard form.22ii--2 31ii+-The complex numbers a + bi and a – bi are called complex conjugates.1.3 Complex Numbers: Add, Subtract, Multiply and Divide Complex NumbersPowers of i: The powers of i form an interesting pattern.Examples: Perform the operation and write your answer in standard form.When n (a natural number) divided by 4 has a remainder of r, thenn ri i=1234iiii====5678iiii====Handy to know!14i101i23i-1.3 Complex Numbers: Add, Subtract, Multiply and Divide Complex NumbersExamples: Perform the operation and write your answer in standard form.3 24 2 1i i- +( )4 22 1i i+Remember that your answer cannot have a power of i greater than 1.Summary of Techniques:We can now solve quadratic equations by:1. Factoringplace in formfactor and use zero-product rule2. Extracting the rootsplace in formtake the square root of both sides3. Completing the squareplace in formcomplete the squaretake the square root of both sides4. Using the quadratic formulaplace in the formuse the quadratic formula2x c=20ax bx c+ + =20ax bx c+ + =2ax bx c+ =242b b acxa- � -=Take half of the x-term coefficientIt goes here( )22 22x a x ax a+ = + +Now square that resultIt goes hereCan use to solve any quad. eq.Can only solve by factoring if this is factorableCan only extract roots if there is no x-term1.3 Complex Numbers: Solve Quadratic Equations in the Complex Number System1.3 Complex Numbers: Solve Quadratic Equations in the Complex Number SystemExamples: Solve each equation in the complex number system. Check your result(s).225 0x + =22 5 0x x- + =checks checks1.3 Complex Numbers: Solve Higher-Order Polynomials in the Complex NumbersExamples: Solve each equation in the complex number system. Check your result(s).21 0x - =checks1.3 Complex Numbers: Solve Higher-Order Polynomials in the Complex NumbersExamples: Solve each equation in the complex number system. Check your result(s).4 23 4 0x x+ - =checks1.3 Complex Numbers: Use the Discriminant to Determine the Nature of SolutionsWe can use the discriminant to determine the nature of the complex solutions to our given quadratic equation:If the discriminant of is:•positive, then there are two different real solutions•zero, then there is a repeated real solution (a double root)•negative, then there are two complex (not-real) number solutions (the two solutions will be complex conjugates)24b ac-20 , 0ax bx c a+ + = �1.3 Complex Numbers: Use the Discriminant to Determine the Nature of Solutions22 4 1 0x x- + =Examples: Use the discriminant to determine the nature of the real solutions of the following quadratic equations.26 2x x+ =24 12 9x x+ =-Independent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect.Read pp. 109-116Homework: pp. 116-117 #9-27 odds, 33-43 odds, 47-59 odds, 69-77 odds1.3 Complex