1. Boundless Lecture Slides Free to share, print, make copies and changes. Get yours at www.boundless.com Available on the Boundless Teaching Platform

2. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: If you have any questions or problems please email: educators@boundless.com http://boundless.com/teaching-platform

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4. The Complex-Number System Addition, Subtraction, and Multiplication Complex Conjugates and Division Complex Numbers Functions, Equations, and Inequalities > Complex Numbers Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_23&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The real number a of the complex number z = a + bi is called the real part of z, and the real number b is often called the imaginary part. The real part is denoted by Re(z) or ℜ (z), and the imaginary part b is denoted by Im(z) or ℑ (z). The Complex-Number System Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/functions-equations-and-inequalities-3/complex-numbers-23/the-complex- number-system-124- 11089?campaign_content=book_196_section_23&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Functions, Equations, and Inequalities > Complex Numbers

6. Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: (a + bi) + (c + di) = (a + c) + (b + d)i. Similarly, subtraction is defined by (a + bi) - (c + di) = (a - c) + (b - d)i. The multiplication of two complex numbers is defined by the following formula: (a + bi)(c + di) = (ac - bd) + (bc + ad)i. Addition, Subtraction, and Multiplication Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/functions-equations-and-inequalities-3/complex-numbers-23/addition- subtraction-and-multiplication-125- 5010?campaign_content=book_196_section_23&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Addition of complex numbers View on Boundless.com Functions, Equations, and Inequalities > Complex Numbers

7. The complex conjugate of the complex number z = x + yi is defined to be x - yi. It is denoted z*. The division of two complex numbers is defined in terms of complex multiplication and real division. Where at least one of c and d is non-zero: (a + bi)/(c + di) = (ac + bd)/(c2 + d2) + [(bc - ad)/(c2 + d2)]i. Division can be defined in this way because of the following observation: (a + bi)/(c + di) = [(a + bi)(c - di)]/[(c + di)(c - di)] = (ac + bd)/(c2 + d2) + [(bc - ad)/(c2 + d2)]i. As shown earlier, c - di is the complex conjugate of the denominator c + di. Complex Conjugates and Division Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/functions-equations-and-inequalities-3/complex-numbers-23/complex- conjugates-and-division-126- 6930?campaign_content=book_196_section_23&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Functions, Equations, and Inequalities > Complex Numbers

8. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

9. Key terms complex of a number, of the form a + bi, where a and b are real numbers and i is the square root of −1. complex conjugate Of a complex number x, the complex number \overline x formed by changing the sign of the imaginary part: The complex conjugate of a + bi is a - bi. complex numbers (complex analysis) A number of the form a + bi, where a and b are real numbers and i denotes the imaginary unit conjugate any of a set of irrational or complex numbers that are zeros of the same polynomial with integral coefficients denominator The number or expression written below the line in a fraction (thus 2 in ½). imaginary of a number, having no real part; that part of a complex number which is a multiple of the square root of -1. imaginary number a number of the form ai, where a is a real number and i the imaginary unit imaginary unit A complex number, usually denoted with i, that is defined as i^2 = -1 parallelogram a convex quadrilateral in which each pair of opposite edges are parallel and of equal length. real number An element of the set of real numbers. The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities

10. Interactive Graph: Complex Number Illustration A complex number can be visually represented as a pair of numbers (a,b) forming a vector on the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and "i" is the imaginary unit, satisfying i2= −1. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Complex Number Illustration." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-example-of-complex-number View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-example-of-complex-numberView on Boundless.com Functions, Equations, and Inequalities

11. Addition of complex numbers Addition of two complex numbers can be done geometrically by constructing a parallelogram. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Complex number." GNU FDL http://en.wikipedia.org/wiki/Complex_number View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Complex_numberView on Boundless.com Functions, Equations, and Inequalities

12. Interactive Graph: Complex Conjugate Geometric representation of z and its conjugate in the complex plane. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Complex Conjugate." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-complex-conjugate View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-complex-conjugateView on Boundless.com Functions, Equations, and Inequalities

13. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities Add: (3 - 5i) + (-4 + 7i)? A) 1 + 2i B) 7 - 12i C) -2 + 3i D) -1 + 2i

14. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities Add: (3 - 5i) + (-4 + 7i)? A) 1 + 2i B) 7 - 12i C) -2 + 3i D) -1 + 2i

15. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities Subtract: (-5 + 3i) - (4 + 7i). A) -1 - 4i B) -1 + 4i C) -9 - 4i D) -9 + 4i

16. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities Subtract: (-5 + 3i) - (4 + 7i). A) -1 - 4i B) -1 + 4i C) -9 - 4i D) -9 + 4i

17. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities Multiply: (4 - 5i)(-2 + 7i). A) 27 + 18i B) 27 + 38i C) -43 + 18i D) -43 + 38i

18. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities Multiply: (4 - 5i)(-2 + 7i). A) 27 + 18i B) 27 + 38i C) -43 + 18i D) -43 + 38i

19. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities Multiply: (3 + 4i)(3 - 4i). A) 9 - 16i B) 9 + 16i C) -7 D) 25

20. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities Multiply: (3 + 4i)(3 - 4i). A) 9 - 16i B) 9 + 16i C) -7 D) 25

21. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities When two complex conjugates a + bi and a - bi are added, the result is: A) 2a B) 2bi C) 2a+2bi D) -2bi

22. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities When two complex conjugates a + bi and a - bi are added, the result is: A) 2a B) 2bi C) 2a+2bi D) -2bi

23. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities When two complex conjugates a + bi and a - bi are multiplied, the result is: A) a2-b2 B) a2+2ab+b2 C) -1 D) a2+b2

24. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities When two complex conjugates a + bi and a - bi are multiplied, the result is: A) a2-b2 B) a2+2ab+b2 C) -1 D) a2+b2

25. Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities Divide 3+2i by 4+6i. A) 12i-5 B) (12-5i)/26 C) 26 D) 24-10i

26. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Functions, Equations, and Inequalities Divide 3+2i by 4+6i. A) 12i-5 B) (12-5i)/26 C) 26 D) 24-10i

27. Attribution Wikipedia. "Complex number." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Complex_numberCC BY-SA 3.0http://en.wikipedia.org/wiki/Complex_number Wiktionary. "imaginary unit." CC BY-SA 3.0 http://en.wiktionary.org/wiki/imaginary+unitCC BY-SA 3.0http://en.wiktionary.org/wiki/imaginary+unit Wiktionary. "parallelogram." CC BY-SA 3.0 http://en.wiktionary.org/wiki/parallelogramCC BY-SA 3.0http://en.wiktionary.org/wiki/parallelogram Wiktionary. "complex numbers." CC BY-SA 3.0 http://en.wiktionary.org/wiki/complex+numbersCC BY-SA 3.0http://en.wiktionary.org/wiki/complex+numbers Wikipedia. "Complex number." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Complex_numberCC BY-SA 3.0http://en.wikipedia.org/wiki/Complex_number Wiktionary. "conjugate." CC BY-SA 3.0 http://en.wiktionary.org/wiki/conjugateCC BY-SA 3.0http://en.wiktionary.org/wiki/conjugate Wiktionary. "denominator." CC BY-SA 3.0 http://en.wiktionary.org/wiki/denominatorCC BY-SA 3.0http://en.wiktionary.org/wiki/denominator Wiktionary. "imaginary." CC BY-SA 3.0 http://en.wiktionary.org/wiki/imaginaryCC BY-SA 3.0http://en.wiktionary.org/wiki/imaginary Wiktionary. "complex conjugate." CC BY-SA 3.0 http://en.wiktionary.org/wiki/complex+conjugateCC BY-SA 3.0http://en.wiktionary.org/wiki/complex+conjugate Wikipedia. "Complex number." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Complex_numberCC BY-SA 3.0http://en.wikipedia.org/wiki/Complex_number Wiktionary. "imaginary number." CC BY-SA 3.0 http://en.wiktionary.org/wiki/imaginary+numberCC BY-SA 3.0http://en.wiktionary.org/wiki/imaginary+number Wiktionary. "real number." CC BY-SA 3.0 http://en.wiktionary.org/wiki/real+numberCC BY-SA 3.0http://en.wiktionary.org/wiki/real+number Wiktionary. "complex." CC BY-SA 3.0 http://en.wiktionary.org/wiki/complexCC BY-SA 3.0http://en.wiktionary.org/wiki/complex Free to share, print, make copies and changes. Get yours at www.boundless.com Functions, Equations, and Inequalities