Before you begin • These slides are used in presentations at workshops. • They are best viewed with a pdf reader like Acrobat Reader (free download). • Make sure that “Single Page View” or “Fit to Window” is selected. • Navigation buttons are provided at the bottom of each screen if needed (see below). • Viewing in web browsers is not recommended. Do not try to print the slides There are many more pages than the number of slides listed at the bottom right of each screen! Apologies for any inconvenience. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 1 / 29 Roots of complex numbers (mα+hs)Smart Workshop Semester 2, 2016 Geoff Coates These slides describe how to find all of the n−th roots of real and complex numbers. Before you start, it helps to be familiar with the following topics: Representing complex numbers on the complex plane (aka the Argand plane). Working out the polar form of a complex number. de Moivre’s Theorem. π π π π Trigonometric ratios for standard first quadrant angles ( 2 , 4 , 3 and 6 ) and using these to find trig ratios for related angles in the other three quadrants. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 2 / 29 What can (mα+hs)Smart do for you? Online Stuff Drop-in Study Sessions presentation slides from Monday, Wednesday, Friday, workshops on many topics 10am-12pm, Ground Floor practice exercises Barry J Marshall Library, teaching weeks and study short videos breaks. and more! Email: geoff[email protected] Workshops Can’t find what you want? See our current Got a question? Workshop Calendar for this Semester’s topics. Drop us a line! ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 3 / 29 Contents Introduction Go Finding roots of complex numbers Go Properties of roots of complex numbers Go Exercises Go Solution to Exercise 1 Go Solution to Exercise 2 Go ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 4 / 29 In the Real Number system the answer is just “2”. Now that we have extended the number realm to include Complex Numbers, it turns out that there are two more answers to this question. In fact, the question “what are the n−th roots of z?” will always have n answers. Introduction Up until now there has been only one answer to the question “what are the cube (or third) roots of 8?” ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 5 / 29 Now that we have extended the number realm to include Complex Numbers, it turns out that there are two more answers to this question. In fact, the question “what are the n−th roots of z?” will always have n answers. Introduction Up until now there has been only one answer to the question “what are the cube (or third) roots of 8?” In the Real Number system the answer is just “2”. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 5 / 29 In fact, the question “what are the n−th roots of z?” will always have n answers. Introduction Up until now there has been only one answer to the question “what are the cube (or third) roots of 8?” In the Real Number system the answer is just “2”. Now that we have extended the number realm to include Complex Numbers, it turns out that there are two more answers to this question. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 5 / 29 Introduction Up until now there has been only one answer to the question “what are the cube (or third) roots of 8?” In the Real Number system the answer is just “2”. Now that we have extended the number realm to include Complex Numbers, it turns out that there are two more answers to this question. In fact, the question “what are the n−th roots of z?” will always have n answers. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 5 / 29 Obviously, 8 is a real number but it can still be expressed as a complex number: z = 8 + 0i Finding roots of complex numbers To find the remaining two cube roots of 8 we need to convert 8 to polar form. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 6 / 29 z = 8 + 0i Finding roots of complex numbers To find the remaining two cube roots of 8 we need to convert 8 to polar form. Obviously, 8 is a real number but it can still be expressed as a complex number: ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 6 / 29 Finding roots of complex numbers To find the remaining two cube roots of 8 we need to convert 8 to polar form. Obviously, 8 is a real number but it can still be expressed as a complex number: z = 8 + 0i ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 6 / 29 8. θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = 0. imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i Now, R is distance (or radius) of this point from (0, 0) so R = Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 8. θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = 0. Now, R is distance (or radius) of this point from (0, 0) so R = Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = 0. 8. Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i Now, R is distance (or radius) of this point from (0, 0) so R = ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 8. θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = 0. Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i Now, R is distance (or radius) of this point from (0, 0) so R = ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = 0. Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i Now, R is distance (or radius) of this point from (0, 0) so R = 8. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 0. Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i Now, R is distance (or radius) of this point from (0, 0) so R = 8. θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 Finding roots of complex numbers To convert z = 8 + 0i to polar form z = R(cos θ + i sin θ), start by representing it on the complex plane: imaginary (y) 8i 6i 4i 2i −8 −6 −4 −2 2 4 6 8 real (x) −2i −4i −6i −8i Now, R is distance (or radius) of this point from (0, 0) so R = 8. θ is the angle the radius makes with the positive x−axis (with anticlockwise as the positive direction) so θ = 0. ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 7 / 29 w 3 = 8(cos 0 + i sin 0) in polar form 1 3 1 w 3 =[ 8(cos 0 + i sin 0)] 3 take cube root to isolate w 1 3 1 1 w = 8 cos 3 × 0 + i sin 3 × 0 by de Moivre’s Theorem = 2 (cos 0 + i sin 0) = 2(1 + i0) = 2 Great! We already knew that! So what’s the point of using polar form? Finding roots of complex numbers So, if w is one of the third roots of 8, we have w 3 = 8 ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 8 / 29 1 3 1 w 3 =[ 8(cos 0 + i sin 0)] 3 take cube root to isolate w 1 3 1 1 w = 8 cos 3 × 0 + i sin 3 × 0 by de Moivre’s Theorem = 2 (cos 0 + i sin 0) = 2(1 + i0) = 2 Great! We already knew that! So what’s the point of using polar form? Finding roots of complex numbers So, if w is one of the third roots of 8, we have w 3 = 8 w 3 = 8(cos 0 + i sin 0) in polar form ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 8 / 29 1 3 1 1 w = 8 cos 3 × 0 + i sin 3 × 0 by de Moivre’s Theorem = 2 (cos 0 + i sin 0) = 2(1 + i0) = 2 Great! We already knew that! So what’s the point of using polar form? Finding roots of complex numbers So, if w is one of the third roots of 8, we have w 3 = 8 w 3 = 8(cos 0 + i sin 0) in polar form 1 3 1 w 3 =[ 8(cos 0 + i sin 0)] 3 take cube root to isolate w ((mα+hs)SmartRootsWorkshop of complexSemester numbers 2, 2016) Contents Prev Next 8 / 29 = 2 (cos 0 + i sin 0) = 2(1 + i0) = 2 Great! We already knew that! So what’s the point of using polar form?