Complex numbers in polar form

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1 remember remember Chapter Complex s The magnitude (or modulus or absolute value) of z = x + yi is the length of the line segment from (0, 0) to z and is denoted by z, x + yi or mod z.. z = x + y and zz = z. y. arg z = θ where tan θ = --. x. z i n, n N produces an anticlockwise rotation of 90n degrees. 5. z = r cos θ + r sin θi = r cis θ in polar form.. Arg z is the angle θ in the range < θ. D Complex s in polar form In the following exercise give arg z or Arg z correct to three decimal places where it is not easily expressed as a multiple of. 1 a Represent z = + i on an Argand diagram. b Calculate the exact distance of z from the origin. Find the modulus of each of the following. 17 a z = 5 + 1i b z = 5 i c z = + 7i d z = i e z = + i f z = ( + i) Complex 1 Mathcad 1 19 If z = + i, w = i and u = + 5i then: i represent each of the following on an Argand diagram ii calculate the magnitude in each case. a z w b u + z c w u d w + z e z + w u f z a Show the points z 1 = + 0i, z = + 5i, z = 7 + 5i and z = 9 + 0i on the complex plane. b Calculate the area of the shape formed when the four points are connected by straight line segments in the order z 1 to z to z to z and back to z 1. 5 a Show the points z = 1 + i, u = and w = + 1i on the complex plane. b Calculate the area of the triangle produced by joining the three points with straight line segments. 0 Find the argument of z for each of the following in the interval [0, ]. (Give exact answers where possible.) a z = + i b z = + i c z = 5 5i d z = + i e z = i f z = 10i g z = i h z = 7 i z = i j z = 55

2 10 Maths Quest 1 Specialist Mathematics 1 7 Convert each of the following into Arguments a b c d e f g h Find the modulus and Argument of each of the following complex s. a i b 5 + 5i c 1 i d + i e 7 10i f i g ( + i) 9 Express each of the following in polar form z = r cis θ where θ = Arg z. a z = 1 + i b z = + i c z = 5 5i d z = 5 15i e z = i f z = i 10 Express each of the following complex s in Cartesian form. a cis b cis -- c 5 cis d cis -- e 7 cis f cis -- g cis 11 1 If z = 50i and w = 5 + 5i the value of z + w is: A B 15 C 17 D 5 E 9 The perimeter of the triangle formed by the line segments connecting the points i, 1 i and + i is: A 1 B 0 C 10 D 17 E 5 1 The Argument of i is: A -- B -- 5 C D -- E -- 1 In polar form, 5i is: A cis -- 5 B cis 5 C cis D 5 cis 5 E 5 cis -- WorkSHEET 15 The Cartesian form of cis is: A i B i C i D i E i

3 Chapter Complex s 19 E Basic operations on complex s in polar form 5 1 Express each of the following in the form r cis θ where θ (, ]. a cis -- cis-- b 5 cis cis -- c cis cis -- d cis cis e 7 cis cis Express the resultant complex s in question 1 in Cartesian form. GC Power of program TI GCprogram Casio Power of Express the following products in polar form. 7 a ( + i)( + i) b ( i)( i) c ( + i)( 1 i) Express each of the following in the form r cis θ where θ (, ]. 5 a 1 cis cis b cis cis -- c 0 cis -- 5 cis -- d cis----- cis e 5 cis cis If z = cis----- and w = cis -- then express each of the following in: i polar form ii Cartesian form. a z b w c z d w 5 If z = 1 i and w = + i, write the following in Cartesian form. a z b w c z d w 5 e f z w 7 Determine ( + i) ( 1 i) in Cartesian form. ( i) Write in the form x + yi. ( i) z w 9 a 5 cis -- cis -- is equal to: A i B 10i C D i E b If = + + ( )i then z is: z ( ) A 1 + i B i C 1 i D + i E 1 i c If z = 1 i and w = + i then is equal to: A + i B C D i E w z

4 10 Maths Quest 1 Specialist Mathematics 10 If z = cis----- and w = cis, find the modulus and the argument of If z = + i and w = i, determine (z + w) 9. z w 1 Find z + w, if z = i and w = i. 1 If z 1 = 5 cis -----, z and, find the modulus and the 5 = cis----- z = 10 cis z argument of 1 z z 1 By finding z if z = cis θ, show that cos θ = cos θ cos θ sin θ + sin θ and that sin θ = cos θ sin θ cos θ sin θ. 15 Using z = r cis θ, verify that zz = z. 1 If z n = (1 + i) n, determine the smallest value of n N so that z n is equal to: ( ) n ( ) n ( ) n i ( ) n i a b c d. Factorisation of polynomials in C A polynomial in z is an expression of the form P(z) = a n z n + a n 1 z n 1 + a n z n a 1 z + a 0, where n N is the degree (highest power) of P(z) and a n (with a n 0) are the coefficients. If a n R, that is, all the coefficients are real, then P(z) is said to be a polynomial over R. Similarly, if at least one of the a n is complex, P(z) is said to be a polynomial over C. For example, P(z) = z 5z + is a polynomial of degree over R and P(z) = iz + z i is a polynomial of degree over C. The fundamental theorem of algebra Firstly recall that R C and the factor theorem, which states: If (x a) is a factor of the polynomial P(x), then P(a) = 0. In 1799 the German mathematician Carl Friedrich Gauss proved that every polynomial over C has a solution that is. That is, if P n (z) is a polynomial of degree n over C, then there exists a z 0 C such that P n (z 0 ) = 0. This important result can be used to show that a polynomial of degree n, with n N, has n solutions. The proof relies on a repeated application of the fundamental theorem of algebra and the factor theorem. Firstly, the fundamental theorem of algebra guarantees that there is a z 0 C such that P n (z 0 ) = 0. The factor theorem states that if P n (z 0 ) = 0 for some z 0 then (z z 0 ) is a factor of P n (z) so that P n (z) = (z z 0 )P n 1 (z), where P n 1 (z) is a polynomial of degree n 1. Now by applying the fundamental theorem of algebra to P n 1 (z) there is a z 1 C such that P n 1 (z 1 ) = 0 and the factor theorem ensures that P n 1 (z) = (z z 1 )P n (z).

1 Maths Quest 1 Specialist Mathematics 9 1 Find the values of a and b (a, b R) if: a (z + 1) is a factor of z iz + aiz + b b (z i) is a factor of az z + biz + 1i c (z + i) is a factor of z + aiz + iz

1 Write down a polynomial of degree, whose coefficients are all real, that has i and as two of its zeros.

5 1 Maths Quest 1 Specialist Mathematics 9 1 Find the values of a and b (a, b R) if: a (z + 1) is a factor of z iz + aiz + b b (z i) is a factor of az z + biz + 1i c (z + i) is a factor of z + aiz + iz + (1 + i)b. 1 Explain why at least one of the zeros of a polynomial of degree n (where n is an odd natural ) is a real. 1 Write down a polynomial of degree, whose coefficients are all real, that has i and as two of its zeros. 15 Find the values of a, (a R) for which ai is a solution to: a P(z) = z + z + z + 10 b P(z) = z + iz 11z i. 1 Factorise z + i over C. 17 a Show that P(1) = 0 for P(z) = z (1 + i)z + (i 1)z + (7 + i)z i. b Find the polynomial Q(z) if P(z) = (z 1)Q(z). c Determine the values of a C, b R if Q(z) is of the form Q(z) = (z a) + b. 1 Factorise z + z + z + 10z + 15 over C given that z + 5 i is a factor. 19 Factorise P(z) = 9z + (9i 1)z + (5 1i)z + 5i over C if P( i) = 0. 0 Determine the value of a R if i is to be a zero of a z a 11 + = z tip! Graphics Calculator tip! Roots of complex s Casio tip removed. 1. To select complex mode, press MODE and select Radian mode; scroll down and select a+bi and press ENTER.. To find the cube roots of z = i, start by finding one of the roots as follows. Press MATH, select :, enter ( i) and press ENTER. So one cube root is z 1 = 1 i.. Since cube roots occur at angles of -----, the second cube root can be found by multiplying z 1 by cis Scrolling shows that this root is i.. The third cube root is found by multiplying z 1 by cis Scrolling shows that this root is 1. 0.i. Note that the cube root was recalled using nd [ENTRY] twice; cis was also recalled using nd [ENTRY] twice and then edited to make it cis

6 Chapter Complex s 15 G Solving equations in C Solve the following quadratic equations over C. a x + x + 5 = 0 b x x + 5 = 0 c x 1x + 19 = 0 d x 1x + 1 = 0 e x x + = 0 Solve the following equations over C. a z z z + 10 = 0 b z z + z = 0 c z 7z + 10z = 0 d z 1z + 5z = 0 e z 0z + z 0 = 0 For f(z) = z, g(z) = z z + 1 and h(z) = z 5z + 5z show that f(z) g(z) = h(z) and hence determine the values of z such that h(z) = 0. Solve these equations over C. a x + 5x + 1 = 0 b z z = 0 c 9z + 5z = 0 d x + 1x + 9 = 0 5 The solutions to the equation (z ) + = 0 are: A z = + i, z = i B z = i, z = + i C z = + i, z = i D z = i, z = + i E z = 9 + 1i, z = 9 1i 5 Find the square roots of each of the following in Cartesian form. a 1 + i b i c 1 + i 7 Find i in Cartesian form. If one of the square roots of a(1 + i) is a cis --, the other square root is: A a cis B a cis C a cis D a cis E a cis Use De Moivre s theorem to solve the following equations, in polar form. a z = i b z = + i c z = + i d z = i e z = 1 i f z + i = Find ( 15i) and determine the value of the sum of the roots. 11 a Find the cube root of. b Show the results on an Argand diagram. 1 Solve the following equations in Cartesian form. a z = 1 b z = 5 c z = d z = 7 1 Find all z satisfying a z 5 = 1 b z + 1 = 0. Express your answers in polar form. GC Roots of program TI GCprogram Casio Roots of WorkSHEET.

AH Complex Numbers.notebook October 12, 2016

AH Complex Numbers.notebook October 12, 2016 Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does