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Subtraction: Notes 4 4 1 4. 1 2i (a+bi)-(c+di )=(a-c)+(b - d)i Multiplication: ( a + b i )( c + d i ) = ( ac - bd ) + ( ad + bc ) i 9 9 1 9. 1 3i Argand Diagram The properties of the complex conjugate y (imaginary axis) (a,b) b a + bi O A = a + bi a Let z = a + b i , the complex conjugate, or briefly conjugate of z is defined by z= a–bi. x (real axis) corresponds to (a,b) Equality of Complex Numbers Given that x and y . ( 5x – y ) + ( x - y ) i = 9 + i . Find the value of The product of a complex number and its conjugate ( a + b i )( a – b i ) = a2 + b2 (A real number) For any complex numbers z1 , z2 , z3 , we have the following algebraic properties of the conjugate operation: (i) z1 z 2 z1 z 2 (ii) z1 z 2 z1 z 2 (iii) z1 z 2 z1 z 2 z (iv) 1 z2 z 1 , provided z2 0 z2 Example Properties of Complex Numbers Addition: ( a + b i )+ ( c + d i ) = ( a + c ) + ( b + d ) i Write 4 3i in the standard form a + bi. 2 5i y De Moivre’s Theorem r y If z = x + i y = r ei , and n is a natural number, then x x zn = (x + i y)n = rn en i In term of trigonometric form Trigonometric Form z = x + i y = r cos + i = r ( cos + i sin ) Exponential Form z = x zn = [r( cos + i sin )]n = rn [cos n + i sin n]. r sin Theorem 2 Given any nonzero complex number z = rei , the equation wn z has precisely n solutions given by + i y = r ( cos + i sin ) = r e i 2k wk n r ei n , k = 0, 1, 2,…., n-1 Theorem 1 or i 1 i 2 If z1 r1e and z2 r2e , the products and the quotients of complex numbers in exponential form are 1. z1 z2 r1ei1 r2ei 2 r1r2ei 1 2 z1 r1ei1 r1 i 1 2 e z2 r2ei 2 r2 360o k 360o k i sin , k = 0, wk n r cos n n 1, 2,…., n-1 or 2k 2k wk n r cos i sin , k = 0, 1, n n 2,…., n-1 Forms of Complex Numbers 1. Express the following complex numbers in the exponential and trigonometric form. a. –1 – 4i b. 3 i c. 3 4i d. 2 + 3i 2. Write in the form of a + bi a. e i b. 3 e i 2 Multiplication and division in exponential form 1. If z1 4e 5 i 18 and a. z1.z2 2. z2 e 17 i 9 , find b. z1 z2 Use the above identities to find z1.z2 and z1 if z2 a. z1 = 2 (cos 90o + i sin 90o) and z2 = 4 (cos 30o + i sin 30o) Powers of Complex Numbers 1. Use De Moivre’s Theorem to find (1 + 2i)8. Leave your answer in the exponential form. 2. Use De Moivre’s Theorem to find ( 3 - 3i)5. Write your answer in the i. trigonometric form ii. standard form 3. By using De Moivre’s Theorem, write each expression in the standard form. a. [ 3 (cos 20o + i sin20o) ]3 b. [ 2 (cos 5 5 4 + i sin )] 8 8 c. (- 3 - i)6 Roots of Complex Numbers 1. Find the fourth roots of 3 7i Leave your answers in the exponential form. 2. Find all the complex cube roots of 27i. Leave your answers in the trigonometric form. 3. Find all the five distinct roots of - 4 + 3i. Leave your answers in the standard form. 4. Find all the sixth distinct roots of -16. Leave your answers in the standard form. 5. Evaluate all the values of form. 4 5 5i . Leave your results in the standard