Chapter 9: Complex Numbers


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1 Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number  definition  argand diagram  equality of comple number 9.3 Algebraic operations on comple number  addition and subtraction  multiplication  comple conjugate  division 9.4 Polar form of comple number  modulus and argument  multiplication and division of comple number in polar form 9.5 De Moivre s Theorem  definition  n th power of comple number  n th roots of comple number  proving some trigonometric identities 9.6 Euler s formula  definition  n th power of comple number  n th roots of comple number  relationship between circular and hyperbolic functions 1
2 9.1 Imaginary numbers Consider: 4 This equation has no real solution. To solve the equation, we will introduce an imaginary number. Definition 9.1 (Imaginary Number) The imaginary number i is defined as: i 1 Therefore, using the definition, we will get, 4 4 4( 1) 4i i Eample: Epress the following as imaginary numbers a) 5 b) 8
3 9. Comple Numbers Definition 9. (Comple Numbers) If is a comple number, then it can be epressed in the form : iy, where, y R and i 1. : real part y : imaginary part Or frequently represented as : Re() = and Im() = y Eample: Find the real and imaginary parts of the following comple numbers (a) 1 3i (b) 4i i i 3 3
4 9..1 Argand Diagram We can graph comple numbers using an Argand Diagram. y Re( ) Eample: Sketch the following comple numbers on the same aes. (a) (c) 3 i (b) 3 i 1 3 i (d) 3 i 3 4 4
5 9.. Equality of Two Comple Numbers Given that where 1 a bi and c di 1, C. Two comple numbers are equal iff the real parts and the imaginary parts are respectively equal. So, if 1, then a c and b d. Eample 1: Solve for and y if given Eample : Solve and y. 3 4i (y ) i. for real numbers Eample 3: Solve the following equation for and y where 5
6 9.3 Algebraic Operations on Comple Numbers Addition and subtraction If 1 a bi and c di are two comple numbers, then 1 a c b di Eample: Given. Find a) Z 1 Z b) Z 1 + Z Multiplication If 1 a bi and c di are two comple numbers, and k is a constant, then (i) a bic di (ii) k1 1 ka kbi ac bd ad bci Eample: Given. Find Z 1 Z. 6
7 9.3.3 Comple Conjugate If Note that a bi then the conjugate of is denoted as Division If we are dividing with a comple number, the denominator must be converted to a real number. In order to do that, multiply both the denominator and numerator by comple conjugate of the denominator. 1 1 iy iy 1 iy iy 7
8 Eample 1: Given that 1 1 i, 3 4i. Find 1, and epress it in a bi form. Eample : Given 1 = + i and = 3 4i, find in the form of a + ib. Eample 3: Given Z =. Find the comple conjugate, Write your answer in a + ib form. Eample 4: Given. Find. 8
9 9.4 Polar Form of Comple Numbers P r y O Modulus of, r y. Argument of, arg () = where y tan. From the diagram above, we can see that r cos y r sin Then, can be written as iy r cos irsin r(cos isin ) rcis ( inpolarform) 9
10 Eample 1: Epress i in polar form. Eample : Epress in polar form. Eample 3 Given that 1 = + i and =  + 4i, find such that. Give your answer in the form of a + ib. Hence, find the modulus and argument of. 10
11 9.5 De Moivre s Theorem The nth Power Of A Comple Number Definition 9.5 ( De Moivre s Theorem) and n R, then If rcos isin n r n cosn isinn Eample 1: a) Write = 1 i in the polar form. Then, using De Moivre s theorem, find 4. b) Use D Moivre s formula to write (1 i) 1 in the form of a + ib. 11
12 9.5. The nth Roots of a Comple Number A comple number w is a nth root of the comple number if w n = or w =. Hence w = 1 = n [ r(cos isin )], 1/ r n k k cos isin n n for k 0,1,,, n 1 Substituting k 0,1,,, n 1 yields the nth roots of the given comple number. Eample 1: Find all the roots for the following equations: (a) (b) 3 i. Eample : Solve and epress them in a + ib form. 1
13 Eample 3: Find all cube roots of. Eample 4: Solve = 0. Sketch the roots on the argand diagram. 13
14 9.5.3 De Moivre s Theorem to Prove Trigonometric Identities DeMoivre s theorem can be used to prove some trigonometric identities. (with the help of Binomial theorem or Pascal triangle.) Eample: Prove that 5 3 cos5 16cos 0cos 5cos and 5 3 sin 5 = 16 sin  0 sin + 5 sin. Solution: The idea is to write (cos θ + i sin θ) 5 in two different ways. We use both the Pascal triangle and De Moivre s theorem, and compare the results. From Pascal triangle, (cos θ + i sinθ) 5 = cos 5 θ + i 5 cos 4 θ sin θ 10 cos θ sin θ i 10 cos θ sin 3 θ + 5 cosθ sin 4 θ + i sin 5 θ. = (cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ) + i(5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ). 14
15 Also, by De Moivre s Theorem, we have (cos θ + i sin θ) 5 = cos 5θ + i sin 5θ. and so cos 5θ + i sin 5θ = (cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ) + i(5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ). Equating the real parts gives cos 5θ = cos 5 θ 10 cos 3 θ sin θ + 5 cos θ sin 4 θ. = cos 5 θ 10 cos 3 θ(1 cos θ) + 5 cos θ(1 cos θ) = cos 5 θ 10 cos 3 θ + 10 cos 5 θ + 5 cos θ 10 cos 3 θ + 5 cos 5 θ. = 16 cos 5 θ 0 cos 3 θ + 5 cos θ. (proved) Equating the imaginary parts gives sin 5θ = 5 cos 4 θ sin θ 10 cos θ sin 3 θ + sin 5 θ = 5(1 sin θ) sin θ 10(1 sin θ) sin 3 θ + sin 5 θ = 5(1 sin θ + sin 4 θ) sin θ 10 sin 3 θ + 10 sin 5 θ + sin 5 θ = 5 sin θ 10 sin 3 θ + 5 sin 5 θ 10 sin 3 θ + 10 sin 5 θ + sin 5 θ = 16 sin 5 θ 0 sin 3 θ + 5 sin θ (proved). 15
16 9.6 Eulers s Formula Definition 9.6 Euler s formula states that e i It follows that e in cos isin cosn isinn From the definition, if is a comple number with modulus r and Arg(), ; then r(cos isin ) i re ( ineuler form) Eample: Epress the following comple numbers in the form of (a) 3 + i (b) 4i i re 16
17 9.6.1 The nth Power Of A Comple Number We know that a comple number can be epress as then i re, i r e 3 3 i 3 r e 4 4 i4 r n r n e e in Eample 1: Given. Find the modulus and argument of 5. Eample : Find in the form of a + ib. Eample 3: Epress the comple number = Then find 1 3i in the form of i re. (a) (b) 3 (c) 7 17
18 9.6. The nth Roots Of A Comple Number The nth roots of a comple number can be found using the Euler s formula. Note that: Then, i re ( k ) 1 1 k i r e, k = 0,1 1 1 k i r e, k = 0,1, 1 n r 1 n e k i n, k = 0,1,,,n 1 Eample 1: Find the cube roots of 1 i. Eample : Given. Find all roots of in Euler form. Eample 3: Solve 3 + 8i = 0 and sketch the roots on an Argand diagram. 18
19 9.6.3 Relationship between circular and hyperbolic functions. Euler s formula provides the theoretical link between circular and hyperbolic functions. Since e i cos isin and i e cos isin we deduce that i i i i e e e e cos and sin. i (1) In Chapter 8, we defined the hyperbolic function by cosh e e and sinh e e () Comparing (1) and (), we have so that e coshi tanh i i tan i e i cos i i e e sinhi isin i. 19
20 Also, so that cosi sini i i e e e e cosh i i e e e e sinh i tani i tanh. Using these results, we can evaluate functions such as sin, cos, tan, sinh, cosh and tanh. For eample, to evaluate cos cos( iy) we use the identity cos( A B) cos Acos B sin Asin B and obtain cos cos cosiy sin siniy Using results in (3), this gives cos cos cosh y isin sinh y. (3) 0
21 Eample: Find the values of a) sinh(3 4 i) b) tan 3i 4 c) sin (1 i) 4 (Ans: a) i b) i c) i) 1
22 Pascal s Triangle In general:
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