1. Complex numbers. Chapter 13: Complex Numbers. Modulus of a complex number. Complex conjugate. Complex numbers are of the form

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1 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple umbers Comple umbers are of the form Sectios 3 & 32 z = + i,, R, i 2 = I the above defiitio, is the real part of z ad is the imagiar part of z The comple umber z = + i ma be represetedithe comple plae as the poit with cartesia coordiates, 0 z=3+2i Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple cojugate Modulus of a comple umber The comple cojugate of z = + i is defied as z = i As a cosequece of the above defiitio, we have Rez = z + z 2 z, Imz =z, z z = i If z ad are two comple umbers, the z + = z +, z = z 2 The absolute value or modulus of z = + i is It is a positive umber z = z z = Eamples: Evaluate the followig i 2 3i

2 2 cotiued Youshouldusethe same rules of algebra as for real umbers, but remember that i 2 = Eamples: # 3: Fid powers of i ad /i Assume z =2+3i ad = 7i Calculatez ad z + 2 Get used to writig a comple umber i the form z =real part+i imagiar part, o matter how complicated this epressio might be Remember that multiplig a comple umber b its comple cojugate gives a real umber Eamples: Assume z =2+3i ad = 7i Fid z Fid z Fid Im z 3 # 3227: Solve 8 5iz i =0 3 I polar coordiates, = r cosθ, where r = = z = r siθ, 0 θ z=+i The agle θ is called the argumet of z Itisdefiedforall z 0, ad is give b arcta if 0 argz =θ = arcta + π if < 0 ad 0 arcta ± 2π π if < 0 ad < 0 Pricipal value Argz Because argz is multi-valued, it is coveiet to agree o a particular choice of argz, i order to have a sigle-valued fuctio The pricipal value of argz, Argz, is such that ta Argz =, with π<argz π Note that Argz Argz=π jumps b 2π whe oe crosses the 0 egative real ais Argz > - π from above

3 Pricipal value Argz cotiued Polar ad cartesia forms of a comple umber Eamples: Fid the pricipal value of the argumet of z = i Fid the pricipal value of the argumet of z = 0 Youeedtobeabletogobackadforthbetweethepolar ad cartesia represetatios of a comple umber z = + i = z cosθ+i z siθ 0 I particular, ou eed to kow the values of the sie ad cosie of multiples of π/6adπ/4 π π Covert cos + i si to cartesia coordiates 6 6 What is the cartesia form of the comple umber such that z =3adArgz =π/4? reads epiθ =cosθ+i siθ, θ R As a cosequece, ever comple umber z 0cabe writte as z = z cosθ+i siθ = z epiθ This formula is etremel useful for calculatig powers ad roots of comple umbers, or for multiplig ad dividig comple umbers i polar form To fid the -th power of a comple umber z 0, proceed as follows Write z i epoetial form, z = z ep iθ 2 The take the -th power of each side of the above equatio z = z ep iθ = z cosθ+i siθ 3 I particular, if z is o the uit circle z =, we have cosθ+i siθ = cosθ+i siθ This is De Moivre s formula

4 cotiued Product of two comple umbers Eamples of applicatio: Trigoometric formulas cos2θ =cos 2 θ si 2 θ, si2θ = 2 siθ cosθ Fid cos3θ ad si3θ i terms of cosθ ad siθ 3 The product of z = r ep iθ ad = r 2 ep iθ 2 is z = r ep iθ r 2 ep iθ 2 = r r 2 ep i θ + θ 2 4 As a cosequece, argz =argz +arg, z = z We ca use Equatio 4 to show that cos θ + θ 2 =cosθ cos θ 2 si θ si θ 2, si θ + θ 2 =siθ cos θ 2 +cosθ si θ 2 5 Ratio of two comple umbers Similarl, the ratio z is give b z = r ep iθ r 2 ep iθ 2 = r r 2 ep i θ θ 2 As a cosequece, z arg =argz arg, z = z Eample: Assume z =2+3i ad = 7i Fid z To fid the -th roots of a comple umber z 0, proceed as follows Write z i epoetial form, with r = z ad p Z z = r ep iθ +2pπ, 2 The take the -th root or the /-th power z = z / = r / ep i θ +2pπ = r ep 3 There are thus roots of z, give b z k = θ +2kπ θ +2kπ r cos + i si i θ +2pπ, k =0,,

5 cotiued cotiued The pricipal value of z is the -th root of z obtaied b takig θ = Argz adk =0 The -th roots of uit are give b 2kπ 2kπ =cos + i si = ω k, k =0,, where ω = cos2π/+i si2π/ I particular, if w is a -th root of z 0,thethe-th roots of z are give b w, w ω, w ω 2,, w ω Eamples: Fid the three cubic roots of Fid the four values of 4 i Give a represetatio i the comple plae of the pricipal value of the eighth root of z = 3+4i If z ad are two comple umbers, the z + z + This is called the triagle iequalit Geometricall, it sas that the legth of a side of a triagle caot be larger tha the sum of the legths of the other two sides More geerall, if z,,, z are comple umbers, the z z z z

Chapter 13: Complex Numbers

Chapter 13: Complex Numbers Sectios 13.1 & 13.2 Comple umbers ad comple plae Comple cojugate Modulus of a comple umber 1. Comple umbers Comple umbers are of the form z = + iy,, y R, i 2 = 1. I the above defiitio, is the real part