[H2 MATH] Chapter 6 - Complex Numbers

SHALYN TAY COPYRIGHTED © A LEVEL H2 MATHEMATICSCOMPLEX NUMBERS

CHAPTERANALYSIS MASTERY EXAM WEIGHTAGE SHALYN TAY COPYRIGHTED ©

COMPLEX NUMBERS & IMAGINARY NUMBER i COMPLEX NUMBER OPERATIONS COMPLEX CONJUGATES COMPLEX ROOTS OF POLYNOMIAL EQUATIONS COMPLEX NUMBERS I SHALYN TAY COPYRIGHTED ©

Complex Numbers SHALYN TAY COPYRIGHTED © A complex numberis of the form: 𝑖ൌെ1 Imaginary Number𝒊 Cartesian Form 𝑧ൌ𝑥൅𝑖𝑦 𝑖ଶൌെ1 𝑖ଷൌെ𝑖 𝑖ସൌ1 𝑖ସ௞ାଵൌെ1ൌ𝑖 𝑖ସ௞ାଶൌെ1 𝑖ସ௞ାଷൌെ𝑖 𝑖ସ௞ൌ1 where x and y are real numbers and 𝑖ൌെ1 x is the real part of z, Re(z)𝑥ൌ0ฺ𝑧ൌ𝑖𝑦is a purely imaginary number y is the imaginary part of z, Im(z)Note that Im(z) does not include 𝒊𝑦ൌ0ฺ𝑧ൌ𝑥is a real number The set of complex numbers is denoted by ℂ

Complex Number Operations SHALYN TAY COPYRIGHTED © Equality of 2 Complex Numbers 𝑥൅𝑖𝑦ൌ𝑎൅𝑖𝑏฻𝑥ൌ𝑎andyൌb Addition of Complex Numbers 𝑥൅𝑖𝑦൅𝑎൅𝑖𝑏ൌ𝑥൅𝑎൅𝑖ሺ𝑦൅𝑏ሻ Subtraction of Complex Numbers 𝑥൅𝑖𝑦െ𝑎൅𝑖𝑏ൌ𝑥െ𝑎൅𝑖ሺ𝑦െ𝑏ሻ Multiplication of Complex Numbers 𝑥൅𝑖𝑦𝑎൅𝑖𝑏ൌ𝑥𝑎൅𝑖𝑥𝑏൅𝑖𝑦𝑎൅𝑖ଶ𝑦𝑏ൌ𝑥𝑎൅𝑖𝑥𝑏൅𝑖𝑦𝑎൅െ1𝑦𝑏ൌ𝑥𝑎െ𝑦𝑏൅𝑖ሺ𝑥𝑏൅𝑦𝑎ሻ Let 𝒊𝟐ൌെ𝟏 Division of Complex Numbers 3൅5𝑖 2െ𝑖 ൌ3൅5𝑖 2െ𝑖ൈ2൅𝑖 2൅𝑖 ൌ6൅3𝑖൅10𝑖൅5𝑖ଶ 4െ𝑖ଶ ൌ6൅13𝑖െ5 4൅1 ൌ1൅13𝑖 5 ൌ1 5൅13 5𝑖 Let 𝒊𝟐ൌെ𝟏 𝑥൅𝑖𝑦 𝑎൅𝑖𝑏ൌ𝑥൅𝑖𝑦 𝑎൅𝑖𝑏ൈ𝑎െ𝑖𝑏 𝑎െ𝑖𝑏

Complex Conjugates SHALYN TAY COPYRIGHTED © The complex conjugateof 𝑧ൌ𝑥൅𝑖𝑦is denoted by 𝑧∗and defined as: Cartesian Form 𝑧∗ൌ𝑥െ𝑖𝑦 where x and y are real numbers and 𝑖ൌെ1 𝑧and𝑧∗are conjugates of each other and known as conjugate pairs Observe that Reሺzሻ= x = Re𝑧∗ While Im𝑧∗= െ𝑦= െImሺzሻ Useful Properties: 1.ሺ𝑧∗ሻ∗ൌ𝑧 2.z൅𝑧∗ൌ2𝑅𝑒𝑧 3.zെ𝑧∗ൌ2𝑖𝐼𝑚𝑧 4.z𝑧∗ൌ𝑥ଶ൅𝑦ଶ 5.zൌ𝑧∗฻𝑧𝑖𝑠𝑟𝑒𝑎𝑙 6.ሺz൅𝑤ሻ∗ൌ𝑧∗൅𝑤∗ 7.ሺz𝑤ሻ∗ൌ𝑧∗𝑤∗ ***Important Result: Complex Roots of Polynomial Equations Non-real roots of a polynomial equation with real coefficientsoccur in conjugate pairs 𝑥ଶെ2𝑥൅2ൌ0has REAL coefficients to which 𝑥ൌ1േ𝑖are conjugate pair solutions 𝑧ଶ൅െ2൅3𝑖𝑧൅ሺ5െ𝑖ሻൌ0has IMAGINARY coefficients therefore the solution does not contain conjugate pairs

ARGAND DIAGRAMS MODULUS & ARGUMENTS POLAR FORM OF COMPLEX NUMBERS MOD & ARG RELATIONSHIP WITH CONJUGATES GEOMETRICAL EFFECT OF MULTIPLYING 2 COMPLEXES COMPLEX NUMBERS II SHALYN TAY COPYRIGHTED ©

Geometrical Representation of Complex Numbers SHALYN TAY COPYRIGHTED © Argand Diagram Example Im Re P ( x, y )y x r 𝜽 modulus argument r is the modulus of the complex number z, denoted by 𝑧 𝒛ൌ𝒓ൌ𝒙𝟐൅𝒚𝟐 𝜃is the argument of the complex number z, denoted by arg(z)whereെ𝝅൏𝒂𝒓𝒈ሺ𝒛ሻ൑𝝅and arg(z) should be given in radians Complex Number addition and subtraction follow the vector parallelogram law of addition and subtraction Im Re 4 3 𝜶 Im Re െ4 -3 𝜶 3൅4𝑖ൌ3ଶ൅4ଶൌ5 tan𝛼ൌ4 3ฺ𝛼ൌ0.927𝑟𝑎𝑑 argሺ3൅4𝑖ሻൌ0.927𝑟𝑎𝑑 െ3െ4𝑖ൌ3ଶ൅4ଶൌ5 tan𝛼ൌ4 3ฺ𝛼ൌ0.927𝑟𝑎𝑑 arg3൅4𝑖ൌെ𝜋൅0.927𝑟𝑎𝑑 measure argument from positive real axis -ve +ve

Complex Numbers Polar Form SHALYN TAY COPYRIGHTED © Im Re P ( x, y )y x r 𝜽 modulus argument Cartesian Form 𝑧ൌ𝑥൅𝑖𝑦 Any complex number can be written as: Polar Form 𝒓cos𝜽൅𝑖sin𝜽 OR 𝒓𝑒௜𝜽 Multiplication & Division of Complex Numbers in Polar Form 𝑧ଵ 𝑧ଶൌ𝑟ଵ𝑒௜ఏభ 𝑟ଶ𝑒௜ఏమൌ𝑟ଵ 𝑟ଶ𝑒௜ሺఏభିఏమሻ𝑧ଵ𝑧ଶൌ𝑟ଵ𝑟ଶ𝑒௜ሺఏభାఏమሻ Other Useful Properties 1.𝑧ଵ𝑧ଶൌ𝑧ଵ𝑧ଶ 2.௭భ ௭మൌ௭భ ௭మ 3.𝑧௡ൌ𝑧௡ 1.𝑎𝑟𝑔𝑧ଵ𝑧ଶൌ𝑎𝑟𝑔𝑧ଵ൅𝑎𝑟𝑔𝑧ଶൌ𝜃ଵ൅𝜃ଶ 2.𝑎𝑟𝑔௭భ ௭మൌ𝑎𝑟𝑔𝑧ଵെ𝑎𝑟𝑔𝑧ଶൌ𝜃ଵെ𝜃ଶ 3.𝑎𝑟𝑔𝑧௡ൌ𝑛𝑎𝑟𝑔𝑧ൌ𝑛𝜃ଵ

Mod & ArgRelationship With Conjugates SHALYN TAY COPYRIGHTED © 𝑧∗ൌ𝑧 𝑎𝑟𝑔𝑧∗ൌെ𝑎𝑟𝑔𝑧 Im Re P ( x, y )y x r 𝜃 P* ( x, -y ) െ𝜃 𝑧𝑧∗ൌ𝑥ଶ൅𝑦ଶൌ𝑧ଶ 𝑧ൌ𝑟𝑒௜ఏ 𝑧∗ൌ𝑟𝑒ି௜ఏ