Geometricreoresentation of comiolexmumbersGeometricrepresentationof complexnumbersImaginaryaxis(yaxis),=a+bi80Real axis (r axis)vectorrepresentation:z=a+ibFCV&ITsha Uni.of Sci&Tech)September2.201911/40MineLilChs
Geometric representation of complex numbers Geometric representation of complex numbers O Real axis (x axis) Imaginary axis (y axis) z = a + bi θ r r cos θ r sin θ vector representation: z = a + ib triangle functions representation: z = r(cos θ + isin θ) polar coordinate representation: z = re iθ Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 11 / 40
lexnumbersationotconGeometricrepresentationof complexnumbersImaginaryaxis(yaxis),=a+birsine00Real axis (r axis)rcosavectorrepresentation:z=a+ibtrianglefunctions representation:z=r(cos+isin0)FCV&ITtha Uni.of Sci &Tech)September2.201911/40MineLiIChs
Geometric representation of complex numbers Geometric representation of complex numbers O Real axis (x axis) Imaginary axis (y axis) z = a + bi θ r r cos θ r sin θ vector representation: z = a + ib triangle functions representation: z = r(cos θ + isin θ) polar coordinate representation: z = re iθ Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 11 / 40
lexmumbersometricrtation otconGeometricrepresentation of complex numbersImaginary axis(yaxis)=a+birsing0Real axis (r axis)rcosovectorrepresentation:z=a+ibtriangle functions representation:z=r(cos+isin0)polarcoordinate representation:z=reie11/40FCV&ITSeptember2.2019MineLlChaigsha Uni.ofSci&Tech)
Geometric representation of complex numbers Geometric representation of complex numbers O Real axis (x axis) Imaginary axis (y axis) z = a + bi θ r r cos θ r sin θ vector representation: z = a + ib triangle functions representation: z = r(cos θ + isin θ) polar coordinate representation: z = reiθ Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 11 / 40
Definition(modulusandargument)Forz=a+biIz/ = r = Va2 + 62(3)is called thenorm or modulus,orabsolutevalueof z(4)Argz=0is called theargument oramplitude of thecomplexnumberz.Remark:Theangle has an infinite number of possible values,includingnegative ones, that differ by integral multipes of 2.The principle value ofisdenoted=argz,where-π<<元Argz=argz+2kπ,k=0.±1±2...<argz<FCV&IT12/40September2.2019MineLilCUni.ofSci&Tech)
Geometric representation of complex numbers Definition (modulus and argument ) For z = a + bi, |z| = r = p a 2 + b 2 (3) is called the norm or modulus, or absolute value of z. Argz = θ (4) is called the argument or amplitude of the complex number z. Remark: The angle θ has an infinite number of possible values, including negative ones, that differ by integral multipes of 2π. The principle value of θ is denoted θ = arg z, where −π < θ ≤ π. Argz = arg z + 2kπ, k = 0, ±1, ±2, · · · , −π < arg z ≤ π Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 12 / 40
mumbersy21=1+2i22=-1+2i1aoQuestionComputetheprincipleargumentof z1,z213/40sha Uni. of Sci & Tech)FCV&ITSeptember2.2019MineLilChal
Geometric representation of complex numbers O x y z1 = 1 + 2i z2 = −1 + 2i θ1 θ2 Question Compute the principle argument of z1, z2. Answer: arg z1 = arctan 2, arg z2 = π − arctan 2 Ming Li (Changsha Uni. of Sci & Tech) FCV & IT September 2, 2019 13 / 40