Complex Numbers Review Reference Mary L Boas Mathematical Methods in the Physical Sciences Chapter 2 14 George Arfken Mathematical Methods for Physicists Chapter 6 The real numbers denoted R are incomplete in the sense that standard operations applied to some real numbers do not yield a real result e g square root 1 It is surprisingly easy to enlarge the set of real numbers producing a set of numbers that is closed under standard operations one simply needs to include 1 and linear combinations of it Thus this enlarged field of numbers called the complex numbers denoted C consists of numbers of the form z a b 1 where a and b are real numbers There are lots of notations for theses numbers In mathematics 1 is called i so z a bi whereas in electrical engineering i is frequently used for current so 1 is called j so z a bj In Mathematica complex numbers are constructed using I for i Since complex numbers require two real numbers to specify them they can also be represented as an ordered pair z a b In any case a is called the real part of z a Re z and b is called the imaginary part of z b Im z Note that the imaginary part of any complex number is real and the imaginary part of any real number is zero Finally there is a polar notation which reports the radius a k a absolute value or magnitude and angle a k a phase or argument of the complex number in the form r The polar notation can be converted to an algebraic expression because of a surprising relationship between the exponential function and the trigonometric functions ej cos j sin Thus there is a simple formula for the complex number z 1 in terms of its magnitude and angle p a2 b2 r z1 a r cos z1 cos b r sin z1 sin z1 a bj z1 cos j sin z1 ej For example we have the following notations for the complex number 1 i 1 i 1 j 1 I 1 1 2 45 2ej 4 Since complex numbers are closed under the standard operations we can define things which previously made no sense log 1 arccos 2 1 sin i The complex numbers are large enough to define every function value you might want Note that addition subtraction multiplication and division of complex numbers proceeds as usual just using the symbol for 1 let s use j z1 a bj z2 c dj Im z1 b a z1 Im a2 b2 r r b Re a Re Figure 1 Complex numbers can be displayed on the complex plane A complex number z a bi may be displayed as an ordered pair a b with the real axis the usual xaxis and the imaginary axis the usual y axis Complex numbers are also often displayed as vectors pointing from the origin to a b The angle can be found from the usual trigonometric functions z r is the length of the vector z1 z2 a bj c dj a c b d j z1 z2 a bj c dj a c b d j z1 z2 a bj c dj ac adj bcj bdj 2 ac bd ad bc j 1 1 a bj a bj a b 1 2 2 2 j 2 2 z1 a bj a bj a bj a b a b a b2 Note in calculating 1 z1 we made use of the complex number a bj a bj is called the complex conjugate of z1 and it is denoted by z1 or sometimes z1 See that zz z 2 Note that in terms of the ordered pair representation of C complex number addition and subtraction looks just like component by component vector addition a b c d a b c d Thus there is a tendency to denote complex numbers as vectors rather than points in the complex plane While the closure property of the complex numbers is dear to the hearts of mathematicians the main use of complex numbers in science is to represent sinusoidally varying quantities in a simple way For example you may remember that the superposition of sinusoidal quantities is itself sinusoidal but with a new amplitude and phase For example in a series RC circuit the voltage across the resistor might be given by A cos t whereas the voltage across the capacitor might be given by B sin t and the voltage across the combination according to Kirchhoff is the sum VR t VC t A cos t B sin t where A B R p B A 2 2 cos t sin t A B A2 B 2 A2 B 2 p A A2 B 2 cos cos t sin sin t where cos 2 A B2 p A2 B 2 cos t Yuck That s a lot of work just to add two sinusoidal waves we seek a simpler method which might not seem overly simple at first glance Note that V R can be written as Im z1 Im z1 z2 z2 z1 Re Re z1 Figure 2 The complex conjugate is obtained by reflecting the vector in the real axis Complex number addition works just like vector addition Re Aej t and VC can be written as Re jBej t so VR t VC t Re A jB ej t Now using the polar form of the complex number A jB p where tan B A A jB A2 B 2 e j we have VR t VC t Re A jB ej t p Re A2 B 2 e j ej t p A2 B 2 Re ej t p A2 B 2 cos t Complex numbers are particularly important for calculations in a c circuits where voltages and currents are all changing sinusoidally at the same frequency We assume each is of the form v t Re V0 ej t i t Re I0 ej t The possibility of phase shifts between these voltages and currents is accounted for by making V0 and I0 complex numbers v t Re V0 ej t Re V0 ej ej t V0 cos t Thus would be the phase shift of this voltage and V rms V0 2 In the case of a capacitor the voltage depends on the stored charge which is the integral of the current Z Z q t I0 j t 1 1 v t I0 ej t dt Re Re e i dt C C C j C So V0 I0 j C i e voltage and current have a linear relationship Playing the role of resistance is Z 1 j C which is called the impedance of the capacitor For resistors capacitors and inductors there is a linear relationship between the complex currents flowing through the device and the complex voltage across the device V0 ZI0 where Z is the complex impedance For resistors Z R for capacitors Z 1 j C and for inductors Z j L The complex numbers V0 I0 and Z can be treated in Kirchhoff s laws exactly as voltages currents and …