Complex number-(E)-02-Theory-II

(iv) Fourth quadrant arg . It is an acute angle and negative. Quadrant x y arg(z) Interval of  I II III IV + – – + + + – – Note :  Argument of the complex number 0 is not defined.  Principal value of argument of a purely real number is 0 if the real number is positive and is if the real number is negative.  Principal value of argument of a purely imaginary number is if the imaginary part is positive and is if the imaginary part is negative. (2) Properties of arguments (i) or 1 or – 1) In general (ii) (iii) or 1 or – 1) (iv) or 1 or – 1) (v) or 1 or – 1) (vi) If then , where (vii) (viii) (ix) (x) or (xi) (xii) where and Note :  Proper value of k must be chosen so that R.H.S. of (i), (ii), (iii) and (iv) lies in  The property of argument is same as the property of logarithm. If arg (z) lies between and inclusive), then this value itself is the principal value of arg (z). If not, see whether arg (z) or . If go on subtracting until it lies between and inclusive). The value thus obtained will be the principal value of arg (z).  The general value of is . Important Tips  If and arg = arg z2.  arg arg i.e., z1 and z2 are parallel.  arg arg where n is some integer.  , where n is some integer.  arg – arg .  If then (i) (arg arg (ii)    If and then are conjugate complex numbers of each other.  or  arg arg arg arg  arg  Amplitude of complex number in I and II quadrant is always positive and in IIIrd and IVth quadrant is always negative.  If a complex number multiplied by i (Iota) its amplitude will be increased by and will be decreased by , if multiplied by –i, i.e. and Complex number Value of argument +ve Re (z) 0 –ve Re (z) +ve Im (z) –ve Im (z) – (z) respectively (iz) –(iz) n. arg (z) arg (z1) + arg (z2) arg (z1) – arg (z2) Example: 24 Amplitude of is (a) (b) (c) (d) Solution: (a) (Since lies on negative imaginary axis) Example: 25 If and , then (a) (b) (c) (d) Solution: (b) Let Since amp =  = – is point image of Trick : must be equal to . Example: 26 Let z, w be complex numbers such that and arg , then arg z equals (a) (b) (c) (d) Solution: (d) , . Example: 27 The amplitude of (a) (b) (c) (d) Solution: (c) For amplitude, Example: 28 If and arg then (a) (b) (c) (d) Solution: [c] and arg , Let , then and and Trick: Since arg here the complex number must lie in second quadrant, so (a) and (b) rejected. Also , which satisfies (c) only. Example: 29 If z and are to non-zero complex numbers such that and arg (z) – arg then is equal to (a) 1 (b) – 1 (c) i (d) – i Solution: (d) .....(i) and arg   .....(ii) From equation (i) and (ii), and   2.8 Square Root of a Complex Number. Let be a complex number such that where x and y are real numbers. Then    .....(i) and .....(ii) [On equating real and imaginary parts] Solving, and  Therefore for b>0 for b<0. Note :  To find the square root of replace i by – i in the above results.  The square root of i is , [Here b = 1]  The square root of – i is , [Here b = –1] Alternative method for finding the square root (i) If the imaginary part is not even then multiply and divide the given complex number by 2. e.g. z = 8 – 15i here imaginary part is not even so write z = (16 – 30i) and let = 16 – 30 i . (ii) Now divide the numerical value of imaginary part of by 2 and let quotient be P and find all possible two factors of the number P thus obtained and take that pair in which difference of squares of the numbers is equal to the real part of e.g., here numerical value of Im(16 – 30i) is 30. Now 30 = . All possible way to express 15 as a product of two are , 3 etc. here = 16 = Re (16– 30i) so we will take 5, 3. (iii) Take i with the smaller or the greater factor according as the real part of a + ib is positive or negative and if real part is zero then take equal factors of P and associate i with any one of them e.g., Re(16 – 30i)  0, we will take i with 3. Now complete the square and write down the square root of z. e.g., Example: 30 The square root of are (a) (b) (c) (d) Solution: (a) Example: 31 equals (a) (b) (c) (d) None of these Solution: (a) Trick: It is always better to square the options rather than finding the square root. 2.9 Representation of Complex Number. A complex number can be represented in the following from: (1) Geometrical representation (Cartesian representation): The complex number is represented by a point P whose coordinates are referred to rectangular axes and which are called real and imaginary axis respectively. Thus a complex number z is represented by a point in a plane, and corresponding to every point in this plane there exists a complex number such a plane is called argand plane or argand diagram or complex plane or gaussian plane. Note :  Distance of any complex number from the origin is called the modules of complex number and is denoted by |z|, i.e.,  Angle of any complex number with positive direction of x– axis is called amplitude or argument of z. i.e., (2) Trigonometrical (Polar) representation : In  OPM, let , then and Hence z can be expressed as where r = |z| and  = principal value of argument of z. For general values of the argument Note :  Sometimes is written in short as .