Chapter-2-Complex number-(E)-01-Theory

2.1 Introduction. Number system consists of real numbers and imaginary numbers ....etc.) If we combine these two numbers by some mathematical operations, the resulting number is known as Complex Number i.e., “Complex Number is the combination of real and imaginary numbers”. (1) Basic concepts of complex number (i) General definition : A number of the form where and is called a complex number so the quantity is denoted by 'i' called iota thus . A complex number is usually denoted by z and the set of complex number is denoted by c i.e., For example, etc. are complex numbers. Note :  Euler was the first mathematician to introduce the symbol i (iota) for the square root of – 1 with property He also called this symbol as the imaginary unit.  Iota (i) is neither 0, nor greater than 0, nor less than 0.  The square root of a negative real number is called an imaginary unit.  For any positive real number a, we have   The property is valid only if at least one of a and b is non-negative. If a and b are both negative then .  If then . (2) Integral powers of iota (i) : Since hence we have , and . To find the value of first divide n by 4. Let q be the quotient and r be the remainder. i.e., where In general we have the following results , where n is any integer. In other words, if n is even integer and if n is odd integer. The value of the negative integral powers of i are found as given below : Important Tips  The sum of four consecutive powers of i is always zero i.e.,  where n is any integer.   Example: 1 If then the value of is (a) 50 (b) – 50 (c) 0 (d) 100 Solution: (c) (since G.P.) Example: 2 If and n is a positive integer, than (a) 1 (b) i (c) (d) 0 Solution: (d) Trick: Since the sum of four consecutive powers of i is always zero.  Example: 3 If then (a) x = 4n, where n is any positive integer (b) x = 2n, where n is any positive integer (c) x = 4n +1, where n is any positive integer (d) x = 2n +1, where n is any positive integer Solution: (a)    Example: 4 is (a) Positive (b) Negative (c) Zero (d) Can not be determined Solution: (d) or – 1 (which is depend upon the value of n). Example: 5 If then (a) 6 (b) 10 (c) – 18 (d) – 15 Solution: (d) Given that;   Now, Example: 6 The complex number is equal to (a) 0 (b) 2 (c) (d) None of these Solution: (d)  2.2 Real and Imaginary Parts of a Complex Number. If x and y are two real numbers, then a number of the form is called a complex number. Here ‘x’ is called the real part of z and ‘y’ is known as the imaginary part of z. The real part of z is denoted by Re(z) and the imaginary part by Im(z). If z = 3 – 4i, then Re(z) = 3 and Im(z) = – 4. Note :  A complex number z is purely real if its imaginary part is zero i.e., Im(z) = 0 and purely imaginary if its real part is zero i.e., Re(z) = 0.  i can be denoted by the ordered pair (0,1).  The complex number (a, b) can also be split as (a, 0) + (0, 1) (b, 0). Important Tips  A complex number is an imaginary number if and only if its imaginary part is non-zero. Here real part may or may not be zero.  All purely imaginary numbers except zero are imaginary numbers but an imaginary number may or not be purely imaginary.  A real number can be written as a + i.0, therefore every real number can be considered as a complex number whose imaginary part is zero. Thus the set of real number (R) is a proper subset of the complex number (C) i.e., R  C.  Complex number as an ordered pair : A complex number may also be defined as an ordered pair of real numbers and may be denoted by the symbol (a,b). For a complex number to be uniquely specified, we need two real numbers in particular order. 2.3 Algebraic Operations with Complex Numbers. Let two complex numbers and Addition : Subtraction : Multiplication : Division : (when at least one of c and d is non-zero) (Rationalization) Properties of algebraic operations with complex numbers : Let and are any complex numbers then their algebraic operation satisfy following operations: (i) Addition of complex numbers satisfies the commutative and associative properties i.e., and (ii) Multiplication of complex number satisfies the commutative and associative properties. i.e., and (iii) Multiplication of complex numbers is distributive over addition i.e., and Note :  is the identity element for addition.  is the identity element for multiplication.  The additive inverse of a complex number is (i.e. – a – ib).  For every non-zero complex number z, the multiplicative inverse of z is . Example: 7 (a) (b) (c) (d) Solution: (d) Example: 8 (a) (b) (c) (d) Solution: (d) Example: 9 The real value of for which the expression is a real number, is [Roorkee 1997; Rajasthan PET 1999] (a) (b) (c) (d) None of these Solution: (c) Given that Since , then  2.4 Equality of Two Complex Numbers. Two complex numbers and are said to be equal if and only if their real parts and imaginary parts are separately equal. i.e.,  and Thus , one complex equation is equivalent to two real equations. Note :  A complex number iff  The complex number do not possess the property of order i.e., is not defined. For example, the statement makes no sense. Example: 10 Which of the following is correct (a) (b) (c) (d) None of these Solution: (d) Because, inequality is not applicable for a complex number. Example: 11 If , then (a) x = 3, y=1 (b) x =1, y=3 (c) x = 0, y=3 (d) x = 0, y=0 Solution: (d) Applying C2 = 0 = 0+ 0 i, Equating real and imaginary parts x = 0, y = 0 Example: 12 The real values of x and y for which the equation is satisfied, are (a) (b) (c) Both (a) and (b) (d) None of these Solution: (c) Given equation  Equating real and imaginary parts, we get .....(i) and .....(ii) Form (i) and (ii), we get and Trick: Put and then we see that they both satisfy the given equation. 2.5 Conjugate of a Complex Number. (1) Conjugate complex number : If there exists a complex number z =  R, then its conjugate is defined as . Hence, we have and . Geometrically, the conjugate of z is the reflection or point image of z in the real axis. (2) Properties of conjugate : If and are existing complex numbers, then we have the following results: (i) (ii) (iii) (iv) In general (v) (vi) (vii) purely real (viii) purely imaginary (ix) purely real (x) (xi) i.e., is purely real i.e., (xii) i.e., either or z is purely imaginary i.e., (xiii) (xiv) (xv) (xvi) If then (xvii) Important Tips  Complex conjugate is obtained by just changing the sign of i.  Conjugate of  Conjugate of  and real or  (3) Reciprocal of a complex number : For an existing non-zero complex number , the reciprocal is given by i.e.,  = . Example: 13 If the conjugate of be 1+i, then (a) (b) (c) (d) Solution: (c) Given that Example: 14 For the complex number z, one from and is (a) A real number (b) An imaginary number (c) Both are real numbers (d) Both are imaginary numbers Solution: (c) Here (Real) and (Real). Example: 15 The complex numbers and are conjugate to each other for (a) (b) (c) (d) No value of x Solution: (d) and are conjugate to each other if or (i) and or (ii) There exists no value of x common in (i) and (ii). Therefore there is no value of x for which the given complex numbers are conjugate. Example: 16 The conjugate of complex number is (a) (b) (c) (d) Solution: (b)  Conjugate Example: 17 The real part of is (a) (b) (c) (d) Solution: (c) = Hence, real part . Example: 18 The reciprocal of is (a) (b) (c) (d) Solution: (c) 2.6 Modulus of a Complex Number. Modulus of a complex number is defined by a positive real number given by where a, b real numbers. Geometrically |z| represents the distance of point P (represented by z) from the origin, i.e. |z| = OP. If |z| = 0, then z is known as zero modular complex number and is used to represent the origin of reference plane. If |z| = 1 the corresponding complex number is known as unimodular complex number. Clearly z lies on a circle of unit radius having centre (0, 0). Note :  In the set C of all complex numbers, the order relation is not defined. As such or has no meaning. But has got its meaning since are real numbers. Properties of modulus (i) and |z| . (ii) and (iii) (iv) (v) . In general (vi) (vii) (viii) or (ix) is purely imaginary or (x) (Law of parallelogram) (xi) , where Important Tips  Modulus of every complex number is a non-negative real number.  i.e.,  and    is always a unimodular complex number if  is always a unimodular complex number if   Thus is the greatest possible value of and is the least possible value of  If the greatest and least values of are respectively and  Example: 19 (a) – 1/2 (b) 1/2 (c) 1 (d) – 1 Solution (c) Trick : Example: 20 If and are different complex numbers with then is equal to (a) 0 (b) 1/2 (c) 1 (d) 2 Solution (c) Example: 21 For any complex number z, maximum value of is (a) 0 (b) 1 (c) 3/2 (d) None of these Solution (b) We know that or , Maximum value of is 1. Example 22 If and , then | z | is equal to (a) 1 (b) 0 or 1 (c) 1 or 2 (d) 2 Solution: (b) or | z | = 1 Example: 23 For if and and then and satisfy (a) (b) (c) (d) Solution: (d) (Given) . Hence, 2.7 Argument of a Complex Number. Let be any complex number. If this complex number is represented geometrically by a point P, then the angle made by the line OP with real axis is known as argument or amplitude of z and is expressed as arg . Also, argument of a complex number is not unique, since if be a value of the argument, so also is where . (1) Principal value of arg (z) : The value of the argument, which satisfies the inequality is called the principal value of argument. Principal values of argument z will be and according as the point z lies in the 1st , 2nd , 3rd and 4th quadrants respectively, where (acute angle). Principal value of argument of any complex number lies between . (i) First quadrant . arg . It is an acute angle and positive. (ii) Second quadrant, arg . It is an obtuse angle and positive. (iii) Third quadrant arg . It is an obtuse angle and negative.