Chapter Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics. – GAUSS • 5.1 Introduction In earlier classes, we have studied linear equations in one and two variables and quadratic equations in one variable. We have seen that the equation x2 + 1 = 0 has no real solution as x2 + 1 = 0 gives x2 = – 1 and square of every real number is non-negative. So, we need to extend the real number system to a larger system so that we can find the solution of the equation x2 = – 1. In fact, the main objective is to solve the equation ax2 + bx + c = 0, where D = b2 – 4ac < 0, which is not possible in the system of real numbers. W. R. Hamilton 5.2 Complex Numbers (1805-1865) Let us denote −1 by the symbol i. Then, we have i2 =−1 . This means that i is a solution of the equation x2 + 1 = 0. A number of the form a + ib, where a and b are real numbers, is defined to be a ⎛−1⎞ complex number. For example, 2 + i3, (– 1) + i 3, 4 +i⎜11 ⎟are complex numbers.⎝⎠For the complex number z = a + ib, a is called the real part, denoted by Re z and b is called the imaginary part denoted by Im z of the complex number z. For example, if z = 2 + i5, then Re z = 2 and Im z = 5. Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d. Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find the values of x and y. Solution We have 4x + i (3x – y) = 3 + i (–6) ... (1) Equating the real and the imaginary parts of (1), we get 4x = 3, 3x – y = – 6, 3 33which, on solving simultaneously, give x = and y = .44 5.3 Algebra of Complex Numbers In this Section, we shall develop the algebra of complex numbers. 5.3.1 Addition of two complex numbers Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the sum z1 + z2 is defined as follows: z1 + z2 = (a + c) + i (b + d), which is again a complex number. For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8 The addition of complex numbers satisfy the following properties: (i) The closure law The sum of two complex numbers is a complex number, i.e., z1 + z2 is a complex number for all complex numbers z1 and z2. (ii) The commutative law For any two complex numbers z1 and z2, z1 + z2 = z2+ z1 (iii) The associative law For any three complex numbers z1, z2, z3, (z1 + z2) + z3 = z1 + (z2 + z3). (iv) The existence of additive identity There exists the complex number 0+ i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0= z. (v) The existence of additive inverse To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity). 5.3.2 Difference of two complex numbers Given any two complex numbers z1 and z2, the difference z1 – z2 is defined as follows: z1 – z2 = z1 + (– z2). For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i and (2 – i) – (6 + 3i) = (2 – i) + ( – 6 – 3i)= – 4 – 4i 5.3.3 Multiplication of two complex numbers Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the product z1 z2 is defined as follows: z1 z2 = (ac – bd) + i(ad + bc) For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28 The multiplication of complex numbers possesses the following properties, which we state without proofs. (i) The closure law The product of two complex numbers is a complex number, the product z1 z2 is a complex number for all complex numbers z1 and z2. (ii) The commutative law For any two complex numbers z1 and z2, z1 z2 = z2 z1. (iii) The associative law For any three complex numbers z1, z2, z3, (z1 z2) z3 = z1 (z2 z3). (iv) The existence of multiplicative identity There exists the complex number 1 + i 0 (denoted as 1), called the multiplicative identity such that z.1 = z, for every complex number z. (v) The existence of multiplicative inverse For every non-zero complex number z = a + ib or a + bi(a ≠ 0, b ≠ 0), we have the complex number a –b 1+i (denoted by or z–1 ), called the multiplicative inverse2 2 22a +ba +bzof z such that z. 1 =1 (the multiplicative identity).z (vi) The distributive law For any three complex numbers z1, z2, z3, (a) z1 (z2 + z3) = z1 z2 + z1 z3 (b) (z1 + z2) z3 = z1 z3 + z2 z3 5.3.4 Division of two complex numbers Given any two complex numbers z1 and z2, z where z2 ≠0 , the quotient 1 is defined byz2 z 11 =z1 zz22 For example, let z1 = 6 + 3i and z2 = 2 – i ⎛ 2 −−() ⎞z1 ⎛ 1 ⎞ Then =⎜(6 +3) i × = (63+i)⎜ 22 +i 2 12 ⎟ z 2 −i ⎟⎜ 1 +−()⎟⎠2 ⎝ ⎠⎝2 +−() 21 ⎛2 +i ⎞ 1163i)⎟ ⎡−+ ( (= (+ ⎜ = 123 i 6 +⎤6)= 9 +12i)⎣⎦⎝5 ⎠55 5.3.5 Power of i we know that 32 422i =ii =−1 i =−i , i =i =−1 =1() ()2 () 2352 62i =()i 2 i =()1 i =i , i =()3 =( ) =−1 , etc.− i −1 −11 ii −21 1 Also, we have i =×= =−i, i == =−1, ii −1 i2 −1 −31 1 ii −4 11i = = ×==i, i = == 1 i3 −ii 1 i41 In general, for any integer k, i4k = 1, i4k + 1 = i, i4k + 2 = –1, i4k + 3 = – i 5.3.6 The square roots of a negative real number Note that i2 = –1 and ( – i)2 = i2 = – 1 Therefore, the square roots of – 1 are i, – i. However, by the symbol −, we would mean i only. Now, we can see that i and –i both are the solutions of the equation x2 + 1 = 0 or x2 = –1. Similarly (3 i)2 =(3)2 i2 = 3 (– 1) = – 3 (−3 i)2 = (−3)2 i2 = – 3 Therefore, the square roots of –3 are 3 i and −3 i . Again, the symbol −3 is meant to represent 3 i only, i.e., −3 = 3 i . Generally, if a is a positive real number, −a = a −1 = ai , We already know that ab for all positive real number a and b. This result also holds true when either a > 0, b < 0 or a < 0, b > 0. What if a < 0, b < 0? Let us examine. Note that ()() a ×b = ab for all real numbers)i2 =−11 1 − (by assuming−= − 1 = 1 = 1, which is a contradiction to the fact that i2 =−1. Therefore, a ×b ≠ab if both a and b are negative real numbers. Further, if any of a and b is zero, then, clearly, a ×b =ab = 0. 5.3.7 Identities We prove the following identity ++ zz , for all complex numbers z1 and z2. Proof We have, (z + z)2 = (z + z) (z + z), (z1 +z2 )2 =z12 z22 212 12 1212= (z1 + z2) z1 + (z1 + z2) z2 (Distributive law) = z2 +zz +zz +z2 (Distributive law)1 21 12 2 = z2 +zz +zz +z2 (Commutative law of multiplication)1 12 12 2 = z2 +2zz +z2 1 122 Similarly, we can prove the following identities: 22(i) z −z 2 =z −2 zz +z(12 )1 122 32 23(ii) (z +z )3 =z +3 zz +3zz +z12 1 12122 32 23(iii) (z −z )3 =z −3zz +3zz −z12 1 12122 22(iv) z–z =(z +z )(z –z )1 2 1212 In fact, many other identities which are true for all real numbers, can be proved to be true for all complex numbers. Example 2 Express the following in the form of a + bi: − ⎛⎞ ⎜⎟5 i 2ii(i) ()i 1 (ii) −i ⎛−1 ⎞3 ⎝ ⎠ ()()⎜ ⎟ −52 − 55 8 ⎝8 ⎠ ()1 i 5 ()iSolution (i) −5i ⎛ ⎞ ⎜ ⎟= i = −1 = = +0 ⎝⎠888 88 ⎛1 ⎞3 1 121i () − i × 5 i2 i .(ii) ()− 2i ⎜⎟= 2 ×i = ()i = ⎝8 ⎠88××8 256 256 Example 3 Express (5 – 3i)3 in the form a + ib. Solution We have, (5 – 3i)3 =53 – 3 × 52 × (3i) + 3 × 5 (3i)2 – (3i)3 = 125 – 225i – 135 + 27i = – 10 – 198i. Example 4 Express (−3 +− 2 )(23 −i)in the form of a + ib Solution We have, (−3 +− 2 )(23 −i)= (−3 + 2 i)(23 −i) 2 −+= (62 )+31 (+22 )i= 6 3 +26i −2 i−+ i 5.4 The Modulus and the Conjugate of a Complex Number Let z = a + ib be a complex number. Then, the modulus of z, denoted by | z |, is defined 22 22to be the non-negative real number a +b , i.e., | z | = a +b and the conjugate of z, denoted as z , is the complex number a – ib, i.e., z = a – ib. For example, += 32 +12 =10 ,3 i25−i = 22 +− ( 5) 2 = 29 , and , −− = 3i – 53 i 3 i 25 i =2 +5 i 35+= −, − iObserve that the multiplicative inverse of the non-zero complex number z is given by 1 a −b −aib z z–1= = +i = = 2222 22 2aiba +ba +ba ++ b z 2 or zz =z Furthermore, the following results can easily be derived. For any two compex numbers z1 and z2 , we have zz 11 =(i) zz =z z (ii) provided z ≠012 1 2 2z z2 2 ⎛⎞ zz1 =1(iii) zz =zz (iv) z ±z =z ±z (v) ⎜⎟ provided z≠ 0.1212 1 212 2zz22⎝⎠Example 5 Find the multiplicative inverse of 2 – 3i. Solution Let z = 2 – 3i 22 z = 2 + 3i andThen z 2 =2 +− ( 3) =13 Therefore, the multiplicative inverse of 23− i is given by z 23+i 23 z–1 == =+ i z 2 13 13 13 The above working can be reproduced in the following manner also, 12 +3i z–1 = = 2 −3i (2 −3 )(2 +ii 3) 23i 2 +3i 23+ = = =+ i 22 −(3 ) i 2 13 13 13 Example 6 Express the following in the form a + ib 5 + 2i (i) (ii) i–35 1− 2i 5 + 2i 5 + 2i 1+ 2i 55+ 2i + 2i −2 ==× 1− 2i 1− 2i 1+ 2iSolution (i) We have, 1−(2i)2 36+ 2i 3(1 +2 2) i + 2i .12 3 = = = 12 +1 11 ii (ii) i−35 == =× = =i3517 2i 2 −ii −ii()i EXERCISE 5.1 Express each of the complex number given in the Exercises 1 to 10 in the form a + ib. ⎛⎞5 i 9 19 391. ()i ⎜−3 ⎟ 2. i +i 3. i −⎝ 5 ⎠ 4. 3(7 + i7) + i (7 + i7) 5. (1 – i) – ( –1 + i6) ⎛12 ⎞⎛ 5 ⎞ ⎡⎛17 ⎞⎛ 1 ⎞⎤ ⎛ 4 ⎞+i −4 +i +i +4 +i −− + i6. ⎜ ⎟⎜ ⎟ 7. ⎢⎜ ⎟⎜ ⎟⎥⎜⎟⎝55 ⎠⎝ 2 ⎠ ⎣⎝33 ⎠⎝ 3 ⎠⎦ ⎝ 3 ⎠ ⎛1 ⎞3 ⎛13 2 ⎝3 ⎠⎝ 3 ⎠8. (1 – i)4 9. ⎜+3i ⎟ 10. ⎜−− i ⎟⎞ Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13. 11. 4 – 3i 12. 53+i 13. – i 14. Express the following expression in the form of a + ib : (3 +i 5 )(3 −i 5 ) (3 + 2 i)−(3 −i 2 ) 5.5 Argand Plane and Polar Representation We already know that corresponding to each ordered pair of real numbers (x, y), we get a unique point in the XY-plane and vice-versa with reference to a set of mutually perpendicular lines known as the x-axis and the y-axis. The complex number x + iy which corresponds to the ordered pair (x, y) can be represented geometrically as the unique point P(x, y) in the XY-plane and vice-versa. Some complex numbers such as 2 + 4i, – 2 + 3i, 0 + 1i, 2 + 0i, – 5 –2i and Fig 5.11 – 2i which correspond to the ordered pairs (2, 4), ( – 2, 3), (0, 1), (2, 0), ( –5, –2), and (1, – 2), respectively, have been represented geometrically by the points A, B, C, D, E, and F, respectively in the Fig 5.1. The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane. Obviously, in the Argand plane, the modulus of the complex number x + iy = x2 + y2 is the distance between the point P(x, y) and the origin O (0, 0) (Fig 5.2). The points on the x-axis corresponds to the complex numbers of the form a + i 0 and the points on the y-axis corresponds to the complex numbers of the form Fig 5.2 0 + ib. The x-axis and y-axis in the Argand plane are called, respectively, the real axis and the imaginary axis. The representation of a complex number z = x + iy and its conjugate z = x – iy in the Argand plane are, respectively, the points P (x, y) and Q (x, – y). Geometrically, the point (x, – y) is the mirror image of the point (x, y) on the real axis (Fig 5.3). Fig 5.3 5.5.1 Polar representation of a complex number Let the point P represent the nonzero complex number z = x + iy. Let the directed line segment OP be of length r and θ be the angle which OP makes with the positive direction of x-axis (Fig 5.4). We may note that the point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P. We consider the origin as the pole and the positive direction of the x axis as the initial line. Fig 5.4 We have, x = r cos θ, y = r sin θ and therefore, z = r (cos θ + i sin θ). The latter is theis said to be the polar form of the complex number. Here r = x2 +y2 =z modulus of z and θ is called the argument (or amplitude) of z which is denoted by arg z. For any complex number z ≠ 0, there corresponds only one value of θ in 0 ≤θ < 2π. However, any other interval of length 2π, for example – π < θ≤ π, can be such an interval.We shall take the value of θ such that – π < θ≤ π, called principal argument of z and is denoted by arg z, unless specified otherwise. (Figs. 5.5 and 5.6) ≤θ< π z =+ Solution Let 1 = r cos θ, 3Example 7 Represent the complex number 1 i 3 in the polar form. = r sin θ By squaring and adding, we get i.e., r = 42= (conventionally, r >0) 13 π Therefore, cos θ = , sin θ = , which gives θ= 22 3 Fig 5.7⎛ ππ⎞ z =2cos +isin Therefore, required polar form is ⎜⎟⎝ 33 ⎠ The complex number z 1 i=+3is represented as shown in Fig 5.7. −16 Example 8 Convert the complex number into polar form.1+i 3 −16 −16 1−i 3× 1+i 31+i 31−i 3Solution The given complex number = –161 – i 3 –161 – i 3) == (( )(= –4 1 – i 3)=– i3 (Fig 5.8).4+ 4 1– (i3)2 1+3 22 2r (cos θ+sin θ)=4 Let – 4 = r cos θ, 43= r sin θ By squaring and adding, we get 22 216 + 48 = r (cos θ +sin θ) which gives r2 = 64, i.e., r = 8 13 Hence cos θ = − , sin θ = 22 Fig 5.8 π 2πθ = π –= 33 ⎛ 2π 2π⎞8cos +i sin Thus, the required polar form is ⎜⎟⎝ 33 ⎠ EXERCISE 5.2 Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2. 1. z = – 1 – i 3 2. z = – 3 + i Convert each of the complex numbers given in Exercises 3 to 8 in the polar form: 3. 1 – i 4. – 1 + i 5. – 1 – i 6. – 3 7. 3 + i 8. i 5.6 Quadratic Equations We are already familiar with the quadratic equations and have solved them in the set of real numbers in the cases where discriminant is non-negative, i.e., ≥ 0, Let us consider the following quadratic equation: ax2 +bx +c =0 with real coefficients a, b, c and a ≠ 0. Also, let us assume that the b2 – 4ac < 0. Now, we know that we can find the square root of negative real numbers in the set of complex numbers. Therefore, the solutions to the above equation are available in the set of complex numbers which are given by b 22−± b −4ac −±b 4ac −bi x = = 2a 2a At this point of time, some would be interested to know as to how many roots does an equation have? In this regard, the following theorem known as the Fundamental theorem of Algebra is stated below (without proof). “A polynomial equation has at least one root.” As a consequence of this theorem, the following result, which is of immense importance, is arrived at: “A polynomial equation of degree n has n roots.” Example 9 Solve x2 + 2 = 0 Solution We have, x2 + 2 = 0 or x2 = – 2 i.e., x = ±−2 = ± 2 i Example 10 Solve x2 + x + 1= 0 Solution Here, b2 – 4ac = 12 – 4 × 1 × 1 = 1 – 4 = – 3 −±− 13 −± 31 iTherefore, the solutions are given by x = = 21 2× Example 11 Solve 5x2 x5 =0++ Solution Here, the discriminant of the equation is 12 −×45 ×5 = 1 – 20 = – 19 Therefore, the solutions are −±− 1 19 −±19 1 i = .25 25 EXERCISE 5.3 Solve each of the following equations: 1. x2 + 3 = 0 2. 2x2 + x + 1 = 0 3. x2 + 3x + 9 = 0 4. – x2 + x – 2 = 0 5. x2 + 3x + 5 = 0 6. x2 – x + 2 = 0 x7. 2x2 ++2 =0 8. 3x2 −2x +33 =0 21 2 x x=0 x + 10x ++ += 9. 10.22 Miscellaneous Examples (3−2 )(2 +3) ii Example 12 Find the conjugate of .(1+2 )(2 i −i) (3−2 )(2 +3) ii Solution We have , (1+2 )(2 i −i) 69+−+ i 4i 6 12+5i 4 −3i = = × 2 −+ +i 4i 2 43i 4 −3i+ 48−36i +20i +15 6316 − i 63 16 = = = − i 16 +9 25 25 25 (3−2 )(2 +3) 6316ii is + iTherefore, conjugate of .(1+2 )(2 i −i) 25 25 Example 13 Find the modulus and argument of the complex numbers: 1+i 1(i) , (ii)1−i 1+i 1+i 1 i1 i1−1+2i++ Solution (i) We have, = ×= =i= 0 + i1−i 1−+i1 i 11 + Now, let us put 0 = rcos θ, 1 = rsin θ Squaring and adding, r2 = 1 i.e., r= 1 so that cos θ = 0, sin θ = 1 π Therefore, θ = 2 1+i πHence, the modulus of is 1 and the argument is 2.1−i 11−i 1−i 1 i = ==−(ii) We have 1+i (1 i)(1 i)1+122+− 11Let = rcos θ, – = rsin θ2211 −1Proceeding as in part (i) above, we get r= ;cosθ = ,sin θ = 2 22 −πTherefore θ = 4 11 −πHence, the modulus of is , argument is .1+i 24 aib+ Example 14 If x+ iy= −, prove that x2 + y2 = 1.aib Solution We have, 22 22(ai)( +b) −+ai − 2a+baiab 2bab b+x+ iy= =22 = 22 22 i(ai)( +b) +ab +−baiab ab + a2 −b22ab So that, x – iy = − i22 22a +ba +b Therefore, 222 22 222(a −b )4ab (a +b )x2 + y2 = (x + iy) (x – iy) = + = = 1222 222 222(a +b )(a +b )(a +b ) Example 15 Find real θ such that 32 sinθ+ i is purely real.12 sinθ−i Solution We have, 32 sinθ (3+ i + 2 sinθ)+ i 2 sinθ)(1 i = 12 sin−i θ (1− 2 sini θ)(1 + 2 sinθ)i 3+6 sini θ+2 sinθ–4sin 2θ 34sin 2θ ii− 8sinθ = = +2 221+ 4sin θ 14sin + θ 14sin θ+ We are given the complex number to be real. Therefore 8sinθ 2 = 0, i.e., sin θ = 014sin θ+Thus θ = nπ, n ∈ Z. i −1 Example 16 Convert the complex number z = in the polar form.ππ cos +i sin 33 i −1Solution We have, z = 1 +22 2 i + 3 −1+ 3i) 31 3 +1−2(i −1) 1− 3i (= ×= = + i 1+ 3i 1−3i 13+ 22 31− 3 +1Now, put = r cosθ , = r sin θ 22 Squaring and adding, we obtain 2 ⎛ 31−⎞2 ⎛ 3 +1⎞22 ⎛⎜(3)2 +1⎞⎟ ×r=+ = ⎝⎠24⎜⎟⎜⎟ ==222⎝ ⎠⎝ ⎠ 44 31− 3 +1 Hence, r=2 which gives cosθ= ,sinθ= 22 22 ππ5πTherefore, θ=+= (Why?)46 12 Hence, the polar form is ⎛ 5π 5π⎞2cos +isin⎜⎟⎝ 12 12 ⎠ Miscellaneous Exercise on Chapter 5 ⎡ 25 3 ⎛⎞⎤18 1⎢i+1. Evaluate: ⎜⎟⎥.i⎢ ⎝⎠⎥⎣⎦ 2. For any two complex numbers z1 and z2, prove that .Re (zz) = Re zRe z– ImzImz121212 ⎛ 12 ⎞⎛3 −4i⎞−3. Reduce ⎜ ⎟⎜⎟to the standard form .14i1+i 5 +i⎠⎝− ⎠⎝ +) 22ai−b 22 ab−= prove that (xy 2 =+ .4. If xiy cid 2 +2−cd 5. Convert the following in the polar form: 17i+ 13i+(i) 2, (ii)( )−ii2 1–2 Solve each of the equation in Exercises 6 to 9. 220 236. 3x−4x+=0 7. x−2x+=0 32 8. 27x2 −10x+=09. 21 2 −28 x += 01 x 10 zz 1++1210. If z1 = 2 – i, z2 = 1 + i, find . 1– 2 +1zz 2(x2 +1) 2 (x +i )11. If a + ib = 2 , prove that a2 + b2 = 22.2x +12x +1 () 12. Let z1 = 2 – i, z2 = –2 + i. Find 12 ⎛⎞⎛zz ⎞ 1 (i) Re⎜⎟, (ii) Im⎜⎟.z zz ⎝1 ⎠⎝11 ⎠12i+13. Find the modulus and argument of the complex number −.13i 14. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i. 1+i 1−i−15. Find the modulus of .1−i 1+i uv . –16. If (x + iy)3 = u + iv, then show that +=4(x2 y2)xy βα– .17. If α and β are different complex numbers with β =1, then find 1 – αβ x18. Find the number of non-zero integral solutions of the equation 1 –i x =2 . 19. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f 2) (g2 + h2) = A2 + B2 ⎛1 +i ⎞m 20. If ⎜1 –i ⎟=1, then find the least positive integral value of m. ⎝ ⎠ Summary A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number. Let z1 = a + ib and z2 = c + id. Then (i) + z = (a + c) + i (b + d)z12(ii) z1 z2 = (ac – bd) + i (ad + bc) For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists the a −b 1+icomplex number 2 2 2 2 , denoted by or z–1, called thea +ba +bz⎛ a2 −b ⎞ multiplicative inverse of z such that (a + ib) ⎜ 22 +i 22 ⎟= 1 + i0 =1 a +ba +b⎝⎠ For any integer k, i4k = 1, i4k + 1 = i, i4k + 2 = – 1, i4k + 3 = – i The conjugate of the complex number z = a + ib, denoted by z , is given by z = a – ib. The polar form of the complex number z = x + iy is r (cosθ + i sinθ), where xy(the modulus of z) and cosθ = , sinθ = . (θ is known as the rr argument of z. The value of θ, such that – π < θ≤π, is called the principal argument of z. A polynomial equation of n degree has n roots. The solutions of the quadratic equation ax2 + bx + c = 0, where a, b, c ∈ R, b−± a ≠ 0, b2 – 4ac < 0, are given by x = . 2a Historical Note The fact that square root of a negative number does not exist in the real number system was recognised by the Greeks. But the credit goes to the Indian mathematicianMahavira(850) who first stated this difficulty clearly. “He mentions in his work ‘Ganitasara Sangraha’ as in the nature of things a negative (quantity) is not a square (quantity)’, it has, therefore, no square root”. Bhaskara, another Indian mathematician, also writes in his work Bijaganita, written in 1150. “There is no square root of a negative quantity, for it is not a square.” Cardan (1545) considered the problem of solving He obtained x = 5 + as the solution of it, which was discarded by him by saying that these numbers are ‘useless’. Albert Girard (about 1625) accepted square root of negative numbers and said that this will enable us to get as many roots as the degree of the polynomial equation. Euler was the first to introduce the symbol i for and W.R. Hamilton (about 1830) regarded the complex number a + ib as an ordered pair of real numbers (a, b) thus giving it a purely mathematical definition and avoiding use of the so called ‘imaginary numbers’. — • —

Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics. – Gauss

### 5.1 Introduction

W. R. Hamilton

(1805-1865)

In earlier classes, we have studied linear equations in one and two variables and quadratic equations in one variable. We have seen that the equation x2 + 1 = 0 has no real solution as x2 + 1 = 0 gives x2 = – 1 and square of every real number is non-negative. So, we need to extend the real number system to a larger system so that we can find the solution of the equation x2 = – 1. In fact, the main objective is to solve the equation ax2 + bx + c = 0, where D = b2 – 4ac < 0, which is not possible in the system of real numbers.

### 5.2 Complex Numbers

Let us denote by the symbol i. Then, we have . This means that i is a solution of the equation x2 + 1 = 0.

A number of the form a + ib, where a and b are real numbers, is defined to be a complex number. For example, 2 + i3, (– 1) + , are complex numbers.

For the complex number z = a + ib, a is called the real part, denoted by Re z and b is called the imaginary part denoted by Im z of the complex number z. For example, if z = 2 + i5, then Re z = 2 and Im z = 5.

Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d.

Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find the values of x and y.

Solution We have

4x + i (3x – y) = 3 + i (–6) ... (1)

Equating the real and the imaginary parts of (1), we get

4x = 3, 3x – y = – 6,

which, on solving simultaneously, give and .

### 5.3 Algebra of Complex Numbers

In this Section, we shall develop the algebra of complex numbers.

5.3.1 Addition of two complex numbers Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the sum z1 + z2 is defined as follows:

z1 + z2 = (a + c) + i (b + d), which is again a complex number.

For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8

The addition of complex numbers satisfy the following properties:

(i) The closure law The sum of two complex numbers is a complex number, i.e., z1 + z2 is a complex number for all complex numbers z1 and z2.

(ii) The commutative law For any two complex numbers z1 and z2, z1 + z2 = z2 + z1

(iii) The associative law For any three complex numbers z1, z2, z3, (z1 + z2) + z3 = z1 + (z2 + z3).

(iv) The existence of additive identity There exists the complex number

0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z.

(v) The existence of additive inverse To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity).

5.3.2 Difference of two complex numbers Given any two complex numbers z1 and z2, the difference z1 – z2 is defined as follows:

z1 – z2 = z1 + (– z2).

For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i

and (2 – i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i

5.3.3 Multiplication of two complex numbers Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the product z1 z2 is defined as follows:

z1 z2 = (ac – bd) + i(ad + bc)

For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28

The multiplication of complex numbers possesses the following properties, which we state without proofs.

(i) The closure law The product of two complex numbers is a complex number, the product z1 z2 is a complex number for all complex numbers z1 and z2.

(ii) The commutative law For any two complex numbers z1 and z2,

z1 z2 = z2 z1.

(iii) The associative law For any three complex numbers z1, z2, z3,

(z1 z2) z3 = z1 (z2 z3).

(iv) The existence of multiplicative identity There exists the complex number 1 + i 0 (denoted as 1), called the multiplicative identity such that z.1 = z, for every complex number z.

(v) The existence of multiplicative inverse For every non-zero complex number z = a + ib or a + bi(a ≠ 0, b ≠ 0), we have the complex number (denoted by or z–1 ), called the multiplicative inverse of z such that

(the multiplicative identity).

(vi) The distributive law For any three complex numbers z1, z2, z3,

(a) z1 (z2 + z3) = z1 z2 + z1 z3

(b) (z1 + z2) z3 = z1 z3 + z2 z3

5.3.4 Division of two complex numbers Given any two complex numbers z1 and z2, where , the quotient is defined by

For example, let z1 = 6 + 3i and z2 = 2 – i

Then =

= =

5.3.5 Power of i we know that

,

, , etc.

Also, we have

In general, for any integer k, i4k = 1, i4k + 1 = i, i4k + 2 = –1, i4k + 3 = – i

5.3.6 The square roots of a negative real number

Note that i2 = –1 and ( – i)2 = i2 = – 1

Therefore, the square roots of – 1 are i, – i. However, by the symbol , we would mean i only.

Now, we can see that i and –i both are the solutions of the equation x2 + 1 = 0 or x2 = –1.

Similarly i2 = 3 (– 1) = – 3

= i2 = – 3

Therefore, the square roots of –3 are and .

Again, the symbol is meant to represent only, i.e., = .

Generally, if a is a positive real number, = = ,

We already know that = for all positive real number a and b. This result also holds true when either a > 0, b < 0 or a < 0, b > 0. What if a < 0, b < 0? Let us examine.

Note that

(by assuming = for all real numbers)

= = 1, which is a contradiction to the fact that .

Therefore, if both a and b are negative real numbers.

Further, if any of a and b is zero, then, clearly, = 0.

5.3.7 Identities We prove the following identity

, for all complex numbers z1 and z2.

Proof We have, (z1 + z2)2 = (z1 + z2) (z1 + z2),

= (z1 + z2) z1 + (z1 + z2) z2 (Distributive law)

= (Distributive law)

= (Commutative law of multiplication)

=

Similarly, we can prove the following identities:

(i)

(ii)

(iii)

(iv)

In fact, many other identities which are true for all real numbers, can be proved to be true for all complex numbers.

Example 2 Express the following in the form of a + bi:

(i) (ii)

Solution (i) = = = =

(ii) = = .

Example 3 Express (5 – 3i)3 in the form a + ib.

Solution We have, (5 – 3i)3 = 53 – 3 × 52 × (3i) + 3 × 5 (3i)2 – (3i)3

= 125 – 225i – 135 + 27i = – 10 – 198i.

Example 4 Express in the form of a + ib

Solution We have, =

= =

5.4 The Modulus and the Conjugate of a Complex Number

Let z = a + ib be a complex number. Then, the modulus of z, denoted by | z |, is defined to be the non-negative real number , i.e., | z | = and the conjugate of z, denoted as , is the complex number a – ib, i.e., = a – ib.

For example, , ,

and , , = 3i – 5

Observe that the multiplicative inverse of the non-zero complex number z is

given by

z–1 = = = =

or z

Furthermore, the following results can easily be derived.

For any two compex numbers z1 and z2 , we have

(i) (ii) provided

(iii) (iv) (v) provided z2 ≠ 0.

Example 5 Find the multiplicative inverse of 2 – 3i.

Solution Let z = 2 – 3i

Then = 2 + 3i and

Therefore, the multiplicative inverse of is given by

z–1

The above working can be reproduced in the following manner also,

z–1 =

=

Example 6 Express the following in the form a + ib

(i) (ii) i–35

Solution (i) We have,

= = .

(ii) =

Exercise 5.1

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

1. 2. 3.

4. 3(7 + i7) + i (7 + i7) 5. (1 – i) – ( –1 + i6)

6. 7.

8. (1 – i)4 9. 10.

Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.

11. 4 – 3i 12. 13. – i

14. Express the following expression in the form of a + ib :

5.5 Argand Plane and Polar Representation

We already know that corresponding to each ordered pair of real numbers (x, y), we get a unique point in the XY-plane and vice-versa with reference to a set of mutually perpendicular lines known as the x-axis and the y-axis. The complex number x + iy which corresponds to the ordered pair (x, y) can be represented geometrically as the unique point P(x, y) in the XY-plane and vice-versa.

Some complex numbers such as 2 + 4i, – 2 + 3i, 0 + 1i, 2 + 0i, – 5 –2i and 1 – 2i which correspond to the ordered pairs (2, 4), ( – 2, 3), (0, 1), (2, 0), ( –5, –2), and (1, – 2), respectively, have been represented geometrically by the points A, B, C, D, E, and F, respectively in the Fig 5.1.

The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane.

Obviously, in the Argand plane, the modulus of the complex number x + iy = is the distance between the point P(x, y) and the origin O (0, 0) (Fig 5.2). The points on the x-axis corresponds to the complex numbers of the form a + i 0 and the points on the y-axis corresponds to the complex numbers of the form 0 + i b. The x-axis and y-axis in the Argand plane are called, respectively, the real axis and the imaginary axis.

Fig 5.2

The representation of a complex number z = x + iy and its conjugate z = x – iy in the Argand plane are, respectively, the points P (x, y) and Q (x, – y).

Geometrically, the point (x, – y) is the mirror image of the point (x, y) on the real axis (Fig 5.3).

Fig 5.3

5.5.1 Polar representation of a complex number Let the point P represent the non-zero complex number z = x + iy. Let the directed line segment OP be of length r and θ be the angle which OP makes with the positive direction of x-axis (Fig 5.4).

Fig 5.4

We may note that the point P is uniquely determined by the ordered pair of real numbers (r, θ), called the polar coordinates of the point P. We consider the origin as the pole and the positive direction of the x axis as the initial line.

We have, x = r cos θ, y = r sin θ and therefore, z = r (cos θ + i sin θ). The latter is said to be the polar form of the complex number. Here is the modulus of z and θ is called the argument (or amplitude) of z which is denoted by arg z.

For any complex number z ≠ 0, there corresponds only one value of θ in 0 ≤ θ < 2π. However, any other interval of length 2π, for example – π < θ ≤ π, can be such an interval.We shall take the value of θ such that – π < θ ≤ π, called principal argument of z and is denoted by arg z, unless specified otherwise. (Figs. 5.5 and 5.6)

Fig 5.5

Fig 5.6

Example 7 Represent the complex number in the polar form.

Solution Let 1 = r cos θ, = r sin θ

By squaring and adding, we get

r2(cos2θ+sin2θ) = 4

i.e., (conventionally, r >0)

Therefore,

Therefore, required polar form is

The complex number is represented as shown in Fig 5.7.

Fig 5.7

Example 8 Convert the complex number into polar form.

Solution The given complex number =

= = (Fig 5.8).

Fig 5.8

Let – 4 = r cos θ, = r sin θ

By squaring and adding, we get

16 + 48 = r2(cos2θ+sin2θ)

which gives r2 = 64, i.e., r = 8

Hence cos θ = , sin θ =

Exercise 5.2

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

1. z = – 1 – i 2. z = – + i

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

3. 1 – i 4. – 1 + i 5. – 1 – i

6. – 3 7. + i 8. i

5.6 Quadratic Equations

We are already familiar with the quadratic equations and have solved them in the set of real numbers in the cases where discriminant is non-negative, i.e., ≥ 0,

Let us consider the following quadratic equation:

ax2 + bx + c = 0 with real coefficients a, b, c and a ≠ 0.

Also, let us assume that the b2 – 4ac < 0.

Now, we know that we can find the square root of negative real numbers in the set of complex numbers. Therefore, the solutions to the above equation are available in the set of complex numbers which are given by

x =

Note At this point of time, some would be interested to know as to how many roots does an equation have? In this regard, the following theorem known as the Fundamental theorem of Algebra is stated below (without proof).

“A polynomial equation has at least one root.”

As a consequence of this theorem, the following result, which is of immense importance, is arrived at:

“A polynomial equation of degree n has n roots.”

Example 9 Solve x2 + 2 = 0

Solution We have, x2 + 2 = 0

or x2 = – 2 i.e., x = = i

Example 10 Solve x2 + x + 1= 0

Solution Here, b2 – 4ac = 12 – 4 × 1 × 1 = 1 – 4 = – 3

Therefore, the solutions are given by x =

Example 11 Solve

Solution Here, the discriminant of the equation is

= 1 – 20 = – 19

Therefore, the solutions are

.

Exercise 5.3

Solve each of the following equations:

1. x2 + 3 = 0 2. 2x2 + x + 1 = 0 3. x2 + 3x + 9 = 0

4. – x2 + x – 2 = 0 5. x2 + 3x + 5 = 0 6. x2 – x + 2 = 0

7. 8.

9. 10.

Miscellaneous Examples

Example 12 Find the conjugate of .

Solution We have ,

= =

= =

Therefore, conjugate of .

Example 13 Find the modulus and argument of the complex numbers:

(i) , (ii)

Solution (i) We have, = = 0 + i

Now, let us put 0 = r cos θ, 1 = r sin θ

Squaring and adding, r2 = 1 i.e., r = 1 so that

cos θ = 0, sin θ = 1

Therefore, θ =

Hence, the modulus of is 1 and the argument is .

(ii) We have

Let = r cos θ, – = r sin θ

Proceeding as in part (i) above, we get

Therefore

Hence, the modulus of is , argument is .

Example 14 If x + iy = , prove that x2 + y2 = 1.

Solution We have,

x + iy = = =

So that, x – iy =

Therefore,

x2 + y2 = (x + iy) (x – iy) = = = 1

Example 15 Find real θ such that

is purely real.

Solution We have,

We are given the complex number to be real. Therefore

= 0, i.e., sin θ = 0

Thus θ = nπ, n ∈ Z.

Example 16 Convert the complex number in the polar form.

Solution We have, z =

= =

Now, put

Squaring and adding, we obtain

=

Hence, which gives

Therefore, (Why?)

Hence, the polar form is

Miscellaneous Exercise on Chapter 5

1. Evaluate: .

2. For any two complex numbers z1 and z2, prove that

Re (z1 z2) = Re z1 Re z2 – Imz1 Imz2.

3. Reduce to the standard form .

4. If prove that .

5. Convert the following in the polar form:

(i) , (ii)

Solve each of the equation in Exercises 6 to 9.

6. 7.

8.

9.

10. If z1 = 2 – i, z2 = 1 + i, find .

11. If a + ib = , prove that a2 + b2 = .

12. Let z1 = 2 – i, z2 = –2 + i. Find

(i) , (ii) .

13. Find the modulus and argument of the complex number .

14. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.

15. Find the modulus of .

16. If (x + iy)3 = u + iv, then show that .

17. If α and β are different complex numbers with .

18. Find the number of non-zero integral solutions of the equation .

19. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that

(a2 + b2) (c2 + d2) (e2 + f 2) (g2 + h2) = A2 + B2

20. If , then find the least positive integral value of m.

### Summary

• A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.• Let z1 = a + ib and z2 = c + id. Then

(i) z1 + z2 = (a + c) + i (b + d)

(ii) z1 z2 = (ac – bd) + i (ad + bc)

• For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists the complex number , denoted by or z–1, called the multiplicative inverse of z such that (a + ib) = 1 + i0 =1

• For any integer k, i4k = 1, i4k + 1 = i, i4k + 2 = – 1, i4k + 3 = – i

• The conjugate of the complex number z = a + ib, denoted by , is given by

= a – ib.

• The polar form of the complex number z = x + iy is r (cosθ + i sinθ), where

r = (the modulus of z) and cosθ = , sinθ = . (θ is known as the argument of z. The value of θ, such that – π < θ ≤ π, is called the principal argument of z.

• A polynomial equation of n degree has n roots.

• The solutions of the quadratic equation ax2 + bx + c = 0, where a, b, c ∈ R,

a ≠ 0, b2 – 4ac < 0, are given by x = .

### Historical Note

The fact that square root of a negative number does not exist in the real number system was recognised by the Greeks. But the credit goes to the Indian mathematician Mahavira (850) who first stated this difficulty clearly. “He mentions in his work ‘Ganitasara Sangraha’ as in the nature of things a negative (quantity) is not a square (quantity)’, it has, therefore, no square root”. Bhaskara, another Indian mathematician, also writes in his work Bijaganita, written in 1150. “There is no square root of a negative quantity, for it is not a square.” Cardan (1545) considered the problem of solving

x + y = 10, xy = 40.

He obtained x = 5 + and y = 5 – as the solution of it, which was discarded by him by saying that these numbers are ‘useless’. Albert Girard (about 1625) accepted square root of negative numbers and said that this will enable us to get as many roots as the degree of the polynomial equation. Euler was the first to introduce the symbol i for and W.R. Hamilton (about 1830) regarded the complex number a + ib as an ordered pair of real numbers (a, b) thus giving it a purely mathematical definition and avoiding use of the so called ‘imaginary numbers’.