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Complex Numbers

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Published in: Mathematics
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SOME BASIC AND IMPORTANT CONCEPTS REGARDING COMPLEX NUMBERS

Brijesh S / Mumbai

10 years of teaching experience

Qualification: B.Tech/B.E. (Mumbai University - 2010)

Teaches: Biology, Chemistry, Algebra, English, Marathi, Mathematics, Science, Special Education, Vedic Maths

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  1. SOME BASICS AND IMPORTANT CONCEPT OF A COMPLEX NUMBER Conjugate of a Complex Number: 1. The conjugate of a complex number z =a+ib is defined as a-ib and is denoted by i Modulus and Argument of a Complex Number: 2. If z = a+ib is a complex number then, a2 + b2 Modulus of z, denoted by mod z or Izl, is defined as Izl = and, Argument (amplitude) 9 or amp of z or arg of z is defined by the equations i.e., b a cose = , sin Izl Izl b = tan a Power and square root of a complex number: 3. Power of i: •2=-1, We have I So, in general, i4n= 1, •411+1 — •4n+3 = -i, where n e N Square root of complex number: In this section, we obtain the square root of a complex number.
  2. Let z = a+ib and w = X+iy be two complex numbers such that w is a square root of z. Hence, Squaring both sides, we get i.e., + 2xyi = a + ib Equating real and imaginary parts, We get, — a , 2xy = b Solving these equations, we get square roots of z. Let's have some examples for more understanding. Find the square root of 8— 6i Solution: • 2 = 8-6i Let — Y2) + 2xyi = 8-6i Compare real parts and imaginary parts, Iz12=lzl :. 2xy = -6 Now, consider the modulus: + y 2 = + 62) = 10 Solving (1) and (3), we get and and From (2), x and y are of opposite sign, (1) (2) (3) :. (x = 3 and y = -1) or (x = -3 ory= 1)
  3. 4. 5. DeMoivre's Theorem: If n is the set of integers, rational numbers or being any fraction, then, ( Cos 9+ isin 9 ( Cos 9- isin 9 Also, ( Cos 9- isin 9 = Cos no + isin no = Cos no - isin no - Cos no + isin no Properties of Complex Numbers: Properties of Addition: ii. iii. Closure Property: The sum of two complex numbers is always a complex number. Commutative property: For any two complex numbers Zl and z2, we have Zl + .72 + Associative property : For any three complex numbers Zl, z2 and z3, we have Existence of additive identity : For any complex number z, we have z +0=0+z=z Existence of additive inverse: Every complex number z = a + ib has —z = -a + i(-b) as its additive inverse as z + (-z) = (-z) + z
  4. ll. Properties of Multiplication : (Here also we have 5 properties for multiplication similar to addition, as mentioned above) Additionally we have a law i.e., Distributive law: For complex numbers Zl, z2, z3,we have Zl(Z2+Z3) - ZIZ2 + Zl Z3 ( + = ZIZ3 + 6. 7. Euler's theorem: The complex number elX can be written as elX = cos x + i sin x (1) from which follows: a) cos x = Re[elX] where Re stands for real and sin x = elX] where 1m stands for imaginary b) The complex conjugate of elX is e so that (2) = cos sin x c) Which leads us to the following important results, the first by adding eq(l) and eq(2), the second by finding their difference. COS X = ( elX+ e-lX)/2 sin x = ( elX- e-lX) / 2i Polar form of complex number: The polar form of a complex number is another way to represent a complex number. i o b
  5. The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of r and where r is the length of the vector and B is the angle made with the real axis. From Pythagorean Theorem: By using the basic trigonometric ratios: a cose=— And sin e = — Multiplying each side by r: r cose=a andrsine=b The rectangular form of a complex number is given by Substitute the values of a and b . = r cos e sin = r(cos 9+i sin 9) In the case of a complex number, r represents the absolute value or modulus and the angle is called the argument of the complex number. This can be summarized as follows: The polar form of a complex number z = a + bi is z , where
  6. and e = tan -l a or 1b or e = tan a —l _ +1800 a a = r cose and b = sin e for a > 0 for a < 0. 8. Solution of a Quadratic Equation in complex number system: Let the given equation be ax2+ bx + c = 0 where a, b, c e R and a . The solution of this quadratic equation is given by Hence, the roots of the equation ax2+ bx + c = __b-l- b2 4ac 0 are and —b a: b2 4ac b2 4ac The expression is called the discriminant. If b2 — 4ac < 0 then the root of the given quadratic equation are not real in nature.
  7. 9. Cube roots of unity: Cube roots of unity are, 1, 1+07 If we consider a = 2 2 then after solving, we get, 2 and, — 2 2 Thus if a and ß are complex cube roots of unity then ß and ß2 = a Let us consider a = w (omega) then ß = . We denote one complex cube root by letter w. So that the other root is (-02 . Thus 1, w, (-02 are the three cube roots of 1. Hence, w = Power of w; u.)3n- 1 and w2- 2 2 2 ii. iii. 3n+1 — 311+2 —

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