Complex Number and Quadratic Equations - Solutions and Study Material

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Complex Number : Let us denote \sqrt{-1} by the symbol i. Then ,we have i^{2} = -1. This means that I is a solution of the equation x^{2}+1=0

A number of the form a + ib, where a and b are real numbers, is defined to be a complex .

Complex Number: 2 + 3i, (-1)+i\sqrt{3} , 4+i\left(\frac{-1}{11}\right) 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 Z_{1} = a + ib and Z_{2} = c + id are equal if a = c and b = d

Example: 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 x = \frac{3}{4} , y = \frac{33}{4}

Algebra of Complex Numbers: In this Section, we shall develop the algebra of complex numbers

Addition of two complex numbers Let Z_{1} = a + ib and Z_{2} = c + id be any two complex numbers .Then , the Z_{1} is define as follows :

Z_{1} +Z_{2} = (a + c) + i(b + d) , which is again a complex number.

For Example (2 + 3i) +(-6 +5i) = (2-6) + I (3 + 5) = (-4 + 8 i)

The addition of complex number satisfy the following properties:

  • The closure law The sum of two complex numbers is a complex number, i.e., Z_{1} +Z_{2} is a complex number for all complex numbers Let Z_{1} and Let Z_{2}
  • The commutative law For any two complex numbers Z_{1} and Z_{2} ,
    Z_{1} +Z_{2} \:=\:Z_{2} +Z_{1}
  • The associative law For any three complex numbers Z_{1},Z_{2},Z_{3},
    Z_{1} +(Z_{2}+ Z_{3})\:=\: Z_{1}+( Z_{2}+ Z_{3})
  • 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
  • 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)

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