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Complex NumbersThe mathematical need to have solution for negative discriminant, these new kind of numbers we call it as complex numbers.According to the quadratic formula , if b ^{2}  4ac ≥ 0 then roots are real but if b ^{2}  4ac < 0 then the roots are not real. For example : x ^{2} + 4 = 0 x ^{2} =  4 ∴ x = $\pm$ √ (4) ∴ x = $\pm$ 2 √ (1) We assume that the square root of (1) is denoted by the symbol ‘i’, called imaginary unit. The symbol ‘i’ was first introduced by the famous Swiss mathematician, Leonhard Euler in 1748, possibly because ‘i’ is the first letter of the Latin word ‘imaginarius’. Thus , for any two real numbers ‘a’ and ‘b’, we can form a new number a + ib . This number a + ib is called a complex number. The set of all complex numbers is denoted by ‘C’. The complexnumbers is of the form ‘a + ib’, where a and b are real numbers and ‘i’ is the imaginary unit which has the property of i ^{2} = 1 .So in a given complexnumbers ‘a’ is a real part and ‘b’ is its imaginary part. Example : 1) 4 + 7i 2) 1/2 + i In the 1st example 4 is a real part and 7 is an imaginary part. In the 2nd example 1/2 is a real part and 1 is an imaginary part. A complexnumber is denoted by single letter such as z, w etc. z = a + ib are denoted as a = Re z and b = Im z . Two complexnumbers z_{1} = a_{1} + ib_{1} and z_{2} = a_{2} + ib_{2} z _{1} = z _{2} , if a _{1} = a _{2} and b _{1} = b _{2} A complexnumber ‘z’ is said to be zero if its both real and imaginary parts are zero. In other words z= a + ib = 0 if and only if a = 0 and b = 0 If z = a + ib, the number a – ib is called the complex conjugate(or simply conjugate) of a + ib and is denoted by z̄ Examples on complex Numbers1) Write the following as complexnumbers.i) √(27) ⇒ √(1 x 27) = √(1) √27 = i√27 ii) 4  √(5) ⇒ 4  √(1 x 5) = 4  √5 √(1) = 4  i√5 2) Write the real and imaginary part of the following numbers. i) 2 + i √2 Solution Let z = 2 + i√2 Re z = 2 and Im z = √2 ii) √3/ 5 i Solution : Let z = √3/5 i = 0 + √3/5 i Re z = 0 and Im z √3/5 3) Find a and b such that 2a + i 4b and 2i represents the same complexnumbers. Solution : 2a + i 4b = 2i 2a + i 4b = 0 + 2i By the definition of equality, 2a = 0 ⇒ a = 0 And 4b = 2 ⇒ b = 2/4 = ½ ∴ a = 0 and b = ½ 4) Find the conjugates of 2 + 5i and √5 Solution : By definition, the complex conjugate is obtained by reversing the sign of the imaginary part of the complexnumbers. Hence the required conjugates are 2 – 5i and √5 (as there is no imaginary part) From complex numbers to Home page Covid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! Covid19 has affected physical interactions between people. Don't let it affect your learning.
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