Mathematics > Complex Numbers and Quadratic Equations

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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.
  • 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.
  • 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.
  • Given any two complex numbers z1 and z2, the difference z1 – z2 is defined as follows: z1 – z2 = z1 + (– z2).
  • 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).
  • Given any two complex numbers z1 and z2, where z2 ≠ 0 , the quotient  z1/z2 is defined by

     

  • 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 √a2+b2, i.e., | z | = √a2 + b2 and the conjugate of z, denoted as z` is the complex number a – ib, i.e., z` = a – ib.

    z z`=|z|2

  • Polar representation the nonzero complex number z = x + iy -- z = r ( cosθ + i sinθ )

    Where r=√x2+y2

Quadratic Equations

Roots of equation ax2+bx+c

 

Or

Examples 

Question

Find the multiplicative inverse of 2 – 3i.

Solution

z = 2 – 3i

z` = 2 + 3i

|z|2= (22+ (-3)2) = 13

z-1 = z`/|z|2=(2+3i)/13 = (2/13)+(3/13)i

 

Question

Represent the complex number z =1+ i √3 in the polar form.

Solution

1 = r cos θ, √3 = r sin θ          

r2 ( cos2 θ + sin2 θ ) = 4

r = 2

cos θ = (1/2), sin θ = (√3/2)

θ=(π/3)

z = 2(cos π/3+i sin π/3)

 

Question

Solve x2 + 2 = 0

Solution

x2 + 2 = 0

or 

x2 = – 2 x

or 

x2 = – 2 x = ± √−2 = ±√ 2 i

  

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