Complex Numbers and Quadratic Equations

Complex Numbers

  • 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.


  •  If a < 0 then \sqrt{a} = \sqrt{|a|} i
  •  The system of complex number includes the system of real numbers.
  •  Every real number is a complex number.
  •  ‘O’ is both purely real and purely imaginary number.
  •  A complex number is an imaginary number if and only if its imaginary part is non-zero. Here real part may or may not be zero.
  •  All purely imaginary numbers except zero are imaginary numbers but an imaginary number may or may not be purely imaginary.
  •  Inequality relation does not hold good in case of complex numbers having non-zero imaginary parts. Ex: “8 + 5i > 3 + 7i” makes no sense.
  •  Two complex numbers are said to be equal if and only if their real parts and imaginary parts are separately equal.

View the Topic in this video From 00:40 To 19:56

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  •  For any n∈N i2n = (i2)n = (−1)n = 1 (if n is even) 
                                                   = −1 (if n is odd)
  •  For any n∈N i2n+1 = (i2n).i = (−1)n.i = i (if n is even) 
                                                         = −i (if n is odd)
  •  The sum of 4 consecutive powers of ‘i’ is ‘o’.
  •  Also for any n∈N, the value of i−n is found out by writing this as \frac{1}{i^{n}} and solving in.