Complex Numbers

• Basics of Complex Numbers   



• Definition of i
  
  i=−1−−−√
• i2=−1
  
  i3=−i
  
  i4=1
• A complex number is any number which can be written as a+ib where a and b are real numbers and i=−1−−−√
  
  a is the real part of the complex number and b is the imaginary part of the complex number.
  
  Example for a complex number : 9 + i2
• if z=a+ib is a complex number, a is called the real part of z and b is called the imaginary part of z.
  
  It can be represented as Re(z) = a and Im(z) = b
• Conjugate of the complex number z=x+iy can be defined as z¯=x−iy
  
  Example : 4+i2¯¯¯¯¯¯¯¯¯=4−i2 and 4−i2¯¯¯¯¯¯¯¯¯=4+2i
• if the complex number a+ib=0, then a=b=0
• if the complex number a+ib=x+iy, then a=x and b=y
• if x+iy is a complex numer, then the non-negative real number x2+y2−−−−−−√ is the modulus (or absolute value or magnitude) of the complex number x+iy. It can be denoted as
  
  | x+iy |=x2+y2−−−−−−√     (Note that modulus is a non-negative real number )
• eiθ=cosθ+isinθ     (Euler Formula)
• Rectangular (Cartesian) Form of Complex Numbers
  
  A complex number when written in the form a+ib, it is in the rectangular (Cartesian) form
• eiθ=cosθ+isinθ     (Euler Formula)