Identities o ( z1 + z2 ) 2 = z1 2 + z2 2 + 2z1z2 o ( z1 - z2 ) 2 = z1 2 + z2 2 - 2z1z2 o ( z1 + z2 ) 3 = z1 3 + z2 3 + 3z1z2 2 + 3z1 2 z2 o ( z1 - z2 ) 3 = z1 3 - z2 3 + 3z1z2 2 - 3z1 2 z2
o z1 2 - z2 2 = ( z1 + z2 ) ( z1 - z2 )
Example : Express ( 5 – 3i ) 3 in the form a + ib .
Solution : ( 5 – 3i ) 3 = 5 3 – 3 × 52 × ( 3i ) + 3 × 5 ( 3i ) 2 – ( 3i ) 3 = 125 – 225i – 135 + 27i = – 10 – 198i .
Modulus & Conjugate of a complex Number Let z = a + ib be a complex number . Modulus of z , denoted by | z |, is defined to be real number ( a 2 + b 2 ) 1 / 2 , | z | = ( a 2 + b 2 ) 1 / 2
Numerical : Find the Modulus of ( 3 – 4i ) Solution : | z | = ( a 2 + b 2 ) 1 / 2 = ( 3 2 + 4 2 ) 1 / 2 = 5
Let z = a + ib be a complex number . the complex number a – ib , i . e ., Z = a – ib . Also Z * = | Z | 2 Or Z – 1 = 1 / | Z | 2 ( Useful to find inverse of a complex number )
Numerical : Find the conjugate of ( 3 + 4i ) Solution : Conjugate = 3-4i
Numerical : Find inverse of ( 3 + 4i ) Z – 1 = 1 / | Z | 2 = ( 3 – 4i )/ 5 = 3 / 5 – 4 / 5i
Argand Plane & Polar representation Complex numbers can represented in 2 forms
o o
Argand Plane Polar Representation
Argand Plane
The complex number x + iy can be represented geometrically as the unique point P ( x , y ) in the XY-plane and vice-versa . Plane with complex number assigned to each of its point is called complex or Argand plane .