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.