Algebra of Complex Numbers
Properties of addition
(1) Sum of two complex numbers is a complex number (closure law)
(4) 0 + 0i denoted by 0 is the identity for addition (0 is the additive identity) since z + 0 = 0 + z = z. For any complex number z.
(5) -z= -a - ib is the negative or additive inverse of z = a + ib
. Since 0 is the additive identity for any complex number z, - z is the additive inverse of z.
- Subtraction: is defined as
- Multiplication of complex numbers
Properties of multiplication
(1) Product of two complex numbers is a complex number (closure law)
(4) 1 + 0i is the multiplicative identity since
(5) multiplicative inverse or reciprocal of z. Since we cannot leave a complex number in the denominator of a rational number, we can simplify by multiplying and dividing by its conjugate.
Note:- Division of a complex number by another is done using the same principle.
Example: If ,prove that
Using property (1) of conjugates
Example 9: If , find the real values ofx and y .
Equating real parts on both sides,
Equating imaginary parts on both sides
Substitute for y from (2)
P represents a variable complex number Z. Find the locus of P if
Let [treat it like point (x, y)]
(Remark: you will learn soon that a complex number can bne represented by a point in a plane.)
This is the locus of
Identities as applied to complex numbers
In fact all the identities which are true for real numbers are true for complex numbers also.