Download XI OPERATIONS WITH COMPLEX NUMBERS

XI OPERATIONS WITH COMPLEX NUMBERS.
A complex number a + bi has a real part (a) and an imaginary part (i). A complex
number a + bi is imaginary if b  0 , and pure imaginary if it is of the form bi.
Of the numbers, 3  5i, 2  i, and 3i , the numbers 3  5i and 2  i are
imaginary, and 3i is pure imaginary.
Two imaginary numbers are equal if and only if the real parts of the numbers are
equal, and if the imaginary parts of the numbers are equal.
If 4  3i  x  y i , then 4 = x and 3 = y.
To find the sum of two complex numbers, add the real parts of the number, and add the
coefficients of the imaginary parts of the number :
(a  bi)  (c  di)   a  c   b  d  i
Example. Find the sum of 2 + 3i and 7 - 4i.
2 + 3i + 7 - 4i = 2 + 7 + (3 - 4)i = 9 - i
To find the product of two complex numbers, multiply the two terms as two binomials :
(a  bi)(c  di)  ac  bci  adi  bdi 2
 ac  i (bc  ad )  bd
 ac  bd  i (bc  ad )
Example. Find the product of (3 - 2i)(2 + 5i)
(3  2i )(2  5i)
 (3)(2)  (3)(5i )  (2)(2i)  (2i)(5i)




6
6
6
16
 15i  4i  10i 2
 11i  10(1)
 11i  10
 11i
To find the quotient of two complex numbers,
a  bi
, multiply the fraction by the
c  di
conjugate c  di of the denominator :
a  bi c  di

=
c  di c  di
(a  bi )  c  di 
ac  bd  (cb  ad )i
(c  di )(c  di) c 2  cdi  cdi  d 2i 2
ac  bd  (cb  ad )i

c2  d 2

Example. Find the quotient of ( 2  i )  ( 3  4i)
Example. Find the reciprocal of 3 - i.
Since the product of a reciprocal and a given number must equal one, the
1
reciprocal of 3 - i must equal
. However, this is not a complex number in simplest
3i
form, so the denominator must be rationalized by using the conjugate of 3 - i which is
equal to 3 + i.
1
3i
3i
3i



2
3  i 3  i 9  3i  3i  i
9  i2
3 i
3 i
3
1



 i
9  (1)
10
10 10