If the two complex numbers 5+a + 4i and 3 +2bi -3i are equal, what is the value of a and b? What......is the additive and multiplicative inverse of the complex number?

Expert Answers
thilina-g eNotes educator| Certified Educator

Let `Z1 = (5+a) + 4i`    and `Z2 = 3+(2b-3)i.`

If two complex number are equal to each other, their real parts and imaginary parts are equal to each other separately.


Analysing the real part,

`(5+a) = 3`

Therefore, `a = -2`


Analysing the imaginary part,

`4 = (2b-3)`

`2b = 7`

`b =7/2`


Therefore `a = -2` and `b =7/2` .

The complex number represented is,


`Z = 3+4i `


The additive inverse of a complex number is the complex number which makes the addition of them equal to 0+0i.

Therefore additive inverse of `3+4i = -3-4i`

Additive inverse is `Z' = -3-4i.`


The multiplicative inverse of a complex number is the complex number, Z such that,

`(3+4i)Z'' = 1`

`Z'' = 1/(3+4i)`

`Z'' = (3-4i)/((3+4i)(3-4i))`

`Z'' = (3-4i)/(9-16i^2)`

`Z'' = (3-4i)/(9+16)`

`Z'' = (3-4i)/25`


Therefore the multiplicative inverse is `Z'' = 1/25(3-4i)`