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?
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`
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)`