If `z` is a complex number, then find the minimum value of `|z|+|z-1|+|2z-3|dot`

Answer

`-1`

Explanation

`z +z ^(-1)=1`
` or z^(2) -z + 1=0`
`rArr z =- omega or -omega^(2)`
For `z = -omega`
`z^(100) + z^(-100) = (-omega)^(100) + (-omega)^(100)`
`= omega+(1)/(omega) = omega + omega^(2)=1`
For `z = -omega^(2)`,
`z^(100) + z^(-100)=- (-omega^(2))^(100) _ (-omega^(2))^(-100)`
` = omega^(200) + (1)/(omega^(200))`
`= omega^(2)+(1)/(omega^(2)) = omega^(2) + omega = -1`