If ` z_(1) =3 -4i and z_(2) = 5 + 7i`,verify
(i) ` |-z_(1)| =|z_(1)|`
(ii) ` |z_(1) +z_(2)| lt |z_(2)|`

Explanation

we have `z_(1)=3-4iand z_(2)=5 +7i`
`(i) -z_(1)= -(3-4i)= - 3 +4i`

`|-z_(1)|=sqrt((-3)^(2)+(4)^(2))`
` =sqrt25 =5`
`and |z_(1)|=sqrt((3)^(2)+(-4)^(2))=sqrt25 =5`
` |-z_(1)|=|z_(1)|`
`(ii) z_(1) +z_(2) = ( 3-4i) + ( 5+7i)`
= 8 + 3i
` |z_(1) +z_(2)|=sqrt((8)^(2)+(3)^(2))=sqrt(64+9) =sqrt73`
`Rightarrow |z_(1) +z_(2)|^(2) =(sqrt(73))^(2) =73`
`and |z_(2)|=sqrt((5)^(2)+(7)^(2))=sqrt74`
`Rightarrow |z_(2)|^(2)=(sqrt74)^(2) =74`
` Rightarrow |z_(1)+Z_(2)|^(2) lt|z_(2)|^(2)`
`Rightarrow |z_(1) +z_(2)| lt |z_(2)|` [ Taking square root on both sides ]