how i^(2) = -1let i = (-1)^(1/2)=(3-4)^(1/2)= {[(3-4)^(2)]^(0.5)}^(0.5)={[(9+16-24)^(0.5)]^(0.5)= {[(1)^(0.5)]^(0.5)=(1)^0.5=1.......then why cant we write i=1 ? then all imaginary number concept will be vanished

Hello student,
Please find the answer to your question below
The proof which you are showing is an analytical proof
These kind of proofs can be done in mathematics to show 2=1,2=5 etc,,,which are actually not true
‘i’ actually represents a rotation of 90 degrees that is why we represent a complex number as (x,y) while plotting on graph
so i²means i multiplied by i which makes the rotation 180 degrees so on a graph paper you can see that this 180 degree rotation makes magnitude negative.hence
i²is treated as -1.