Fault in new cartesian sign conventions

If we analye, while proving mirror,lens or lens makers formula we make use of sign coventions but when we solve numericals we again make use of sign conventions why? This creates problem for new cartesians sign conventions

INERTIAL OR NON INERTIAL FRAME

WHEN A BODY FALLS TOWARDS EARTH , IT SLIGHTLY MOVES TOWARDS WEST & EARTH ROTATES EAST TO WEST, DOES THIS MEAN EARTH IS NON INERTIAL FRAME OF REFRENCE.

NON INERTIAL FRAME OF REFRENCE

IS EARTH INERTIAL OR NON INERTIAL FRAME OF REFRENCE

find mathematical error

¡(iota)= -¡
proof

¡ = √-1
=√(-1×-1×-1)
=√-1 ×√-1 ×√-1
=¡× ¡× ¡
=-1 ס
» ¡ =-¡

WE KNOW THAT

2<3

Multiplying 2 from both sides

2^2 < 3×2
Subtracting 3^2 from both sides

2^2 - 3^2 < 3×2 -3^2

(2-3) (2+3) < 3(2-3) using a^2 -b^2 & taking 3 common

Canceling 2-3 from both sides
2+3< 3

5<3