but isn't something wrong with this? how can we derive |a|=|b| from just a=|b|?
there is nothing wrong. we can derive |a| = |b| from a = |b| i.e.
a = |b| => |a| = |b| (but not the other way around).
you just need to consider the range of numbers that the first equation implies for 'a'. in the first equation (a = |b|) the right side ( |b| ) is 0 or greater than 0, therefore the left side (a) is also 0 or greater, so "a" can't be a negative number
(a >= 0).
let me explain in a different way:
1) original equation:
a = |b| implies that:
I. 'b' can be anything
II. a is 0 or greater than 0 i.e. 'a' cant be negative (a>=0).
b= -0.6 and a is 0.6
b = 0 and a is 0
b = 10 and a is 10
2) the new equation:
|a| = |b| says that absolute value
of 'a' is equal to absolute value of 'b'. why is this ok?
because from the first equation we already know that 'a' is not negative
, and since absolute value of non-negative numbers are the same as the number, the new equation true also.
remember always pay attention to range of values a variable can take. for example in X = 1/Y you should beware that y can never be 0.