In a book I am following, it is assumed that the temperature resistance relationship is linear.
I would expect, then, that the resistance at a given temperature would be given by the formula
$$ R_{t} = mt + R_{0} $$
Where m is increase in resistance per unit temperature. Therefore by taking another known resistance at temperature A we could calculate this m
$$ m = \frac{R_{A} - R_{0}}{A - 0} $$
However the book (and everywhere else) defines the temperature coefficient as the above gradient divided by the resistance at temperature 0.
$$ \alpha = \frac{R_{A} - R_{0}}{R_{0}(A - 0)} $$
and then the formula for resistance at temperature t as
$$ R_{t} = \alpha R_{0}t + R_{0} \equiv R_{0}(\alpha t + 1) $$
My question is: what is the purpose of dividing by resistance at temperature 0 just to multiply by it again later, and what does this alpha coefficient actually 'mean' (in the same way I understand gradient m 'means' resistance increase by increasing temperature by 1 unit)?