# Doubt in Amplitude Modulation

I am doing a course in Communication Engineering and on reading Amplitude Modulation in my textbook I found that a typical AM is defined by the following equation;

$$s(t) = (1+k_{a}m(t))A_{m}cos(\omega t)$$ The reasoning which the book gives is that $$-1<k_{a}m(t)<1$$, so that $$1+k_{a}m(t) > 0$$ But why do this and instead just make $$0<k_{a}m(t)<1$$, because including a 1 would only make things worst by increasing transmission power and it also does not supress the carrier.

You could make $k_a$ vary in the range 0 to 1, if you also modified $A_m$, and interpretted $k_a$ as something other than the modulation that you put on the carrier with a conventional modulator, and recovered from the carrier with a conventional detector.
So you see it's easier to interpret $k_a$ conventionally, because then it maps onto what's typically happening as we use the system.
$k_a m(t)$ is defined to be in the range -1..1 because in practice the signal $m(t)$ used to modulate the carrier $A_m \cos(\omega t)$ actually is an AC signal centered at 0V (e.g. an audio signal).