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.
It doesn't matter what equation you use to describe AM, the actual physical modulation itself has an unsuppressed carrier, and a certain amount of power. It's a very simple modulation, one of the earliest, and as such is inefficient, but easy to use.
If you want a suppressed carrier, that's a different type of modulation, which needs a more complicated demodulator to read it.
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).
