Does the number of teeth in the gears of a bicycle matter when determining its gain ratio? When talking about bicycles, it is usual to measure their gears by the number of teeth they have. This measure is then used to calculate the gear ratio: the ratio between the number of teeth of the front gear and those of the back gear. This in turn is calculated for each front-back gear combination available on a bike to describe the range of its achievable gains.
My physics knowledge is very limited, but when I imagine what determines those gains I come to the same conclusions as this answer: the gain is determined by the radiuses of the crank, the chain wheel (or front gear), the rear wheel, and the sprocket (or rear gear).
Of course, if the distance between teeth has to be the same for all gears (due to the chain being the same) more teeth means bigger radius and vice versa.
However, few ever talk about radiuses, most discuss only about the number of teeth. This often leads some to some claim that the number of teeth plays a significant role, thus that different numbers of teeth would lead to different gains despite the radiuses being the same.
This feels odd to me: I have the feeling that having more teeth merely serves to distribute forces among them to prevent them from wearing out, but I'm unable to confirm it because I don't know how to approach the problem from a physics standpoint.
Does the number of teeth (on fixed radiuses) actually matter? If so, what are the principles behind it?
 A: The number of teeth is indeed the critical factor in a well designed gear, as this will give the actual ratio of the distances moved. Of course the size of the teeth must match the chain, otherwise there would be serious energy losses, resistance to motion, and wearing of both the chain and the teeth. This means that the circumference of the gear wheels must be strictly proportional to the the number of teeth, and likewise the radius will be proportional to the circumference. But the very fact of teeth means that neither the radius nor the circumference are as easy to state, or as precise to measure, as the number of teeth, and in a well designed gear the radius is determined by the number of teeth.
A: What matters is the distance your foot travels (mostly on the down stroke for recreational riders) divided by the distance the rear tire travels...for your mechanical leverage (inverse gain).
So if the gears are 1:1, then that is going to depend on the crank-arm length divided by the rear wheel diameter (plus some variation for tire thickness).
Regarding the gears, the ratio of the distance between the chain and BB spindle in front and the distance between the chain and rear axle determines your mechanical advantage. That ratio is equal to the ratio of the number of teeth on the sprocket divided by the number of teeth on the cog....
...unless you are running Biopace® or some other non-circular gearing, then there is additional leverage:

A: You're right. Ratios are not determined by teeth per se. The ratios have to do with distance travelled by the gears (circumference of the gears).
The teeth are sized to fit the chain, the gear sizes are chosen to allow whole numbers of teeth along their circumferences. So there are only certain sizes that work.
But, in theory you could have 4 teeth on a gear, with large spacing between them. The force would only be divided by the number of teeth locked into the chain at any one time, most likely increasing wear and tear of the teeth. Also, you would probably have higher chance of slippage and other problems.
NB. Most industries have standardised gear or teeth sizes, so it's easier to just talk in number if teeth.
A: What matters is not the absolute number of teeth, but the ratio between the numbers of teeth on different gears of the same bicycle, which is the same as the ratio of the radii.
Provided that a tooth has a triangular form with base size of $l$, the number of teeth on a gear with radius $R$ is $n=2\pi R/l$, wheres the ration of the numbers of teeth on two gears is
$$
\frac{n_1}{n_2}=\frac{2\pi R_1}{l}\frac{l}{2\pi R_2}=\frac{R_1}{R_2}
$$
A: For two gear to mesh, they must have a common gear module $$m = \frac{\text{no. of teeth}}{\text{diameter}} = \frac{z_i}{d_i}$$
So two meshed gears must have $$ \frac{z_1}{d_1} = \frac{z_2}{d_2}$$
and if the gear ratio is described by the ratio of diameters or the ratio of teeth, it does not matter as they are both the same
$$ \gamma = \frac{d_2}{d_1} = \frac{z_2/m}{z_1/m} = \frac{z_2}{z_1} $$
