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Defined the Gear Ratio ($i$) as the ratio between the number of teeth on the pinion (driving gear) ($N_{1}$) and the number of teeth on the gear (driven gear) ($N_{2}$).
Why is this better, if better at all, having non-integer values of $i$? Could anyone please illustrate using arguments exploring the fabrication and performance points of view?
Gear ratios can be referred to as hunting and non-hunting ratios.
In a non-hunting ratio, any one gear tooth will contact its corresponding gears' teeth in the same place every time.
In a hunting ratio, the number of teeth in the gear set have no common factors between them. This way as the pitch diameters 'roll' over one another it takes a very long time for the same teeth to once again make contact. As previously mentioned, this is much better for wear, vibration and noise.
It should be fairly obvious that any gear can only have an integer number of teeth. Also for gears to mesh properly the tooth profile on all meshing gears needs to be similar if not identical. So, in practice the limitation is that individual gears need integer teeth and there is some pretty much fixed ratio between the number of teeth and radius which rules out some ratios for a given tooth profile and size.
However it is entirely possible to have non integer gear ratios.
Spur gears are usually designed on the module system, this is based on the ratio of N teeth to the diameter of the gear and thus eliminates the irrational pi from the specification.
This only applies to conventional spur gears though, with pulleys you can have any arbitrary ratio you want and it is certainly not impossible to fudge tooth sizes even in spur gear or chain drives to get arbitrary ratios at the expense of increased backlash and/or wear.
Non-integer gear ratio reduces gear noise and improves service life. Micromo has done a nice job of explainining why [1]. In short: Assume a gear tooth on gear A is faulty. Let's say the gear ratio is 3:1 (See picture 1). Then, this faulty tooth engages with three locations on the larger gear (gear B) at every rotation of the gear. Imagine this in the long run and it will cause wear only on those three locations. Now imagine the gear ratio was 3.444:1 (See picture 2). then the engaging locations on the larger gear (gear B) keeps shifting at every rotation (see the picture where the numbers on the wheel shows engagement at that number of rotation of gear A), hence wear is distributed among all gear B teeth and each tooth takes less wear in the long run thus more service life.
The primary reason is it promotes longer gear life. If the gear ratio were an integer number and a tooth was damaged, that tooth would wear on the same few opposing teeth until failure. A non integer ratio causes a damaged tooth to wear across all opposing gear teeth. This spreads the effective wear out, vice focusing it like an integer ratio would.
reading these answers i think there is some misunderstanding re: what is meant by integer ratios. For most gear sets (whether spur, helical, etc) the ratio is the ratio of the number of teeth which is obviously an integer ratio.
Example a typical example differential might have a 41 tooth ring and 11 tooth pinion for a ratio of 41:11. That is typically expressed in decimal which is 3.72727272...:1 which is typically rounded off to 3.73:1. The actual ratio remains a ratio of integers.
if you have a transmission with a sequence of gears you would multiply all the integer ratios and still have an overall integer ratio.
A bit aside to the question, in this example common teeth repeat contact only every 11 revs of the big gear, and every tooth combination meshes an equal number of times. The reason cited by @ahipsterpeterpan may be why Chevy chose prime numbers for the tooth count.
