Actually, the meter was originally defined as one ten-millionth of the length of the meridian through Paris from pole to the equator. So in essence, the meter was one forty-millionth of the circumference of the earth. But it was soon figured out that the earth isn’t perfectly spherical due to rotation, so they had to nix that definition.
As Nametag said, a meter is currently defined differently. It is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
Reaistically, what happens with units is that their definitions change. For instance, it used to be that the inch (or foot, or whatever) was defined in terms of one physical artifact, and the meter was defined in terms of another. If one considers there to be an exact conversion between the two (even if it’s unknowable to humans), then that conversion factor is almost certainly irrational, since there are far more irrational numbers than rationals.
Then, someone decided that it was silly to have two separate standards, so they took the best approximation they had to the conversion factor, and then changed the definition. In this case, they noticed that 1 inch was awfully close to 2.54 centimeters, so they just said “From now on, we’ll define that 1 inch equals exactly 2.54 centimeters.”. The result of this is that the old inch and the new inch aren’t exactly the same (although they’re as close as it was practical to make them). Nowadays, most units are defined in this way relative to each other, so most conversion factors between modern units are, in fact, rational.
The only way the conversion could be irrational is if each is known to infinite accuracy and that is plainly impossible. They could be given independent definition and measured to, say, 15 place accuracy (that is about as accurate as ANYTHING has ever been measured), so the conversion would also be known to 15 place accuracy. Well, any 15 place decimal is a rational number. Not only that but a finite decimal. You might say that tomorrow they could achieve 16 place accuracy and the conversion factor would now change. That is true, but in fact, the Imperial units are now all given in terms of exact, finite decimal conversions.
The meter was originally defined as 1/10,000,000 of the distance from the north pole to the equator on a lone through Paris. They attempted to measure this and store on a platinum-iridium bar stored in a temperature controlled vault in a suburb of Paris called Sevres. Of course they got it wrong. They then defined it as the length of that bar (actually as the distance between two lines inscribed on the bar). But that is not very convenient and it was finally defined in terms of the wavelength of some precisely defined radiation. I am sure someone can find the actual definition. I think it involves a partticular transition in cesium. So now anyone with a good clock can measure it anywhere. Clocks have been built accurate to 15 places and better ones are in the works.
A book I own claims that the number two is defined as the cardinality of the set consisting of a pair of platinum-iridium balls stored in said vault.
I have an example for you: convert frequency (hertz = 1/s) to angular velocity (radian/s). They’re both listed on the NIST page Zut referenced.
They differ by a factor of 2*pi.
From my point of view, the use of rational conversion factors is a mere convienience, in the spirit that Chronos described. Rational numbers are, well, you know, so rational to work with.