Bath Tub Mathematics

As Fiorella wallowed in the tub, she contemplated numbers, which, not being mathematically inclined, is unusual for her.  On the other hand, patterns are her cup of tea, and it was the patterns of numbers that struck her as interesting.  For instance, the fact that every odd number can become an even number by adding or subtracting "one."  And that every even number can become an odd number by the same process.   And that odd numbers can become even numbers by doubling them, but only even numbers that were originally odd numbers can become odd by division.

And then she wondered if a number system could be set up using only odd numbers, with the evens represented in relation to them, like more than one/less than three, more than three/less than five, etc.  What do you think?