Fraction

A fraction is a concept which measures magnitude.

Is 3/4 = 6/8? Well yes, in the obvious sense. But no, in the sense that 3/4 means that something was divided into four pieces and we are considering 3 of them, whereas 6/8ths means that something was divided into 8 pieces and we are considering 6 of them. Those two situations are not literally equivalent.

If you want to retain that information, you should identify whatever you are seeing as a pair, like $(3,4)$ or $(6,8)$. As soon as you take a quotient, you "lose" that information.

Ratio

A ratio is [TODO]

Ratios are very fundamental. Any measurement at all involves ratios.

A natural number is a set of several things, thought of in relation to a single thing.

A fraction is a set of one or more things, thought of in relation to a set of one or more things.

Just like how I defined an integer as an ordered pair of natural numbers, I could define a fraction as an ordered pair of natural numbers. The difference is that these pairs have a different equivalence relation, and different addition / multiplication operations.

Rational numbers

Rational numbers, which we denote $\mathbb {Q}$, is concept of signed differences between ratios.