Difference between fractions and rational numbers?

Question

What is the difference between fractions and rational numbers? is $\frac{\pi}{1}$ a fraction?

Answer 1

Even without touching on irrational numbers, there is a huge difference between rational numbers and fractions. The difference is somewhat analoguous to the difference between a natural number and the decimal system.

Rational numbers are points on the number line. They are quantities. They are numbers. They exist (as far as any number can exist) even without us humans to study them and write about them. If an alien species landed on earth and talked to us, they would almost certainly know what a rational number is (although they may not use the same word).

A fraction is one way (of several) that we humans have invented / discovered that lets us describe rational numbers in a nice way. It's a piece of terminology. The aliens may use fractions (possibly written in a different way) or they may not. Maybe they like to work in terms of repeating decimal expansions, and have never even thought of using division as a main way to describe numbers.

Finally, I'll leave with this: $\frac12$ and $\frac24$ are different fractions, but they represent the same rational number.

Answer 2

A rational number is a number which can be expressed as a fraction of two integer values.

$\cfrac \pi1$ is a fraction, but it does not evaluate to a rational expression.

A fraction which evaluates to a rational expression is defined by an expression that can be expressed as fraction of two integer values.