Q:

What is the definition of value?

Imagine a number line. Each number has a value, but how is value defined? For rational numbers this is self evident, but what about real numbers? For example, we know that $\pi>e$, because $\pi>3$ while $e<3$. So here we have a rational number between them. This however doesn't tell us the value of either $\pi$ or $e$. Do we have to use asymptotically approaching rational numbers to define the value of a real number through a procedure with an infinite number of steps? Or is there a different method?

A:

Yes, as you say, in general, we have to use asymptotically approaching rational numbers to define the value of a real number through a procedure with an infinite number of steps. Of course, because rational numbers, such as $3$, are also real numbers, sometimes a real number can be defined finitely.

