What is the difference between two real numbers?

Question

Let $x$ and $y$ be real numbers. What does the difference between x and y mean?

1. $|x-y|$ or $|y-x|$?
2. $x-y$?
3. $y-x$?

To me, only the first case makes any sense whatsoever. However, I cannot find a formal definition.

Answer 1

Difference between $x$ and $y$ means either $(x-y)$ or $(y-x)$. If nothing is mentioned any one the two can be considered.

Though in general, in most cases, absolute value should be taken, i.e., $|x-y|$ or $|y-x|$.

Answer 2

This question has come up a couple of times at the Ask Dr. Math forum, e.g., here and here. The "official" answer is that the difference between two numbers is generally understood to be non-negative. For a nice example, see this "find the difference" NRICH problem.

There are, of course, settings where "difference" refers to straight subtraction, so that a difference can be negative. Usually it's clear whether "difference between $x$ and $y$" means $|x-y|$ or $x-y$, but occasionally it isn't. (You might, for example, see if you can solve the NRICH problem by interpreting "difference" as "subtract the number on the right from the number on the left.")