Difference of two variables, $x$ and $y$

Question

Let's say you have two variables, $x$ and $y$, where $x$ is less than $y$. What would be "the difference of $x$ and $y$"? I think it should be $y - x$, since if you plot them on a number line, $x + (y - x) = y$. But my math teacher says it is $x - y$. Which one is right?

Answer 1

So Let $r$ be the difference between $x$ and $y$. As $x < y$ this implies that $x+r=y$. Note that $(x+r)-x= (y)-x$. So $r=y-x$. In this situation $r$ is strictly positive.

Let's look at it in a different way. Suppose $x-y$ was the difference between the two (i.e. $x-y=r$). As $x<y$ we would have $x\geq x-x = 0 > x-y$. This would mean that $r$ is strictly negative. Would that seem odd to you?

I would say that the difference is $r=|x-y|$ so regardless of whether $x\le y$ or $x\ge y$ we have a strictly positive number. But who knows maybe there is a specific reason why your teacher wanted it to be $x-y$.

Answer 2

I use the definition of difference as $|y-x|$, since $x<y$, $|y-x|=y-x$.

when in doubt, you might want to ask the definition of "difference" that is being used.