# Are there clear, formal definitions for “terms” in subtraction operation?

I tutor children of all ages in Mathematics and I've noticed so many different words thrown around regarding binary operations, particularly with subtraction. For example, when working with a 2nd grade student the other day, her homework was trying to help her "visualize" subtraction and asked "how far apart are 5 and 8?" for her to get 3, as in the solution for $8-5=?$. Now at her grade, she only knows Natural numbers and therefore at her age, subtraction is understood as the larger number coming first (i.e. the minuend) and the smaller number coming second (i.e. the subtrahend) for the "answer" (i.e. the difference) always being positive. Since the concept has been so drilled into her that the "bigger number goes on top" when she does subtraction, she is unlikely to even try attempting $5-8$ because that just doesn't seem "possible" for her.

Now normally I wouldn't have even given something like this a second thought, but as I've taken higher and higher math and have become absolutely enthralled with Number Theory, I find myself going crazy with little things like this. I have looked all over for the past two hours and I cannot find any real consensus over proper phrasing of subtraction, as well as the specific meanings of subtrahend, minuend, and difference. I've seen everything from "subtraction means $minuend-subtrahend=difference$" with no further specifications on which each of those words mean or what set(s) they are defined on, to simply specifying that a "difference" is always positive, and others saying that the minuend is ALWAYS greater than the subtrahend (from the Latin word origins), and all they way to classifying the subtraction operation as a partial function

\begin{aligned} & f: \Bbb N \times \Bbb N \to \Bbb N \\ & f(x,y)=x-y \\ & \text{where it is defined only when x>y} \\ \end{aligned}

Then later that day, I was working with an Algebra I student preparing for her semester exam and she had to create and solve a system of equations from something like "the sum of two numbers is 16 and the difference between them is 8" so she did

\left\{ \begin{aligned} x+y &=16 \\ x-y &=8 \\ \end{aligned} \right.

\begin{align} \text{solving to get $x=12$ $and$ $y=4$} \end{align}

In this problem, it didn't end up affecting her answer by arbitrarily choosing $x$ as the minuend and $y$ as the subtrahend.

But what if she gets a problem in a month that says "the difference between a number and its square is 20. What number(s) can it be?" and she tries

\begin{align} x-x^2&=20 \\ x^2-x+20&=0 \\ \end{align}

Now since at this point she hasn't learned about complex solutions, she will be stumped. But when I read "the difference between x and y" I think of that as $|x-y|$

If applied to the previous exercise

\begin{align} &|x-x^2|=20 \\ &x=(-4) \; or \; x=5 \\ \end{align}

Perhaps I'm way overthinking this and have gone a bit overboard on this post (especially seeing as I've spent the past 3 hours trying to figure out this formatting), but as I reach higher math while tutoring primary and secondary school children of various ages, I can't help but be frustrated with seemingly very little consistency in math education some of my students come in and their material, in an effort to help them understand the concepts by attempting to "dumb down" things, sometimes ends up presenting all sorts of things that are clearly not true. Now I'm not talking about saying things taught to the students wouldn't hold true in non-Euclidean geometry, for unary operations, in various high level pure mathematics, etc. These are things that can very well cause problems even before Algebra I.

• I think context is key here...when I see "difference" between $x$ and $y$ it means $x-y$--especially since one can make sense of subtraction even in contexts where the absolute value does not make sense. Some of this might become more clear for you if you learn some basic abstract algebra, esp. what an abelian group is. – Eric Auld Jan 29 '16 at 18:24
• I'm well aware what an abelian group is. This question wasn't meant to deal with math education, really. I'm just curious on the definitions of minuend, subtrahend, and difference as they apply to elementary arithmetic. – Brendan Feb 1 '16 at 0:41