Inverse operations and functions are wonderful things. Without them, solving equations would be much more challenging. Yet inverse operations can also be odd beasts. This is the first of several postings on operations/functions and their inverses.
The first arithmetic operation we all learned was addition. It seems to arise fairly naturally from the counting numbers (1, 2, 3, etc.), and the set of counting numbers is closed when using addition: no matter which two counting numbers you add, you will always get another member of the set. Once addition was mastered, the world was our oyster.
We also figured out, or were taught, that addition is commutative
However, as we began to combine all sorts of numbers we also realized that sometimes we had to “take away”, “un-combine”, or “subtract”. In un-doing addition, we defined “subtraction” and were forced to confront some new ideas which did not trouble us when addition was our sole concern.
Subtraction is not commutative:
Nor is subtraction associative:
The lack of these two properties makes it inflexible as an operator when compared to addition. Furthermore, when we subtract two counting numbers:
we can get an answer (negative two) that is not a counting number. Subtraction introduced us to zero and the negatives of the counting numbers, leading us to a new concept and phrase in our vocabulary: “the integers”, defined to include the counting numbers, zero, and the negatives of all the counting numbers.
The greater complexity/sophistication introduced by the integers has a bright side though. It turned out that subtracting an integer had exactly the same result as adding the negative of that integer:
This allowed us to turn subtraction problems into an addition problems if we so chose, so that the commutative and associative properties could once again be used to re-arrange the problem for our convenience:
I wonder what percent of people conceive of (5 – 3) as:
– “five take away three”, or
– “what I must add to three to get to five”, or
– “five plus a negative three”
These are all valid, but different, mental models for subtraction. I think of the first as a conceptual perspective, the second as an inverse perspective, and the third as avoidance. It is interesting to note that even though subtraction is not conceptually challenging to grasp, the inverse perspective still takes the most words to describe. It is also interesting to note that most people I have seen working algebra problems practice avoidance when faced with subtraction… treating it as the addition of a negative gives me so many more options when manipulating an expression, that I have little incentive to treat subtraction any other way.
So, in pondering the simplest of the inverse operations and attempting to generalize a bit, it appears that inverse functions can:
– not share convenient properties with the “forward” version of the function, like being commutative or associative
– have ranges that force us to expand the universe of numbers that we had been habitually using before being introduced to the inverse function
– be more challenging to describe verbally than their “forward” sibling
And one last point to ponder that will become more relevant with other operations: addition and subtraction are dimensionally-closed operations. They act within the dimension(s) of the quantities you are adding or subtracting. For example, dimensionless numbers are added along a number line, inches along the edge of a ruler, surface areas on the surface of a plane, volumes within a common 3-dimensional space, etc.
To your list of ways to conceive of 5-3, I’ll add:
(1) How much more is 5 than 3?
(2) How far is 5 from 3 on the number line? and
(3) 3 is part of 5; what is the rest?
I’ll be curious to see how you work on multiplication (and its inverse) and exponentiation (and its inverse). Your three summary observations should still hold for these operations-what about the multiple conceptions (as you have for 5-3)?
Could you perhaps expand on how you see your three additions as different from “what must I add to three to get to five”?
To me , they seem to be more similar to one another than the three in my original list – so, I may not be seeing your perspective, and would like to better understand it.
I started on my multiplication and division post immediately after this one, and it is still a work in progress. To my surprise, I find myself reconsidering my mental foundation of multiplication (and have not even begun to consider division yet), so it is not turning out to be as straightforward as I had assumed in this era of blogs and “ain’t no repeated addition”! So I’ll be curious to see how it comes out too…
And yes, my summary comments were intended to apply broadly. The multiple conceptions will probably get just as interesting with the other operations as attempting to describe them has become.
Thanks for your comments, and I look forward to your response.
See my extended reply here:
Thanks for thinking with me!
Great stuff – Thank you! Now I need to find some time to think it all through… perhaps after April 15th!