Imagine that you’re looking so intently for a “pot of gold” that you don’t see a “bowl of diamonds” dancing in front of your eyes.
That’s my sense of what happens in algebra when we talk about the Commutative Property of Addition. Focused on it, we fail to see a very important related property. In fact, this other property is so neglected that it has no name! At least I have never heard a name for it. If anyone has heard a name for it, please let me know, and I’ll spread the word.
As a reminder, the commutative property of addition tells us the obvious fact that when we add two numbers, the order in which we add them makes no difference. For example, 5 + 3 = 3 + 5, and 21 + 13 = 13 + 21.
Often instructors will ask students: does this commutative property also work for subtraction? So students will start to consider whether or not 8 – 3 = 3 – 8. No, they conclude. The commutative property does not work for subtraction. End of story.
But gosh, that should not be the end of the story. It should be the beginning of a new story! Why? Because there’s something very interesting about 8 – 3 and 3 – 8. Sure, the two differences are not equal. But take a look: 8 – 3 = 5, and 3 – 8 = – 5. The differences are opposites. In other words, it is starting to look as if:
a – b = – (b – a)
Well, let’s try another such problem to see if this opposites pattern happens again. How about now we try another pair of integers, one positive (8), the other negative (– 3).
Is 8 – (– 3) = – [(–3) – 8] (?)
Well, 8 – (– 3) = 11, and –[(–3) – 8] = – [– 11] = 11
So this has worked again.
One more time, to test all possibilities of positives and negatives. Let’s see if this also works if we start with two negative numbers: (– 8) and (– 3).
Does (– 8) – (– 3) = – [(– 3) – (– 8)]
Well, (– 8) – (– 3) = – 8 + 3 = – 5
And (– 3) – (– 8) = – 3 + 8 = + 5
So yes, once again the differences you get are opposites.
So this means that if we widen our vision beyond the classic commutative property, there’s another gem of a property to be learned and used. This property says that:
a – b = – (b – a)
Since I’ve never heard a name for this property, I’ll just give it a name. I’ll call it the Opposite Differences Property.
In my next post, I’ll share some info on some of the nice ways we can use this property.
Comments on: "An Unstated (but Useful) Algebraic Property" (2)
This is great. How is this not a recognized property?
Hi Shana, Always good to hear from you.
I agree. Isn’t that weird that this is un-named? There’s a related property for the “near-commutativity” of division, too. That: a/b = 1/(b/a)
In my view, that’s equally important, and it, too, appears to be un-named. I don’t get it, although I would say that there are just “gaps” in the world.
Another one, from the world of English: there’s no word for “his or her.” For example, we have to say, It’s time for everyone to take his or her seat. The reason: everyone is grammatically singular. That’s why we can’t say: .. for everyone to take their seat. But shouldn’t there be a single word for “his or her”? My proposal there: make up a new word: “hiser,” pronounced “hizzer,” rhymes with scissor. We’d say: It’s time for everyone to take hiser seat. Doesn’t sound too bad, does it? If you like it, spread it around. Maybe it’ll catch on …