Augends, Addends and Commutative Property of Addition
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original photo from Richard Masoner Edited by Brian
Not sure what got me thinking about this today, but I was musing about the commutative property, and how it applies to addition and multiplication. (Yeah, it was a pretty boring morning.)
Specifically, I was thinking about the word, “augend”. The augend of an addition problem is the first of the series of addends. It’s not a word that is usually taught, and I was wondering why not.
You should be aware that addition and multiplication have the commutative property, and subtraction and division don’t. That just means 4+6 is the same as 6+4, and 3*7 is the same as 7*3, but 8-2 and 2-8 are not the same, nor are 9/3 and 3/9.
So it doesn’t really matter which of the numbers is placed where in addition, so you don’t need to specify which is the augend, and which is not. You can call them all addends. Same with multiplication - you can call them all multiplicands. (Not so with minuends, subtrahends, dividends and divisors, though - they don’t commute.)
But I think we should teach about augends. There are several good reasons why, and you can read about them at a lesson I just put up at Names of the numbers in basic arithmetic operations. That lesson is all about why we give the different parts of arithmetic problems their names, (like dividend, divisor, and quotient) and why it makes sense to learn them.
Now that I feel like I’ve cleared this all up for you and me, I’ve got something that I’m not so clear on. Maybe some kind reader has some insight about it she or he’d like to share. It’s this:
Since you must differentiate the names of the parts of subtraction or division problems, what happens if a problem has more than two terms, like 8-3-2? Is there a name for the third term? What if there are fourth or fifth terms?
I assume that they are called, “the second (secondary?) subtrahend, the third (tertiary?) subtrahend, and so on, but I’m not sure.
Anybody got any insights?
You may want to check Names of the numbers in basic arithmetic operations first, though.
By the way, if anybody can write me and tell me why I chose the image that I used for this post, I’ll send them a free Math Mojo e-booklet. (Use the contact me near the top right navigation bar on this page.)
Update: You don’t have to write about that anymore - we have a winner! Mark (see below) got the booklet.
To clarify: The big dog in the picture is “Doggy Daddy,” and the little dog at the door of the train is “Auggie Doggy.” (They are Hannah-Barbera cartoon figures.) They are about to enter a commuter train. Get it? Augend/Auggie, commutative property/commuter train? (Groan.)
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Comment by Mark Schooley
Are you saying that’s Augie Doggie? How do you know it’s not Doggy Daddy? ;)
Brian Sez:
Well, Hallelujah! Someone got it! The big picture is Doggy Daddy, but if you look at the door train, Augie is about to get on. Daddy is running for the train.
I just sent you an e-mail asking you which booklet you’d like.
Thanks for playing “Quiz the Whiz”
Comment by Richard Masoner
I’m glad you like the photo and thanks for the attribution! I’m a math nerd — I’ll have to check out your blog.
Comment by blackhawkmath
There are no such things as secondary or tertiary subtrahends. This is because subtraction is a binary operation, it only works with two inputs. So when there is more than one operator, you must identify the principal operator, which is:
*the last one performed
*the only one outside of grouping symbols (parentheses) when all possible grouping symbols are used to help indicate the correct order of operations intended.
So, in your example, “8-3-2″ which is the minuend and which is the subtrahend is determined by which operator (subtraction sign or dash) is the principal operator.
If we put in parentheses that correspond to the correct order of operations, then ‘(8-3)’ is the minuend and ‘2′ is the subtrahend. The minuend also happens to be an indicated difference (which is just another way to write the number 5).
If we insert parentheses thus: ‘8 - (3 - 2)’ we get a completely different problem and answer, since subtraction is not associative. In this expression, 8 is the minuend and ‘(3 - 2)’ is the subtrahend.
The four operations (+, -, *, and /) are all binary, and so each operator should be considered separately after its two inputs are determined, regardless of how many other operators the inputs contain. One should always be aware of the correct order of operations, as this can be very difficult in some cases!
Professor Homunculus sez:
Thanks so much for a clear and informative answer, Blackhawk!
Comment by Al Aunchman
After all these years (30) of struggling to teach children math, I finally realize why it is so difficult. A brief perusal of some of the mathematical girations you go through to multiply two numbers together explains a lot of why kids are poor at math. Commutative and associative properties are more easily understood when you have the basic tools to work with without adding zeros then subtracting the number from your cousins name on your mother’s side of the family. Teach the basics by rote then progress to the more abstract. Simple to complex seems to work.
Professor Homunculus sez:
Al, I’m sorry you’ve come to that conclusion. If you’ve been teaching math for 30 years, you surely have some insights. But I can’t see see how you’d say, “simple to complex” seems to work. May I ask where it seems to work? And if it does, why is it a struggle for you, and why is it so difficult? Have you been teaching with the “girations” (sic) you say I use to make it so frustrating?
I’m not quite sure I understand the logic of your position.
I’m not being facetious (facetious would be, “… subtracting the number from your cousins name on your mother’s side of the family…”).
I’ve thought a lot about what you said here, and will add more to this reply at my next posting (probably today, Mar. 28, ‘08)
(Caveat - Although I’m not being facetious here, the post will be pretty facetious. Please don’t take it personally. It’s a sickness.)
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