Math Mojo - Making Math Meaningful
return to Math Mojo
home page

A Little Look at the Commutative Law of Addition
or Why it Doesn't Matter Which Column you Add First

You probably know that the basic operations of arithmetic are:

Each of these (and the higher operations as well) have certain "properties."
That means that each has certain ways that it deals with particular "laws" of mathematics.

What is nice about these laws is that most of them are simply descriptions of things that you know happen.

For example, with addition, you know that if you have 5 things, and you get another 3 of them, you will end up with 8 things.
You also know that if you started out with 3 things, and you added 5, you would still end up with 8 things.

This kind of law is simply a description of reality. It is called a "descriptive" law. After you learn this lesson, if you want to know more about how mathematical laws work, you can go to the lesson entitled "Introduction to the Basic Laws of Mathematics."

Now we will use the above law (the one about 5 + 3 = 8, as well as 3 + 5 = 8) to explain why it doesn't matter in which order you add the columns in addition on an abax (or on paper, or anywhere else, for that matter). Here is an example:

Example:  12 + 34

When you are adding 12 + 34. You are really adding 1 group of tens, and 2 groups of ones to 3 groups of tens and 4 groups of ones.
You know that when you get to the ones, no matter when you get to them, 2 ones and 4 ones will always be 6 ones.

You also know that when you get to the tens, no matter when you get to them, 1 ten and 3 tens will always be 4 tens.

So either way you do it, you will always end up with 4 tens and 6 ones, or 6 ones and 4 tens, both of which will always be 46.

 

But what if you have to carry? Won't that mess it up?

Not at all! For example:

If you are adding, let's say, 59 + 74, you are really adding 5 groups of tens, and 9 groups of ones to 7 groups of tens and 4 groups of ones.

You know that when you get to the ones, no matter when you get to them, 9 ones and 4 ones will always be 13 ones. And 13 ones will always be 1 group of tens and 3 groups of ones. That makes 13.

You also know that when you get to the tens, no matter when you get to them, 5 tens and 7 tens will always be 12 tens, and that will always be 1 group of hundreds and two groups of tens. That makes 120.

Now we get to the point (finally!) You know that 120 + 13 is always 133, and 13 + 120 is also 133. So what's the difference what order you add them in?

There is no difference, except that schools want you to do it one way, and it makes more sense to do it the other way. It is truly unfortunate that schools make you choose the worse of two choices. That is an example of a very bad "prescriptive" law. (You will understand what that means if you go to the lesson highlighted above.)

The two examples below should make it clear that no matter which column you add first, you will still get the same answer.

 

If you do it like this: you get the same answer that you
would if you did it like this:
5
9
5
9
+
7
4
+
7
4
1
3
  ones (do them first)
1
2
  tens (do them first)
1
2
  tens (do them second)
1
3
  ones (do them second)
1
3
3
1
3
3
  

Here's a larger example, which will show, by the same reasoning, that it doesn't matter what order you add columns in, no matter how many columns there are:

If you add this example like this: you get the same answer that you would if you did it like this:
6
2
7
6
2
7
1
3
1
  
1
3
1
  
+
3
9
8
+
3
9
8
1
6
ones (do them first)
1
6
tens (do them first)
1
4
tens (do them second)
1
4
ones(do them second)
1
0
hundreds (do them third)
1
0
hundreds (do them third)
1
1
5
6
1
1
5
6
or like this:

or like this:
6
2
7
6
2
7
1
3
1
  
1
3
1
  
+
3
9
8
+
3
9
8
1
0
hundreds (do them first)
1 4  
tens (do them first)
1
4
tens (do them second)
    1 6
ones(do them second)
1 6 ones (do them third)
1
0
hundreds (do them third)
1
1
5
6
1
1
5
6

or any other order you like.


An interesting point was brought up by a reader. He asked about adding, say, 999 + 1. When doing it with the above method, wouldn't it be shorter to do it from right-to -left than from left-to-right, because you would have to carry twice.

It is a very good question. As it turns out, there is a way, using the "one-ahead"( or "looking beyond") method. (Don't go looking for those terms in math books. I made them up! They come from my training in magic, and in Eduction (Edux)). The one-ahead method is taught extensively in the "Counting and Adding on an Abax with Math Mojo" booklet.

I will make it available in a booklet on basic addition (without an abax) eventually, but you can get a good Idea of how it works if you go to the interactive lessons on subtraction at this site. In the last lesson, it shows how to subtract from left-to-right if there are multiple columns in which you have to carry.

If you learn and understand that, you can use almost exactly the same Idea for addition. Go learn it and see for yourself! To get to that lesson, click here.

 

By the way, the law which describes that it doesn't matter which order you add in is called the Commutative Law of Addition, or the Commutative Property of Addition.

Why did they give it that funky name? It actually makes sense. The number can commute back and forth of the addition sign. 8+6 is the same as 6 + 8. You may know that to "commute" is to go travel back and forth, like commuting from home to work.

back to top of page

I want to know more about the Basic Laws of Mathematics

Copyright 2001- 2003 by Brian Foley
report typographical errors or broken links to
webmaster@mathmojo.com