Building Simple Words: Addition

Because there is an order to the natural numbers, we can lay them out on a number line. It looks like this:

0---1---2---3---4---5---6---7---8---9---10--->

Starting with a number on the number line, we can get to another number on the number line by stepping to the right on the line. This process is called addition. We take two numbers and add them together to form another number. For this reason, addition is called a binary (meaning two) operation. The two numbers are called the addends and the result of the operation is called the sum.

We work our way from left to right along the number line. For example, let's add 3 + 2.

Start on 3 on the number line and step to the right 2 times.

0---1---2---3---4---5---6---7---8---9---10--->

We stop on 5. So 3 + 2 = 5.

We can build a table of the results of adding the numbers 0 to 9 and memorize the results

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

To add two numbers, we find the first number on the top row and follow the column down to the row with the second number on the left. Where the column and row meet is the sum of the two numbers. For example, 7 + 6 = 13. Start with 7 on the top and follow the column down to the row that starts with 6. Where the column and row meet is 13.

Adding larger numbers requires lining up the positions in columns. All of the ones line up under each other; all of the tens line up under each other; all of the hundreds line up under each other and so on. Then you add down the columns starting with the ones column and move to the left. If the sum of the column is more than 9, you keep the ones value from the sum and add 1 to the column on the left. The number that is "carried" to the column on the left is called the carry.

Let's add 1357 + 246.
Line the numbers up in columns

1	3	5	7
+	2	4	6

Starting with the ones column adding down the column: 7+6=13. Keep the 3 in the ones column and add 1 to the tens column. The carry is in color.

		1
1	3	5	7
+	2	4	6
			3

Now add down the tens column: 1+5+4=10. Keep the 0 in the tens column and add 1 to the hundreds.

	1
1	3	5	7
+	2	4	6
		0	3

Now add down the hundreds column: 1+3+2=6.

1	3	5	7
+	2	4	6
	6	0	3

Since there is no number in the thousands column in the second number, we fill in the position with a 0. Now add down the thousands column: 1+0=1

1	3	5	7
+	2	4	6
1	6	0	3

Adding more than two numbers at a time, you still line up the numbers in columns and add down the columns starting with the ones column. You can only add two numbers at a time, but you can keep a running total as you add your way down the column. When you get to the bottom, keep the ones value and carry the rest of the number to the columns on the left. If your carry is larger than 9, the carry overflows to the columns on the left, carrying to more than one column if you have to.

Addition is the simplest way to form larger numbers. As you can see from the addition table, there are several ways to get to other numbers: 3+2=5 and 4+1=5. You might also notice that changing the order of the numbers doesn't make a difference: 3+2=2+3. This is called commutativity and is a basic property of the natural numbers.

When adding more than tow numbers, we use parenthesis around numbers to indicate the order we want to add. Numbers inside of parenthesis get added first, then that sum gets added to the numbers on the outside of the parenthesis. 3+(4+2) means add 4+2 then add 3 to that sum.
3+(4+2)=
3+ 6 = 9.
This way we can group addition together.

Another basic property of the natural numbers is associativity. This means the grouping doesn't matter. If we add more than two numbers at a time, it doesn't matter how we group the numbers together:

3+(4+2)=(3+4)+2
3+ 6 = 7 +2
9 = 9

These basic properties will follow us all the way through the alphabet. As we add more types numbers to our alphabet, those numbers will also be commutative and associative. Remember, commutative means order and associative means group. An easy way to remember it is that when a governor commutes a sentence he changes the court's order and an association is a group of people. This ability of numbers to commute and associate is a very powerful tool when we start looking at much more complicated forms of math.

In addition, 0 is a very special number. As you can see, anytime we add 0 to a number, the number doesn't change. 0 is called the identity element, because addition by 0 doesn't change a number's identity. This will become an important tool to use later on. The ability to change a number's look without changing its value is very useful.