Adding Whole Numbers: Properties of Addition

Learning Outcomes

Identify and use the identity property of addition
Identify and use the commutative property of addition
Add multiple-digit numbers using columns for place value

Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.

Imagine filling in the table below by adding each row number along the left side to each column number across the top. You can use this table for reference, but it will make your work faster and easier if you have the sums memorized.

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

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.

Identity Property of Addition

The sum of any number [latex]a[/latex] and [latex]0[/latex] is the number.

[latex]\begin{array}{}\\ a+0=a\\ 0+a=a\end{array}[/latex]

example

Find each sum:

[latex]0+11[/latex]
[latex]42+0[/latex]

Solution

1. The first addend is zero. The sum of any number and zero is the number.	[latex]0+11=11[/latex]
2. The second addend is zero. The sum of any number and zero is the number.	[latex]42+0=42[/latex]

try it

Look at the pairs of sums:

[latex]2+3=5[/latex]	[latex]3+2=5[/latex]
[latex]4+7=11[/latex]	[latex]7+4=11[/latex]
[latex]8+9=17[/latex]	[latex]9+8=17[/latex]

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.

Commutative Property of Addition

Changing the order of the addends [latex]a[/latex] and [latex]b[/latex] does not change their sum.

[latex]a+b=b+a[/latex]

example

Add:

[latex]8+7[/latex]
[latex]7+8[/latex]

Show Solution

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part 2 just by recognizing that the addends were the same as in part 1, but in the reverse order. As a result, both sums are the same.

try it

example

Add: [latex]28+61[/latex]

Show Solution

try it

In the previous example, the sum of the ones and the sum of the tens were both less than [latex]10[/latex]. But what happens if the sum is [latex]10[/latex] or more? Let’s use our [latex]\text{base - 10}[/latex] model to find out.

The graphic below shows the addition of [latex]17[/latex] and [latex]26[/latex] again.

An image containing two groups of items. The left group includes 1 horizontal rod with 10 blocks and 7 individual blocks 2 horizontal rods with 10 blocks each and 6 individual blocks. The label to the left of this group of items is

When we add the ones, [latex]7+6[/latex], we get [latex]13[/latex] ones. Because we have more than [latex]10[/latex] ones, we can exchange [latex]10[/latex] of the ones for [latex]1[/latex] ten. Now we have [latex]4[/latex] tens and [latex]3[/latex] ones. Without using the model, we show this as a small red [latex]1[/latex] above the digits in the tens place.

When the sum in a place value column is greater than [latex]9[/latex], we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, [latex]10[/latex] ones for [latex]1[/latex] ten or [latex]10[/latex] tens for [latex]1[/latex] hundred.

Add whole numbers

Write the numbers so each place value lines up vertically.
Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than [latex]9[/latex], carry to the next place value.
Continue adding each place value from right to left, adding each place value and carrying if needed.

example

Add: [latex]43+69[/latex].

Show Solution

try it

example

Add: [latex]324+586[/latex].

Show Solution

try it

example

Add: [latex]1,683+479[/latex].

Show Solution

When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.

try it

example

Add: [latex]21,357+861+8,596[/latex].

Show Solution

This example had three addends. We can add any number of addends using the same process as long as we are careful to line up the place values correctly.

try it

Watch the video below for another example of how to add three whole numbers by lining up place values.

Write the numbers so the digits line up vertically.	[latex]\begin{array}{c}\hfill 43\\ \\ \hfill \underset{\text{____}}{+69}\end{array}[/latex]
Add the digits in each place. Add the ones: [latex]3+9=12[/latex]
Write the [latex]2[/latex] in the ones place in the sum. Add the [latex]1[/latex] ten to the tens place.	[latex]\begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{____}}{+69}\\ \hfill 2\end{array}[/latex]
Now add the tens: [latex]1+4+6=11[/latex] Write the [latex]11[/latex] in the sum.	[latex]\begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{____}}{+69}\\ \hfill 112\end{array}[/latex]

Write the numbers so the digits line up vertically.
Add the digits in each place value. Add the ones: [latex]4+6=10[/latex] Write the [latex]0[/latex] in the ones place in the sum and carry the [latex]1[/latex] ten to the tens place.
Add the tens: [latex]1+2+8=11[/latex] Write the [latex]1[/latex] in the tens place in the sum and carry the [latex]1[/latex] hundred to the hundreds
Add the hundreds: [latex]1+3+5=9[/latex] Write the [latex]9[/latex] in the hundreds place.

Write the numbers so the digits line up vertically.	[latex]\begin{array}{c}\hfill 1,683\\ \\ \hfill \underset{\text{______}}{+479}\end{array}[/latex]
Add the digits in each place value.
Add the ones: [latex]3+9=12[/latex]. Write the [latex]2[/latex] in the ones place of the sum and carry the [latex]1[/latex] ten to the tens place.	[latex]\begin{array}{c}\hfill 1,6\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 2\end{array}[/latex]
Add the tens: [latex]1+7+8=16[/latex] Write the [latex]6[/latex] in the tens place and carry the [latex]1[/latex] hundred to the hundreds place.	[latex]\begin{array}{c}\hfill 1,\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 62\end{array}[/latex]
Add the hundreds: [latex]1+6+4=11[/latex] Write the [latex]1[/latex] in the hundreds place and carry the [latex]1[/latex] thousand to the thousands place.	[latex]\begin{array}{c}\hfill \stackrel{1}{1},\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 162\end{array}[/latex]
Add the thousands [latex]1+1=2[/latex] . Write the [latex]2[/latex] in the thousands place of the sum.	[latex]\begin{array}{c}\hfill \stackrel{1}{1},\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 2,162\end{array}[/latex]

Write the numbers so the place values line up vertically.	[latex]\begin{array}{c}\hfill 21,357\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\end{array}[/latex]
Add the digits in each place value.
Add the ones: [latex]7+1+6=14[/latex] Write the [latex]4[/latex] in the ones place of the sum and carry the [latex]1[/latex] to the tens place.	[latex]\begin{array}{c}\hfill 21,3\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 4\end{array}[/latex]
Add the tens: [latex]1+5+6+9=21[/latex] Write the [latex]1[/latex] in the tens place and carry the [latex]2[/latex] to the hundreds place.	[latex]\begin{array}{c}\hfill 21,\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 14\end{array}[/latex]
Add the hundreds: [latex]2+3+8+5=18[/latex] Write the [latex]8[/latex] in the hundreds place and carry the [latex]1[/latex] to the thousands place.	[latex]\begin{array}{c}\hfill 2\stackrel{1}{1,}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 814\end{array}[/latex]
Add the thousands [latex]1+1+8=10[/latex] . Write the [latex]0[/latex] in the thousands place and carry the [latex]1[/latex] to the ten thousands place.	[latex]\begin{array}{c}\hfill \stackrel{1}{2}\stackrel{1}{1,}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 0814\end{array}[/latex]
Add the ten-thousands [latex]1+2=3[/latex] . Write the [latex]3[/latex] in the ten thousands place in the sum.	[latex]\begin{array}{c}\hfill \stackrel{1}{2}\stackrel{1}{1,}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 30,814\end{array}[/latex]

Module 1: Historical Counting Systems

Learning Outcomes

Add Whole Numbers Without Models

Identity Property of Addition

example

try it

Commutative Property of Addition

example

try it

example

try it

Add whole numbers

example

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example

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example

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example

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Candela Citations

1.
Add.	[latex]8+7[/latex]
	[latex]15[/latex]

2.
Add.	[latex]7+8[/latex]
	[latex]15[/latex]