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Study Guides > Prealgebra

Adding Whole Numbers

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.
+ 00 11 22 33 44 55 66 77 88 99
00 00 11 22 33 44 55 66 77 88 99
11 11 22 33 44 55 66 77 88 99 1010
22 22 33 44 55 66 77 88 99 1010 1111
33 33 44 55 66 77 88 99 1010 1111 1212
44 44 55 66 77 88 99 1010 1111 1212 1313
55 55 66 77 88 99 1010 1111 1212 1313 1414
66 66 77 88 99 1010 1111 1212 1313 1414 1515
77 77 88 99 1010 1111 1212 1313 1414 1515 1616
88 88 99 1010 1111 1212 1313 1414 1515 1616 1717
99 99 1010 1111 1212 1313 1414 1515 1616 1717 1818
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 aa and 00 is the number. a+0=a0+a=a\begin{array}{}\\ a+0=a\\ 0+a=a\end{array}
 

example

Find each sum:
  1. 0+110+11
  2. 42+042+0
Solution
1.  The first addend is zero. The sum of any number and zero is the number. 0+11=110+11=11
2. The second addend is zero. The sum of any number and zero is the number. 42+0=4242+0=42
 
    Look at the pairs of sums:
2+3=52+3=5 3+2=53+2=5
4+7=114+7=11 7+4=117+4=11
8+9=178+9=17 9+8=179+8=17
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 aa and bb does not change their sum. a+b=b+aa+b=b+a
 

example

Add:
  1. 8+78+7
  2. 7+87+8

Answer: Solution  

1.
Add. 8+78+7
1515
2.
Add. 7+87+8
1515

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.    

example

Add: 28+6128+61.

Answer: Solution To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.

Write the numbers so the ones and tens digits line up vertically. \begin{array}{c}\hfill 28\\ \\ \hfill \underset{\text{____}}{+61}\end{array}
Then add the digits in each place value. Add the ones: 8+1=98+1=9 Add the tens: 2+6=82+6=8 \begin{array}{c}\hfill 28\\ \\ \hfill \underset{\text{____}}{+61}\\ \hfill 89\end{array}

    In the previous example, the sum of the ones and the sum of the tens were both less than 1010. But what happens if the sum is 1010 or more? Let’s use our base - 10\text{base - 10} model to find out.The graphic below shows the addition of 1717 and 2626 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, 7+67+6, we get 1313 ones. Because we have more than 1010 ones, we can exchange 1010 of the ones for 11 ten. Now we have 44 tens and 33 ones. Without using the model, we show this as a small red 11 above the digits in the tens place. When the sum in a place value column is greater than 99, we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, 1010 ones for 11 ten or 1010 tens for 11 hundred.

Add whole numbers

  1. Write the numbers so each place value lines up vertically.
  2. 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 99, carry to the next place value.
  3. Continue adding each place value from right to left, adding each place value and carrying if needed.
 

example

Add: 43+6943+69.

Answer: Solution

Write the numbers so the digits line up vertically. \begin{array}{c}\hfill 43\\ \\ \hfill \underset{\text{____}}{+69}\end{array}
Add the digits in each place. Add the ones: 3+9=123+9=12
Write the 22 in the ones place in the sum. Add the 11 ten to the tens place. \begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{____}}{+69}\\ \hfill 2\end{array}
Now add the tens: 1+4+6=111+4+6=11 Write the 11 in the sum. \begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{____}}{+69}\\ \hfill 112\end{array}

 

try it

Add: 35+9835+98.

Answer: 35+98=13335+98=133

  Add: 72+8972+89.

Answer: 72+89=16172+89=161

 
 

example

Add: 324+586324+586.

Answer: Solution

Write the numbers so the digits line up vertically. ...
Add the digits in each place value. Add the ones: 4+6=104+6=10 Write the 00 in the ones place in the sum and carry the 11 ten to the tens place. ...
Add the tens: 1+2+8=111+2+8=11 Write the 11 in the tens place in the sum and carry the 11 hundred to the hundreds ...
Add the hundreds: 1+3+5=91+3+5=9 Write the 99 in the hundreds place. ...

 

try it

Add: 456+376456+376.

Answer: 456+376=832456+376=832

  Add: 269+578269+578.

Answer: 269+578=847269+578=847

 
 

example

Add: 1,683+4791,683+479.

Answer: Solution

Write the numbers so the digits line up vertically. \begin{array}{c}\hfill 1,683\\ \\ \hfill \underset{\text{______}}{+479}\end{array}
Add the digits in each place value.
Add the ones: 3+9=123+9=12. Write the 22 in the ones place of the sum and carry the 11 ten to the tens place. \begin{array}{c}\hfill 1,6\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 2\end{array}
Add the tens: 1+7+8=161+7+8=16 Write the 66 in the tens place and carry the 11 hundred to the hundreds place. \begin{array}{c}\hfill 1,\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 62\end{array}
Add the hundreds: 1+6+4=111+6+4=11 Write the 11 in the hundreds place and carry the 11 thousand to the thousands place. \begin{array}{c}\hfill 1,\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 162\end{array}
Add the thousands 1+1=21+1=2 . Write the 22 in the thousands place of the sum. \begin{array}{c}\hfill 1,\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{______}}{+479}\\ \hfill 2,162\end{array}

  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.  

example

Add: 21,357+861+8,59621,357+861+8,596.

Answer: Solution

Write the numbers so the place values line up vertically. \begin{array}{c}\hfill 21,357\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\end{array}
Add the digits in each place value.
Add the ones: 7+1+6=147+1+6=14 Write the 44 in the ones place of the sum and carry the 11 to the tens place. \begin{array}{c}\hfill 21,3\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 4\end{array}
Add the tens: 1+5+6+9=211+5+6+9=21 Write the 11 in the tens place and carry the 22 to the hundreds place. \begin{array}{c}\hfill 21,\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{_______}}{+8,596}\\ \hfill 14\end{array}
Add the hundreds: 2+3+8+5=182+3+8+5=18 Write the 88 in the hundreds place and carry the 11 to the thousands place. \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}
Add the thousands 1+1+8=101+1+8=10 . Write the 00 in the thousands place and carry the 11 to the ten thousands place. \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}
Add the ten-thousands 1+2=31+2=3 . Write the 33 in the ten thousands place in the sum. \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}

  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.   Watch the video below for another example of how to add three whole numbers by lining up place values. https://youtu.be/N3I6OiO5mKI

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