Lesson 8 SUBTRACTING WHOLE NUMBERS

8,345  
− 5,872 
Here are the expanded forms:
8 thousands  + 3 hundreds  + 4 tens  + 5 ones 
− 5 thousands  − 8 hundreds  − 7 tens  − 2 ones 
2 thousands  + 4 hundreds  + 7 tens  + 3 ones 
2 ones from 5 ones are 3 ones. No problem.
Now we cannot take 7 tens from 4 tens. Therefore we will consider 10 tens added to the 4 tens, and we will take 7 from 14. To compensate, we will add 1 hundred to 8  the next bottom number  making it 9. Because 1 hundred is equal to the 10 tens we added to 4. (Lesson 2.)
Continuing:
9 hundreds from 13 hundreds are 4 hundreds.
6 thousands from 8 thousands are 2 thousands.
In practice, say:
8,345  
− 5,872  
2,473 
"2 from 5 is 3."
"7 from 14 is 7."
"9 from 13 is 4."
"6 from 8 is 2."
Here is the rule:
1.  How do we subtract by adding to both numbers? 
− 8,345  
− 5,872  
− 2,473  
Write the smaller number under the larger, taking care to align the same units. Then, starting with the ones on the right, subtract each digit on the bottom from the corresponding digit on top. When the bottom digit is greater, consider the top digit increased by 10. To compensate, add 1 to the next bottom digit. 

Example 1.  5,312 
− 2,579  
2,733 
"9 from 12 is 3."
"8 from 11 is 3."
"6 from 13 is 7."
"3 from 5 is 2."
The top digits 5312 never change. That is the simplicity of this method. Only a bottom digit might change.
Finally, since subtracting is finding what number to add to the smaller number, the student should always check the answer by adding.
3 + 9 is 12. 3 + 7 is 10, plus 1 is 11. And so on.
Example 2.  6,000 
−1,926  
4,074 
"6 from 10 is 4."
"3 from 10 is 7."
"10 from 10 is 0."
"2 from 6 is 4."
Example 3. 30.21 − .86
Solution. The smaller number is .86 Write it on the bottom, and align the decimal points:
30.21 
−.86 
29.35 
"6 from 11 is 5."
"9 from 12 is 3."
"1 from 10 is 9."
"1 from 3 is 2."
Example 4. 2.1 − .867
Solution. .867 is the smaller number. Write it on the bottom. Align the decimal points.
Note that we must add 0's onto the right of 2.1 (Lesson 3). Both numbers must have the same number of decimal digits.
On checking: .867 plus 1.233 is 2. 100.
Example 5. Subtract .698 from 5
Solution. This problem means
5 − .698
.698 is the smaller number. Write it below 5.000:
Subtraction by regrouping
First, here is a simple example where we do not have to regroup, or borrow:
48  =  4 tens + 8 ones 
− 12  = −  1 ten − 2 ones 
36  =  3 tens + 6 ones 
2 ones from 8 ones are 6 ones.
1 ten from 4 tens are 3 tens.
(Lesson 2)
In practice, simply say:
48 
− 12 
36 
"2 from 8 is 6. 1 from 4 is 3."
But say that we have
42  =  4 tens + 2 ones 
− 18  = −  1 ten − 8 ones 
We cannot take 8 ones from 2 ones. We need more ones. Therefore we will "borrow" 1 of the 4 tens and regroup it with the 2 ones:
The larger number is now 3 tens and 12 ones.
8 ones from 12 ones are 4 ones.
1 ten from 3 tens are 2 tens.
In practice, say:
"8 from 12 is 4." (Because 8 plus 4 is 12.) "1 from 3 is 2."
2.  How do we subtract by regrouping, or borrowing? 
When the digit in the bottom number is larger (6 is larger than 2), decompose 1 unit of higher place value into 10 units of the next lower place value, and regroup with those lower units. 

We cannot take 9 from 4.
Therefore, we will borrow 1 from 3  making it 2  and regroup that