Longhand Subtraction

Make Math Work

Terms You Might Need
The number being subtracted from is called the minuend.
The number being subtracted is called the subtrahend.
The result of a subtraction is called the difference.

Introduction
Small subtractions are easy to remember, such as 4 - 2 or 8 - 5. We don't figure these out, we simply remember the differences. As numbers get larger, they would be harder to remember. Did you ever memorize 154 - 31 or 9123 - 1534?

How We Could Do It
We could start with the first number and count backwards through the rest. For example, we could start with 154, and then count backwards by one 31 times to subtract 31 from 154. We would count backwards 31 times from 154: 153, 152, 151, 150, 149, ... all the way down to 123. This works, but we have to admit, it's time-consuming and impractical. It could take hours to subtract 114,657 from 522,342.

An Easier Way
We could use a calculator. A calculator is a great solution; they are fast and accurate. But what if you had to subtract two really, really large numbers, like 389,475,123,985,234,232,101 from 758,302,375,192,845,629,943? Try that on a calculator. You probably won't be able to subtract it because most calculators do not allow such large numbers.

The Longhand Way
We can use a pen and paper. We write the numbers onto the paper, one underneath the other, and subtract them column by column. This way, for two big numbers, we are only subtracting smaller one-digit numbers at a time. Since we have memorized all of the one-digit subtractions starting from 0 - 0 = 0 all the way to 9 - 9 = 0, subtracting in columns should be easy. Let's try one to make this clearer. Let's subtract 31 from 154.

Step 1
Write the two numbers onto the paper. Write them in such a way that the digits line up underneath each other. Since 154 has more digits than 31, write them in such a way too that the digits on the right line up underneath each other. The larger number, the minuend, must always be the upper number. Try it:

1	5	4
	3	1

Step 2
Subtract the column on the far right and write the difference below the subtrahend. 4 - 1 = 3:

1	5	4
	3	1
		3

Step 3
Subtract the next column to the left and write the difference below the subtrahend. 5 - 3 = 2:

1	5	4
	3	1
	2	3

Step 4
Subtract the next column to the left and write the sum below the subtrahend. In this case this is the last column on the left; our subtraction will be finished after this. Since the digit underneath 1 is blank, we consider the blank as the number 0. 1 - 0 = 1:

1	5	4
	3	1
1	2	3

The subtraction is now complete: 154 - 31 = 123. If you don't believe it, start with 154 and count down by one 31 times.

It Gets A Little More Complicated
What do we do, if while subtracting column by column, we find a column where the subtrahend (the lower number) is larger than the minuend (the upper number)? For example, if we were to subtract 28 from 157, in the right-most column we would be subtracting 7 - 8, which, if we were counting backwards, is not possible. We solve this problem by using a process called borrowing. We will take a portion from the column to the left. Let's try it:
Step 1
Again, write the two numbers onto the paper. Write them in such a way that the digits line up underneath each other. Since 157 has more digits than 28, write them in such a way too that the digits on the right line up underneath each other. The larger number, the minuend, must always be the upper number. Try it:

1	5	7
	2	8

Step 2
We need to subtract 8 from 7, which is not possible. So, we will borrow from the minuend column to the left, which in this example, contains 5. We will take 1 from the 5 and place the 1 to the left of 7 to make 17. We'll cross out the 7 and write 17 above it.

		17
1	5	7
	2	8

Step 3
Now we subtract 8 from 17, for a difference of 9.

		17
1	5	7
	2	8
		9

Step 4
Since we borrowed 1 from the 5 of the middle column, we subtract the borrowed 1 from the 5 for a difference, or result, of 4. To show this we cross out the 5 and write 4 above it.

	4	17
1	5	7
	2	8
		9

Step 5
Now in the middle column we subtract 2 from 4, for a difference of 2: 4 - 2 = 2. Remember our new minuend is 4 since we had to borrow from it to complete the subtraction in the column to the right. Write the difference below the subtrahend.

	4	17
1	5	7
	2	8
	2	9

Step 6
Move one column to the left; this is the last column. We will consider the blank in this column of the subtrahend as zero (0). Our subtraction will be 1 - 0 = 1. This last column has not not been affected by borrowing. Again, write the difference below the subtrahend.

	4	17
1	5	7
	2	8
1	2	9

The subtraction is now complete: 157 - 28 = 129. You can double-check this on a calculator.

Make Math Work Can Help
The Subtract command will subtract two numbers showing the borrowing if it were necessary. You can try these examples:
* This is our first lesson from above: Subtract 154, 31
* This is our borrowing lesson from above: Subtract 157, 29
* This will subtract two larger numbers: Subtract 148153, 97538