Subtracting Whole Numbers

Subtracting whole numbers is the inverse operation of adding whole numbers. With subtraction, instead of adding two numbers to get a sum, you are removing one number from another to get a difference.

Key Terms in Subtraction

Minuend: The number you are subtracting from (the larger number when dealing with whole numbers).
Subtrahend: The number being subtracted (the smaller number when dealing with whole numbers).
Difference: The result of the subtraction.

For example, in the subtraction 8 - 3 = 5:

8 is the minuend (larger number)
3 is the subtrahend (smaller number)
5 is the difference

Single Digit Subtraction Examples

Take a look at these simple subtraction problems. These are subtractions with one digit and are usually fairly easy to do.

- 4

- 5

- 3

- 2

Multi-digit Subtraction with Regrouping

Things start getting more complicated when you have more than one digit and the digit in the ones place of the subtrahend (smaller number) is bigger than the digit in the ones place of the minuend (larger number). A good example of this is 32 - 17 since 7 is bigger than 2.

Let's solve 32 - 17 using expanded notation and regrouping:

Expanded Notation

First, let's write out our numbers in expanded notation:

32 = 30 + 2
17 = 10 + 7

Regrouping in Expanded Notation

We can't subtract 7 from 2, so we need to regroup:

32 = 30 + 2
32 = 20 + 12  (We've taken 10 from 30 and added it to 2)

Subtracting in Expanded Notation

Now we can subtract:

    32 = 20 + 12
  - 17 = 10 +  7
    ———————————————
    15 = 10 +  5

Vertical Subtraction with Regrouping

Now, let's see how this same process looks using vertical subtracion:

Set up the problem:

3 2

- 1 7

Regroup: Take 1 ten from the tens place of 32. Now we have 2 tens and 12 ones:

2 12

3 2

- 1 7

Subtract in each column:
In the ones place: 12 - 7 = 5
In the tens place: 2 - 1 = 1

2 12

3 2

- 1 7

1 5

So, 32 - 17 = 15

Using expanded notation helps us understand why regrouping works. In vertical subtraction, we're doing the same thing, just in a more compact form.

Three-Digit Subtraction with Multiple Regrouping

Let's solve 523 - 268 using vertical subtraction with regrouping:

Set up the problem:

5 2 3

- 2 6 8
We can't subtract 8 from 3 in the ones place, so we need to regroup. We borrow 1 ten from the tens place:

5 1 13

5 2 3

- 2 6 8
Now we can't subtract 6 from 1 in the tens place, so we need to borrow again from the hundreds place:

4 11 13

5 2 3

- 2 6 8
Now we can subtract in each column:
In the ones place: 13 - 8 = 5
In the tens place: 11 - 6 = 5
In the hundreds place: 4 - 2 = 2

4 11 13

5 2 3

- 2 6 8

2 5 5

So, 523 - 268 = 255

This example demonstrates how to handle multiple regrouping operations when subtracting larger numbers. The process is the same as with two-digit numbers, but we may need to borrow from multiple columns to complete the subtraction.

Four-Digit Subtraction with Multiple Regrouping

Let's solve 4000 - 2398 using vertical subtraction with regrouping:

Set up the problem:

4 0 0 0

- 2 3 9 8
We can't subtract 8 from 0 in the ones place, so we need to regroup. We borrow 1 from the tens place, but it's 0, so we continue to the hundreds place, which is also 0. We keep going to the thousands place:

3 9 9 10

4 0 0 0

- 2 3 9 8
Now we can subtract in each column:
In the ones place: 10 - 8 = 2
In the tens place: 9 - 9 = 0
In the hundreds place: 9 - 3 = 6
In the thousands place: 3 - 2 = 1

3 9 9 10

4 0 0 0

- 2 3 9 8

1 6 0 2

So, 4000 - 2398 = 1602

This example demonstrates how to handle multiple regrouping operations when subtracting from a number with zeroes. We had to borrow across multiple columns, turning 4000 into 3000 + 900 + 90 + 10. The process is the same as with smaller numbers, but we need to be careful when borrowing from zeroes.