# 3 ways to decompose numbers

## Table of contents:

Number decomposition helps younger students understand the arrangement and relationships between the digits of the same number and between the numbers in an operation. You can decompose a number into hundreds, tens, and ones, or by separating the numbers into several addends.

## Steps

### Method 1 of 3: Decompose into Hundreds, Tens, and Ones

#### Step 1. Understand the difference between tens and ones

When you see a number with two digits and no decimal point, the digits represent the tens and the ones. The tens are on the left and the ones on the right.

• The number in the units place can be read exactly as it appears. The only numbers that are considered units are those from 0 to 9 (zero, one, two, three, four, five, six, seven, eight, and nine).
• The number in the tens place is apparently of the same type as the one in the ones place. However, if they are analyzed separately, the ten actually has a 0 to the right, so it is a higher number than the one in the ones place. The numbers that are considered tens are: 10, 20, 30, 40, 50, 60, 70, 80 and 90 (ten, twenty, thirty, forty, fifty, sixty, seventy, eighty and ninety).

#### Step 2. Separate the two-digit number into two parts

A two-digit number will always have a part of ones and another part of tens. To decompose the number, you will have to separate it into those two parts.

• Example: decompose the number 82.

• 8 is the tens digit, so this part of the number can be separated and written as 80.
• The 2 is the ones digit, so this part of the number can be separated and written as 2.
• To write the result, you must express it as follows: 82 = 80 + 2.
• Also, keep in mind that a number written in normal form is expressed in its "standard form", but a decomposed number is expressed in expanded form.

### Based on the example above, "82" would be the standard form and "80 + 2" would be the expanded form

#### Step 3. Enter the hundreds

A three-digit number without a decimal point will always have units, tens, and hundreds. The hundreds are to the left of the number, the tens in the center, and the ones to the right.

• Units and tens work exactly the same as for two-digit numbers.
• The number in the hundreds place is apparently the same as the number in the ones place, but when analyzed separately, a hundred actually has two leading zeros. The numbers that are considered hundreds are: 100, 200, 300, 400, 500, 600, 700, 800 and 900 (one hundred, two hundred, three hundred, four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred).

#### Step 4. Separate the three-digit number into three parts

A three-digit number will always have ones, tens, and hundreds. To decompose a number of this size, you will have to separate it into its three parts.

• Example: decompose the number 394.

• The 3 is in the hundreds place, so this part can be separated and written as 300.
• The 9 is in the tens place, so this part of the number can be separated and written as 90.
• The 4 is in the ones place, so this part of the number can be separated and written as 4.
• You must write the final result as follows: 394 = 300 + 90 + 4.
• If you write the number as 394, it will be expressed in its standard form. If you write it as 300 + 90 + 4, it will be expressed in expanded form.

#### Step 5. Apply this decomposition pattern to numbers with infinite digits

You can decompose larger numbers by following the same procedure.

• A digit, located anywhere in a number, can be expressed separately, replacing the digits on the right with zeros. This is applicable to any number, regardless of the number of digits that compose it.
• Example: 5,394,128 = 5,000,000 + 300,000 + 90,000 + 4,000 + 100 + 20 + 8

#### Step 6. Understand how decimals work

You can decompose a decimal number, but each number after the comma must also be expressed separately with a decimal point.

• The tenths place is used when there is a single digit to the right of the decimal point.
• The hundredths place is used when there are two digits to the right of the decimal point.
• The thousandths place is used when there are three digits to the right of the decimal point.

#### Step 7. Separate a decimal number into several parts

If you have a number with digits to the left and to the right of the decimal point, you will need to decompose it by separating both sides.

• Note that all the digits that appear to the left of the decimal point must be decomposed the same as if there were no comma.
• Example: decompose the number 431, 58

• The 4 is in the hundreds place, so it should be expressed separately as: 400
• The 3 is in the tens place, so it should be expressed separately as: 30
• The 1 is in the ones place, so it should be expressed separately as: 1
• The 5 is in the tenths place, so it should be expressed separately as: 0, 5
• The 8 is in the hundredths place, so it should be expressed separately as: 0, 08
• To write the final result, you must express it as follows: 431, 58 = 400 + 30 + 1 + 0, 5 + 0, 08

### Method 2 of 3: Decompose into Multiple Addends

#### Step 1. Understand the concept

When we decompose a number into several addends, what we do is separate it into other numbers (addends) that can be added to obtain the original value.

• When one addend is subtracted from the original number, the result is equal to the second addend.
• When both addends are added, the result is equal to the original number.

#### Step 2. Practice with a simple number

It's easier to understand the concept by practicing with single-digit numbers (numbers that have only units).

### You can use the principles learned in the section "Decomposing into Hundreds, Tenths, and Units" to decompose higher numbers, but since there are so many possible combinations of addends in multi-digit numbers, this method is not very practical for such cases

#### Step 3. Work with all possible combinations of sums

To decompose a number into addends, you just have to write all the possible ways to get the original value using smaller numbers and sums.

• Example: decompose the number 7 into its different addends.

• 7 = 0 + 7
• 7 = 1 + 6
• 7 = 2 + 5
• 7 = 3 + 4
• 7 = 4 + 3
• 7 = 5 + 2
• 7 = 6 + 1
• 7 = 7 + 0

#### Step 4. Use objects if necessary

For some people who are learning this concept, it may be helpful to use visual examples to illustrate the process in a practical way.

• Start by gathering the original number of something. For example, if the number is seven, you can start by collecting seven pieces of candy.

• Separate the candies into two groups, placing one aside. Count the candies that are left in the second group and explain that the seven at the beginning have broken down into two groups of one and six.
• Keep separating candies into two different groups, removing one more from the first group and adding it to the second. Each time you make a move, count the number of candies in both groups.
• You can do this with many different objects, such as small candles, squares of paper, colored clothespins, buckets or buttons.

### Method 3 of 3: Decompose to Solve Features

#### Step 1. Look at a simple sum

You can combine the two decomposition methods to separate this type of operation in different ways.

### This is easy enough for decomposing simple operations, but not as practical when used to decompose long operations

#### Step 2. Decompose the numbers of the operation

Look at the operation and separate the numbers into tens and ones. If necessary, you can separate the units into smaller groups.

• Example: decompose and solve the operation: 31 + 84.

• You can decompose 31 into: 30 + 1.
• You can decompose 84 into: 80 + 4.

#### Step 3. Modify and rewrite the operation more easily

The operation can be rewritten by expressing each decomposed component separately, or by combining certain decomposed components to see the whole operation more clearly.

### Example: 31 + 84 = 30 + 1 + 80 + 4 = 30 + 80 + 5 = 100 + 10 + 5

#### Step 4. Solve the operation

Once you have rewritten the operation to simplify it and make more sense, you just have to add the numbers and find the result.