# Composing & Decomposing Numbers

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Composing and decomposing numbers helps us become more familiar with the number system. We will look at what it means to compose and decompose a number through definition and example.

## Place Value

Suppose you have \$123.45. Notice that each digit in the amount \$123.45 represents a different amount of money. That is, each digit has the following value.

• The 1 is worth \$100, or 1 one hundred dollar bill.
• The 2 is worth \$20, or 2 ten dollar bills.
• The 3 is worth \$3, or 3 one dollar bills.
• The 4 is worth \$0.40, or 4 dimes.
• The 5 is worth \$0.05, or 5 pennies.

In the same way that the digits in a dollar amount represent different amounts of money, the digits in any number have different values. We call these values place values. The image shows a place value chart and demonstrates the value of a digit based on how many places the digit falls to the right or left of the decimal point in a number.

To illustrate this, consider the number 2,957.41. Each digit in this number has a different place value.

• The 2 falls 4 units to the left of the decimal point in the thousands place, so it has a place value of 2 thousands, or 2,000.
• The 9 falls 3 units to the left of the decimal point in the hundreds place, so it has a place value of 9 hundreds, or 900.
• The 5 falls 2 units to the left of the decimal point in the tens place, so it has a place value of 5 tens, or 50.
• The 7 falls one unit to the left of the decimal point in the ones place, so it has a place value of 7 ones, or 7.
• The 4 falls one unit to the right of the decimal point in the tenths place, so it has a place value of 4 tenths, or 4/10 = 0.4.
• Lastly, the 1 falls 2 units to the right of the decimal point in the hundredths place, so it has a place value of 2 hundredths, or 2/100 = 0.02.

Being familiar with place value allows us to break a number down or put a number together given various place values of the number. These two phenomena are called decomposing and composing, respectively. Let's discuss both of these actions and see how they take place.

## Composing a Number

Let's consider money again to introduce this concept. Suppose I give you 1 one hundred dollar bill, 3 ten dollar bills, 8 one dollar bills, and 7 pennies, and then I ask you how much money you have all together. To find this, you would add up the different amounts of each denomination I gave you. That is, you have 1 one hundred dollar bill worth \$100, 3 ten dollar bills worth \$30, 8 one dollar bills worth \$8, and 7 pennies worth \$0.07. If you add this all together, you get the following.

\$100 + \$30 + \$8 + \$0.07 = \$138.07

Thus, you figure out that you have a total of \$138.07. Now let's put this in terms of regular numbers. Suppose I asked what number would come from 2 hundreds, 5 tens, 1 one, and 7 tenths. In the same way that we figured out how much money we had based on the different denominations, we are going to figure this number out based on the place values given. The 2 hundreds are worth 200, the 5 tens are worth 50, the 1 one is worth 1, and the 7 tenths is worth 7/10 or 0.7. To find the number, we add all these place values together.

200 + 50 + 1 + 0.7 = 251.7

The number described is 251.7. Adding up given place values, as we did, is called composing a number. Composing can be defined as making a whole from parts For instance, a musical composer composes a musical piece from notes. The musical piece is the whole made, and the musical notes are the parts that make it up. The definition of compose makes it easy to remember that composing a number is just putting the number together from its parts.

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