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General Studies Math: Help & Review8 chapters | 85 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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.

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.

Now let's look at the flip side of composing. One last time, let's consider money. Suppose I tell you I have $38.19, and the money is composed of bills and coins corresponding to each digit's value. For instance, the 3 falls in the tens spot, so I have 3 ten dollar bills. Similarly, the 8 is in the ones spot, so I have 8 one dollar bills. The 1 is in the ten cents spot, so I have 1 dime, and the 9 is in the one cent spot, so I have 9 pennies. Thus, I have 3 tens worth $30, 8 ones worth $8, 1 dime worth $0.10, and 9 pennies worth $0.09. Putting this all together, I see that my dollar amount of $38.19 can be broken down as follows.

$38.19 = $30 + $8 + $0.10 + $0.09

Now consider doing this for a regular number. That is, break the number down into the sum of its place values. To do this, consider the number 4,771. We have the following.

- The 4 in the thousands spot has place value 4,000.
- The 7 in the hundreds spot has place value 700.
- The 7 in the tens spot has place value 70.
- The 1 in the ones spot has place value 1.

Thus, we can break the number 4,771 down as follows.

4,771 = 4,000 + 700 + 70 + 1

Breaking a number down into the sum of its place values is called **decomposing** the number. Decomposing can be defined as breaking down. For instance, certain compounds decompose in the presence of light. That is, they break down chemically into different parts. This definition makes it easy to remember that decomposing a number is breaking it down into the sum of its place values.

The **place value** of a digit in a number has to do with where the digit falls in relation to the decimal point. We can **decompose** a number by breaking it down into the sum of its place values, and we can **compose** a number by putting its place values together (or summing them up). Being familiar with place value, decomposing numbers, and composing numbers allows us to better understand the number system and how numbers work together.

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General Studies Math: Help & Review8 chapters | 85 lessons

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