Place Value and Expanded Form

Decomposition & Expanding

Consider the number 52,361 (can also be written 52 361 without the comma).

The expanded form of this number can be written as:

50,000 + 2,000 + 300 + 60 + 1

This process is called decomposition. Numbers are decomposed when they are represented as a composition of two or more smaller numbers.

Composing numbers is the opposite of decomposing numbers. The following example illustrates this process.

50,000 + 2,000 + 300 + 60 + 1 = 52,361

The number 52, 361 can also be expanded as follows:

(5 x 10,000) + (2 x 1000) + (3 x 100) + (6 x 10) + (1 x 1)

Place Value

The decomposition or expansion of 52, 361 makes use of place and place value.

In this example, the digit 6 is in the tens place and represents 6 tens (6 x 10) to give a place value of 60. In this way, the place (or position of a digit in a number) determines the place value.

Consider the number 333. The expanded form of this number can be written as:

300 + 30 + 3 OR (3 x 100) + (3 x 10) + (3 x 1)

All of the digits are 3. But each 3 has a different place value depending on its place. The red 3 is in the hundreds place, and has a place value of 300 (3 x 100). The blue 3 is in the ones place, and has a place value of 3 (3 x 1).

A place value chart can help illustrate the decomposition or expanded form of a number.

A zero in the number means that there are no groups of that size in the number. For the number 30,040 in the above table, there are 0 one thousands, 0 hundreds, and 0 ones. These zeros help us to keep the other digits in the correct place.

You’ll notice from the table that each position (i.e. tens, ones, hundreds, millions, ten millions, etc.) is 10x (10 times) bigger than the position before it. For example, ten is 10x bigger than one, and ten thousand is 10x bigger than one thousand.

Similarly, one hundred (100) is 100 bigger than one (1). If you look at the place value chart, the hundreds column is the second column away from the ones column so you are multiplying by 10 twice or multiplying by 100.

You can write the expanded or decomposed form of a number in more than one way. Let’s take the number 45,413 from the previous table and work with it. See the table below for a few other ways that you can represent the same number.

From the place value table, you can see that the number 45,413 could correctly be written in expanded form in the following ways:

(40 x 10,000) + (5 x 1,000) + (4 x 100) + (1 x 10) + (3 x 1)

(45 x 1,000) + (4 x 100) + (1 x 10) + (3 x 1)

(45 x 1,000) + (41 x 10) + (3 x 1)

We can also decompose numbers containing decimals using the notion of place value. Take a look at the following place value chart:

From the table, we see that you could write the number 2.604 in expanded form in the following ways:

2 + 0.6 + 0.004

(2 x 1) + (6 x 0.1) + (4 x 0.001)

Comparing Whole Numbers

We can use our knowledge of place value to help us compare numbers. Let’s use place value to compare and order the following three numbers from least to greatest:

42,231

42,330

43,330

To illustrate the place of each number, we’ll colour the ten thousands place in red, the thousands place in blue, the hundreds place in green, the tens place in purple, and the ones place in orange.

42,231

42,330

43,330

If we compare the ten thousands of each number, they are all the same - so we can’t use the ten thousands to help us determine the order.

But, when we look at the thousands place, we see that the third number has a 3 in the thousands place, whereas the first and second both have a 2 in the thousands place. This means that the third number will be bigger than the first two.

42,231

42,330

43,330 (greatest)

To compare the first two numbers, let’s now look at the hundreds place. We see that the second number has a 3 in the hundreds place (for a place value of 300), whereas the first has a 2 (for a place value of 200). This means that the second number is greater than the first. This also means that the first number is the smaller value. We have the following order now:

42,231 (least)

42,330

43,330 (greatest)

Now, when you compare the ones of the three numbers, you might see that the greatest number (42,330) has a 0 in the ones place whereas the smallest number has a 1 in the ones place.

When we are comparing numbers, we always want to start by comparing the greatest place values. This makes sense because a 9 in the ones place (for a place value of 9) is smaller than a 1 in the tens place (for a place value of 10).

Comparing Decimal Numbers

We can use the same process for comparing decimals. Let’s order the following 3 numbers, this time from greatest to least:

0.032

0.0332

0.32

You will notice that the 3rd number only has 3 digits, whereas the 1st number has 4 and the second has 5 digits. To make comparison by place value easier, let’s line up the decimal points.

In these examples, I will use orange to designate the ones place, blue for the tenths place, purple for the hundredths place, red for the thousandths place, and green for the ten thousandths place.

0.032

0.0332

0.32

When we compare the ones place, we don’t get any information that helps us compare the numbers as they all have a 0 in the ones place.

When we move on to the tenths place, we see that the first two numbers have a 0 in the tenths place, whereas the third number has a 3. This means that this number is the greatest.

Even if the other two numbers has a 9 in the hundredths place and the third number had a 0, we would be comparing a place value of 0.09 and a place value of 0.30, so the third number would still be bigger.

0.032

0.0332

0.32 (greatest)

Let’s pretend for a second that we wanted to compare the ten thousandths place, even though we know it has no bearing on the order of the numbers now.

0.032

0.0332

0.32

We might think that the 1st and the 3rd numbers don’t have digits in the ten thousandths place. But consider the following four numbers and their place value chart expressions:

29.8

29.80

29.800

29.08

Looking at the first three numbers in the chart, we can see that they are actually all the same number. The first number does not have 0’s in the hundredths or thousandths place, but there isn’t anything there at all - so we can treat it as a 0.

You can think of this as being the same as someone saying something is $15 or something is $15.00. They are both the same price.

However, the fourth number (29.08) is actually smaller than the first three, because while the first three have an 8 in the tenths place (for a place value of 0.8), the fourth number has a 0 in the tenths place (for a place value of 0).

You can think of the fourth number as being $29.08, whereas the first three are all equivalent to $29.80 which is $0.72 greater than the fourth number.

Scientific Notation (or "Standard Form")

Like expanded form, scientific notation (sometimes called “standard form”) is just another way to represent numbers. Scientific notation is especially useful for really big or really small numbers. The following examples show how scientific notation can be used to represent a big number.

400 = 4 x 10²

34,093 = 3.4093 x 10⁴

The red part of the number in scientific notation is called the digit, while the purple part represents the power of 10.

For the second example above, 10⁴ was chosen as the power of 10 because 10⁴ = 10 x 10 x 10 x 10 = 10,000. If we calculate 3.4093 times 10,000 (3.4093 x 10,000), we get the number 34,093.

The following example illustrates how we can use scientific notation to represent really small numbers.

0.0045092 = 4.5092 x 10^-3

(take note of the negative sign in front of the 3 in the exponent of the power of 10)

For this example, 10^-3 was chosen because 10^-3equals 0.001. If we calculate 10^-3 x 4.5092, we get the number 0.0045092.

To very quickly decide which power of 10 you should be using to multiply your digit, you can use the following trick:

Special case: The number 3.50, for example, is written as 3.50 x 10⁰ in scientific notation because 10⁰ = 1 and 3.50 x 1 = 3.50

Scientific Notation and Expanded Form

We can now use our knowledge of scientific notation to make the expanded forms of large or small numbers a little neater.

Let’s look at the number 43,450,004 together:

One of the ways we could write this number before is:

(43 x 1,000,000) + (4 x 100,000) + (5 x 10,000) + (4 x 1)

But now we can “tidy up” the expanded form by using scientific notation:

(43 x 10⁶) + (4 x 10⁵) + (5 x 10⁴) + (4 x 10⁰)

Check Your Understanding

Decompose the following numbers in at least three different ways:

1. 54,093

2. 3,300.6

3. 45,092,229,001

4. 1.576

5. 0.0304

Compose the following numbers:

1. 500 + 30 + 6

2. (3 x 1000) + (4 x 100) + (3 x 10)

3. (47 x 1000) + (32 x 10)

Order the following sets of numbers from greatest to least. Explain how you did so using place value.

1. 452, 450, 350

2. 0.0030, 0.0031, 0.0029, 0.030

Write the following numbers in scientific notation (standard form):

1. 430

2. 98,798,398

3. 3094.4

4. 0.00506

5. 0.00000089

*See solutions to provided practice problems on Solutions Page*

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