Place Value Lesson

When we begin learning basic arithmetic, our study involves the use of the group of numbers known as the whole numbers.
Whole Numbers: {0,1,2,3,4,5,...}
The whole numbers begin with a 0, and increase in increments of 1 indefinitely. There is no largest whole number. This is shown using an ellipsis or three dots "..." after the 5. This simply tells us that the pattern of increasing by 1 will continue forever. The numbers 0 through 9 have a special name, these numbers are referred to as the "digits".
Digits: {0,1,2,3,4,5,6,7,8,9}
Our number system relies on the digits along with place value to construct each number. Single-digit numbers are very easy to understand. The number 5, simply has one digit and its value is 5. When we look at multi-digit numbers, the situation is a bit more complex. In the number 53, the 5 no longer has a value of simply 5, it now represents a value of 50. To understand why we look to our place value system. Under the place value system, a digit obtains its value based on its position or placement in a number. Let's take a look at an example:
Observe how the value of the digit 8 changes in each number:
8 - The 8 means 8 ones, or just 8
83 - The 8 means 8 tens, or 80
859 - The 8 means 8 hundreds, or 800
8032 - The 8 means 8 thousands, or 8000
We can see from our example, that changing the position or placement of a digit, changes its value. Generally, we learn place value with a visual aid known as a place value chart: If we start at the rightmost position of the place value chart, we see it begins with the ones' place. As we move left, we are simply multiplying by ten to get to the next place. Moving to the left of the ones' place is the tens' place, since 1 x 10 = 10. Then we come across the hundreds' place (10 x 10 = 100). This pattern continues out indefinitely. We simply keep multiplying the previous place by ten to obtain the next place to the left.

• 1 - Ones
• 1 x 10 = 10 » Tens
• 10 x 10 = 100 » Hundreds
• 100 x 10 = 1000 » Thousands
• 1000 x 10 = 10,000 » Ten Thousands
• 10,000 x 10 = 100,000 » Hundred Thousands
• 100,000 x 10 = 1,000,000 » Millions

Once we understand how to build a place value chart, using one is very easy. We simply line our number up starting with the rightmost digit going into the rightmost place of the place value chart. We then keep filling in the chart one digit at a time moving left. Once this is done, we can easily give the place for each digit in a given number. Over time, this can be done without a place value chart, but it helps with memorization in the beginning. Let's try a few examples.
Example 1: Write 759 in the place value chart, and give the place of each digit.

• 7 - Hundreds
• 5 - Tens
• 9 - Ones

Example 2: Write 26,041 in the place value chart, and give the place of each digit.

• 2 - Ten Thousands
• 6 - Thousands
• 0 - Hundreds
• 4 - Tens
• 1 - Ones

It is important to note how zero is used as a placeholder. The 0 in the number 26,041, is used to tell us we have 0 hundreds. Without the 0 involved, the number will collapse down to 2641, which is not the same value.
Example 3: Write 8,352,194 in the place value chart, and give the place of each digit.

• 8 - Millions
• 3 - Hundred Thousands
• 5 - Ten Thousands
• 2 - Thousands
• 1 - Hundreds
• 9 - Tens
• 4 - Ones

We can also achieve the same result without a place value chart. We begin by writing a 1 (1's) above the rightmost digit. We can then multiply by 10 as we move to the left (1 x 10 = 10 » 10's, 10 x 10 = 100 » 100's, 100 x 10 = 1000 » 1000's,...). Let's try an example.
Example 4: Give the place of each digit for the number 2509, without a place value chart.

• 2 - Thousands
• 5 - Hundreds
• 0 - Tens
• 9 - Ones

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