Place values of whole numbers
By now, you are familiar with numbers from zero to one thousand and beyond!
If we were to take a number like 3156, we know that each digit has its own value. Even though the number “6” is greater than “5”, in this number that has been provided, the value of “5” is greater than “6”. Similarly, the number “1” is lesser in value than “5” or “6”, but as the digit is placed in front of “5” and “6” its value is greater than the other two digits. Therefore, place value determines the value of each digit based on their position in a number.
Base ten blocks or “Dienes Blocks” is a tool that could be used to help you understand place values better. It is a system of blocks each representing the different place values.
Place value by models (cubes)
Take a look at the cubes and the structures it creates.
- Each small cube is a unit.
- A combination of these units gives us the rod. Each rod has ten units.
- Then we have the flat which is a combination of ten rods. In other words it is a combination of a hundred units.
- Next, we have the cube which is a combination of ten flats. In other words it is a combination of a thousand ones.
By now, we are familiar with two-digit, three-digit, and even four-digit numbers. We usually write the numbers in their standard form.
For instance, the following numbers are written in the standard form.
The numbers are:
38
383
3,838
83,838
The position of the digits placed next to each other gives us the place value of each of these digits. A place value chart helps to determine the value of each of these digits that are placed next to each other. Another observation is the comma that is placed immediately before the last three digits of the number. This comma divides the sets of numbers when written in the standard form. The comma appears after a set of three numbers from the left side of the number. Each of these groups of three numbers is called a period.
As shown before, the value of each place value is shown in the form of the base ten blocks. First we have the units place where the digits are placed as they are. Then the rod represents the tens place, where any digit that takes this place is multiplied by ten. Then comes the plane that represents the hundreds place. Each digit in this place is multiplied by one-hundred. These three place values come under the ones period. Then comes the cube which denotes the ones place of the thousands period. Each digit in this place is multiplied by one-thousand. The cycle repeats in which each period has a ones place, a tens place and a hundreds place, and so on.
Each number can be expressed in different forms. The most familiar representation is the standard form. For instance, let’s look at the number 384.
This is the standard form. The same number when written in words looks like this, three hundred and eighty-four.
Now, the next form of expressing numbers is called the expanded form. Here, each number is expressed as a sum of its digits:
300 + 80 + 4 = 384
Solved Examples
Example 1:
Write the following numbers such that they are expressed as ten times the number and 1/10th of the number. Also place these numbers in a place value chart.
1. 9000
2. 500
Answer:
1. Ten times 9000 = 9000 x 10 = 90,000
\(\frac{1}{10}\) times 9000 = 9000 x \(\frac{1}{10}\) = 900
2. Ten times 500 = 500 x 10 = 5,000
\(\frac{1}{10}\) times 500 = 500 x \(\frac{1}{10}\) = 50
Example 2:
There are 8000 students in School A. There are ten times the number of students in school A, in school B. There is only one-tenth the number of students in school A, in school C. Figure out the number of students in School B and school C and place the numbers in a place value table.
Answer:
Ten times the value of 8000 = 8000 x 10 = 80,000
One-tenth the value of 8000 = 8000 x \(\frac{1}{10}\) = 800
Example 3:
Compare the value of 4 in the following sets of numbers:
1. 5432, 4567
2. 9984, 3647
Answer:
To understand the value of 4 in the first set of numbers, let’s first expand the numbers:
5432 = 5000 + 400 + 30 + 2
4567 = 4000 + 500 + 60 + 7
The value of 4 in the number 5432 is 400 and the value of 4 in 4567 is 4000. The value of 4 in 4567 is 10 times more than the value of 4 in 5432.
9984 = 9000 + 900 + 80 + 4
3647 = 3000 + 600 + 40 + 7
The value of 4 in the number 9984 is 4 itself as it is in the units place and the value of 4 in 3647 is 40. The value of 4 in 3647 is 10 times more than the value of 4 in 9984.
The value of a digit in a number depends on its place value.
In a number that has the same digit, the value of the numbers increases as it moves towards the left of this number. For example, 555555, the value of each digit increases as we move towards the left of the number.