Whole Numbers and Place Value

Learning Outcomes

  • Identify whole numbers from a list of numbers
  • Identify counting numbers from a list of numbers
  • Model whole numbers
  • Identify the place value of a digit

 

Identify Counting Numbers and Whole Numbers

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.

Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects: [latex]1,2,3,4,5,\dots[/latex] and so on. These are called the counting numbers. The notation “…” is called an ellipsis, which is another way to show “and so on,” or that the pattern continues endlessly. Counting numbers are also called natural numbers.

Counting Numbers or Natural Numbers

The counting numbers start with [latex]1[/latex] and continue.
[latex]1,2,3,4,5\dots[/latex]

Counting numbers and whole numbers can be visualized on a number line as shown below.

The numbers on the number line increase from left to right, and decrease from right to left.

An image of a number line from 0 to 6 in increments of one. An arrow above the number line pointing to the right with the label “larger”. An arrow pointing to the left with the label “smaller”.
The point labeled [latex]0[/latex] is called the origin. The points are equally spaced to the right of [latex]0[/latex] and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.

Whole Numbers

The whole numbers are the counting numbers and zero.
[latex]0,1,2,3,4,5\ldots[/latex]

 

We stopped at [latex]5[/latex] when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.

example

Which of the following are

[latex]1[/latex]. counting numbers?

[latex]2[/latex]. whole numbers?

[latex]0,\frac{1}{4},3,5.2,15,105[/latex]

Solution

  1. The counting numbers start at [latex]1[/latex], so [latex]0[/latex] is not a counting number. The numbers [latex]3,15,\text{and }105[/latex] are all counting numbers.
  2. Whole numbers are counting numbers and [latex]0[/latex]. The numbers [latex]0,3,15,\text{and }105[/latex] are whole numbers.

The numbers [latex]\frac{1}{4}[/latex] and [latex]5.2[/latex] are neither counting numbers nor whole numbers. We will discuss these numbers later.

 

Try It

 

Model Whole Numbers

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. The number [latex]537[/latex] has a different value than the number [latex]735[/latex]. Even though they use the same digits, their value is different because of the different placement of the [latex]3[/latex] and the [latex]7[/latex] and the [latex]5[/latex].

Money gives us a familiar model of place value. Suppose a wallet contains three $100 bills, seven $10 bills, and four $1 bills. The amounts are summarized in the image below. How much money is in the wallet?

An image of three stacks of American currency. First stack from left to right is a stack of 3 $100 bills, with label “Three $100 bills, 3 times $100 equals $300”. Second stack from left to right is a stack of 7 $10 bills, with label “Seven $10 bills, 7 times $10 equals $70”. Third stack from left to right is a stack of 4 $1 bills, with label “Four $1 bills, 4 times $1 equals $4”.
Find the total value of each kind of bill, and then add to find the total. The wallet contains $374.

An image of “$300 + $70 +$4” where the “3” in “$300”, the “7” in “$70”, and the “4” in “$4” are all in red instead of black like the rest of the expression. Below this expression there is the value “$374”. An arrow points from the red “3” in the expression to the “3” in “$374”, an arrow points to the red “7” in the expression to the “7” in “$374”, and an arrow points from the red “4” in the expression to the “4” in “$374”.
Base-10 blocks provide another way to model place value, as shown in the image below. The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of [latex]10[/latex] ones, and the hundreds square (also known as a “flat”) is made of [latex]10[/latex] tens, or [latex]100[/latex] ones.

An image with three items. The first item is a single block with the label “A single block represents 1”. The second item is a horizontal rod consisting of 10 blocks, with the label “A rod represents 10”. The third item is a square consisting of 100 blocks, with the label “A square represents 100”. The square is 10 blocks tall and 10 blocks wide.
The image below shows the number [latex]138[/latex] modeled with base-10 blocks.

We use place value notation to show the value of the number [latex]138[/latex].

An image consisting of three items. The first item is a square of 100 blocks, 10 blocks wide and 10 blocks tall, with the label “1 hundred”. The second item is 3 horizontal rods containing 10 blocks each, with the label “3 tens”. The third item is 8 individual blocks with the label “8 ones”.

An image of “100 + 30 +8” where the “1” in “100”, the “3” in “30”, and the “8” are all in red instead of black like the rest of the expression. Below this expression there is the value “138”. An arrow points from the red “1” in the expression to the “1” in “138”, an arrow points to the red “3” in the expression to the “3” in “138”, and an arrow points from the red “8” in the expression to the “8” in 138.

Digit Place value Number Value Total value
[latex]1[/latex] hundreds [latex]1[/latex] [latex]100[/latex] [latex]100\phantom{\rule{1 em}{0ex}}[/latex]
[latex]3[/latex] tens [latex]3[/latex] [latex]10[/latex] [latex]30\phantom{\rule{1 em}{0ex}}[/latex]
[latex]8[/latex] ones [latex]8[/latex] [latex]1[/latex] [latex]+\phantom{\rule{.5 em}{0ex}}8\phantom{\rule{1 em}{0ex}}[/latex]
[latex]\text{Sum =}138\phantom{\rule{1 em}{0ex}}[/latex]

example

Use place value notation to find the value of the number modeled by the base-10 blocks shown.

An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is one horizontal rod containing 10 blocks. The third item is 5 individual blocks.

 

Try it

 

 

try it

Use place value notation to find the value of the number modeled by the base-10 blocks shown.

An image consisting of three items. The first item is two squares of 100 blocks each, 10 blocks wide and 10 blocks tall. The second item is three horizontal rods containing 10 blocks each. The third item is 7 individual blocks.

 

Identify the Place Value of a Digit

By looking at money and base-10 blocks, we saw that each place in a number has a different value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

Just as with the [latex]\text{base - 10}[/latex] blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it.
The chart below shows how the number [latex]5,278,194[/latex] is written in a place value chart.

A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.

  • The digit [latex]5[/latex] is in the millions place. Its value is [latex]5,000,000[/latex].
  • The digit [latex]2[/latex] is in the hundred thousands place. Its value is [latex]200,000[/latex].
  • The digit [latex]7[/latex] is in the ten thousands place. Its value is [latex]70,000[/latex].
  • The digit [latex]8[/latex] is in the thousands place. Its value is [latex]8,000[/latex].
  • The digit [latex]1[/latex] is in the hundreds place. Its value is [latex]100[/latex].
  • The digit [latex]9[/latex] is in the tens place. Its value is [latex]90[/latex].
  • The digit [latex]4[/latex] is in the ones place. Its value is [latex]4[/latex].

example

In the number [latex]63,407,218[/latex]; find the place value of each of the following digits:

  1. [latex]7[/latex]
  2. [latex]0[/latex]
  3. [latex]1[/latex]
  4. [latex]6[/latex]
  5. [latex]3[/latex]

 

try it

 

The video below shows more examples of how to determine the place value of a digit in a number.