Whole Numbers and Place Value

Learning Outcomes

Identify the whole numbers from a list of numbers
Identify the counting numbers from a list of numbers
Use place value notation to find the place value of a number modeled by base-10 blocks
Identify the place value of a digit in a number

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:

1,2,3,4,5,\dots

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

The counting numbers start with

1

and continue.

1,2,3,4,5\dots

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

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

The point labeled

0

is called the origin. The points are equally spaced to the right of

0

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.

0,1,2,3,4,5\ldots

We stopped at

5

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

1

. counting numbers?

2

. whole numbers?

0,\frac{1}{4},3,5.2,15,105

Solution

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

The numbers

\frac{1}{4}

and

5.2

are neither counting numbers nor whole numbers. We will discuss these numbers later.

Try It

Which of the following are

1

. counting numbers

2

. whole numbers?

0,\frac{2}{3},2,9,11.8,241,376

Answer:

$2, 9, 241, 376$
$0, 2, 9, 241, 376$

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

537

has a different value than the number

735

. Even though they use the same digits, their value is different because of the different placement of the

3

and the

7

and the

5

. 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

An image of three stacks of American currency. First stack from left to right is a stack of 3 $100 bills, with label

Find the total value of each kind of bill, and then add to find the total. The wallet contains $374. An image of

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

10

ones, and the hundreds square is made of

10

tens, or

100

ones.

An image with three items. The first item is a single block with the label

The image below shows the number

138

modeled with base-10 blocks. We use place value notation to show the value of the number

138

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

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

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.

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.

Answer: There are $2$ hundreds squares, which is $200$ . There is $1$ tens rod, which is $10$ . There are $5$ ones blocks, which is $5$ . An image of

Digit	Place value	Number	Value	Total value
$2$	hundreds	$2$	$100$	$200\phantom{\rule{1 em}{0ex}}$
$1$	tens	$1$	$10$	$10\phantom{\rule{1 em}{0ex}}$
$5$	ones	$5$	$1$	$+\phantom{\rule{.5 em}{0ex}}5\phantom{\rule{1 em}{0ex}}$
				$215\phantom{\rule{1 em}{0ex}}$

The

\text{base - 10}

blocks model the number

215

https://youtu.be/KcXIxM6h9hk

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.

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.

Answer: 237

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

\text{base - 10}

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

5,278,194

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

example

In the number

63,407,218

; find the place value of each of the following digits:

Answer: Write the number in a place value chart, starting at the right. A figure titled

The $7$ is in the thousands place.
The $0$ is in the ten thousands place.
The $1$ is in the tens place.
The $6$ is in the ten millions place.
The $3$ is in the millions place.

try it

For each number, find the place value of digits listed:

27,493,615

Answer:

Ten Millions
Tens
Hundred Thousands
Millions
Ones

Try It

For each number, find the place value of digits listed:

519,711,641,328

Answer:

Billions
Ten Thousands
Tens
Hundred Thousands
Hundred Millions

The video below shows more examples of how to determine the place value of a digit in a number. https://youtu.be/0-4rLGvVEOI

Whole Numbers and Place Value

Learning Outcomes

Identify Counting Numbers and Whole Numbers

Counting Numbers

Whole Numbers

example

Try It

Model Whole Numbers

example

try it

Identify the Place Value of a Digit

example

try it

Try It

Licenses & Attributions

CC licensed content, Shared previously

CC licensed content, Specific attribution

Studienführer > Prealgebra

Whole Numbers and Place Value

Learning Outcomes

Identify Counting Numbers and Whole Numbers

Counting Numbers

Whole Numbers

example

Try It

Model Whole Numbers

example

try it

Identify the Place Value of a Digit

example

try it

Try It

Licenses & Attributions

CC licensed content, Shared previously

CC licensed content, Specific attribution