1.1 Introduction to Whole Numbers

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 and whole numbers can be visualized on a number line as shown in Figure 1.2.

Figure 1.2 The numbers on the number line increase from left to right, and decrease from right to left.

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.

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.

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 $7$ and the $5.$

Money gives us a familiar model of place value. Suppose a wallet contains three $\text{\$100}$ bills, seven $\text{\$10}$ bills, and four $\text{\$1}$ bills. The amounts are summarized in Figure 1.3. How much money is in the wallet?

Figure 1.3

Find the total value of each kind of bill, and then add to find the total. The wallet contains $\text{\$374}.$

Base-10 blocks provide another way to model place value, as shown in Figure 1.4. 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.

Figure 1.4

Figure 1.5 shows the number $138$ modeled with $\text{base-10}$ blocks.

Figure 1.5 We use place value notation to show the value of the number $138.$

Example 1.2

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Solution

There are $2$ hundreds squares, which is $200.$

There is $1$ tens rod, which is $10.$

There are $5$ ones blocks, which is $5.$

Digit	Place value	Number	Value	Total value
$2$	hundreds	$2$	$100$	$200$
$1$	tens	$1$	$10$	$10$
$5$	ones	$5$	$1$	$+5$
				$215$

The $\text{base-10}$ blocks model the number $215.$

Try It 1.3

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Try It 1.4

Use place value notation to find the value of the number modeled by the $\text{base-10}$ blocks shown.

Identify the Place Value of a Digit

By looking at money and $\text{base-10}$ 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.

Figure 1.6 shows how the number $\mathrm{5,278,194}$ is written in a place value chart.

Figure 1.6

• The digit $5$ is in the millions place. Its value is $\mathrm{5,000,000}.$
• The digit $2$ is in the hundred thousands place. Its value is $\mathrm{200,000}.$
• The digit $7$ is in the ten thousands place. Its value is $\mathrm{70,000}.$
• The digit $8$ is in the thousands place. Its value is $\mathrm{8,000}.$
• 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 1.3

In the number $\mathrm{63,407,218};$ find the place value of each of the following digits:

1. ⓐ $7$
2. ⓑ $0$
3. ⓒ $1$
4. ⓓ $6$
5. ⓔ $3$

Solution

Write the number in a place value chart, starting at the right.

• ⓐ 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.

Use Place Value to Name Whole Numbers

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period followed by the name of the period without the ‘s’ at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.

So the number $\mathrm{37,519,248}$ is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.

Notice that the word and is not used when naming a whole number.

Example 1.4

Name the number $\mathrm{8,165,432,098,710}$ in words.

Solution

Begin with the leftmost digit, which is 8. It is in the trillions place.	eight trillion
The next period to the right is billions.	one hundred sixty-five billion
The next period to the right is millions.	four hundred thirty-two million
The next period to the right is thousands.	ninety-eight thousand
The rightmost period shows the ones.	seven hundred ten

Putting all of the words together, we write $\mathrm{8,165,432,098,710}$ as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Example 1.5

A student conducted research and found that the number of mobile phone users in the United States during one month in $2014$ was $\mathrm{327,577,529}.$ Name that number in words.

Solution

Identify the periods associated with the number.

Name the number in each period, followed by the period name. Put the commas in to separate the periods.

Millions period: three hundred twenty-seven million

Thousands period: five hundred seventy-seven thousand

Ones period: five hundred twenty-nine

So the number of mobile phone users in the Unites States during the month of April was three hundred twenty-seven million, five hundred seventy-seven thousand, five hundred twenty-nine.

Use Place Value to Write Whole Numbers

We will now reverse the process and write a number given in words as digits.

Example 1.6

Write the following numbers using digits.

• ⓐ fifty-three million, four hundred one thousand, seven hundred forty-two
• ⓑ nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine

Solution

ⓐ Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period.

Put the numbers together, including the commas. The number is $\mathrm{53,401,742}.$

ⓑ Identify the words that indicate periods.

Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Then write the digits in each period.

The number is $\mathrm{9,246,073,189}.$

Notice that in part ⓑ , a zero was needed as a place-holder in the hundred thousands place. Be sure to write zeros as needed to make sure that each period, except possibly the first, has three places.

Example 1.7

A state budget was about $\text{\$77}$ billion. Write the budget in standard form.

Solution

Identify the periods. In this case, only two digits are given and they are in the billions period. To write the entire number, write zeros for all of the other periods.

So the budget was about $\text{\$77,000,000,000.}$

Round Whole Numbers

In $2013,$ the U.S. Census Bureau reported the population of the state of New York as $\mathrm{19,651,127}$ people. It might be enough to say that the population is approximately $20$ million. The word approximately means that $20$ million is not the exact population, but is close to the exact value.

The process of approximating a number is called rounding. Numbers are rounded to a specific place value depending on how much accuracy is needed. $20$ million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have $\mathrm{19,700,000}$ as a result. Had we rounded to the ten thousands place, we would have $\mathrm{19,650,000}$ as a result, and so on. The place value to which we round to depends on how we need to use the number.

Using the number line can help you visualize and understand the rounding process. Look at the number line in Figure 1.7. Suppose we want to round the number $76$ to the nearest ten. Is $76$ closer to $70$ or $80$ on the number line?

Figure 1.7 We can see that $76$ is closer to $80$ than to $70.$ So $76$ rounded to the nearest ten is $80.$

Now consider the number $72.$ Find $72$ in Figure 1.8.

Figure 1.8 We can see that $72$ is closer to $70,$ so $72$ rounded to the nearest ten is $70.$

How do we round $75$ to the nearest ten. Find $75$ in Figure 1.9.

Figure 1.9 The number $75$ is exactly midway between $70$ and $80.$

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, $80.$ So, $75$ rounded to the nearest ten is $80.$

Now that we have looked at this process on the number line, we can introduce a more general procedure. To round a number to a specific place, look at the number to the right of that place. If the number is less than $5,$ round down. If it is greater than or equal to $5,$ round up.

So, for example, to round $76$ to the nearest ten, we look at the digit in the ones place.

The digit in the ones place is a $6.$ Because $6$ is greater than or equal to $5,$ we increase the digit in the tens place by one. So the $7$ in the tens place becomes an $8.$ Now, replace any digits to the right of the $8$ with zeros. So, $76$ rounds to $80.$

Let’s look again at rounding $72$ to the nearest $10.$ Again, we look to the ones place.

The digit in the ones place is $2.$ Because $2$ is less than $5,$ we keep the digit in the tens place the same and replace the digits to the right of it with zero. So $72$ rounded to the nearest ten is $70.$

Example 1.8

Round $843$ to the nearest ten.

Solution

Locate the tens place.
Underline the digit to the right of the tens place.
Since 3 is less than 5, do not change the digit in the tens place.
Replace all digits to the right of the tens place with zeros.
	Rounding 843 to the nearest ten gives 840.

Example 1.9

Round each number to the nearest hundred:

1. ⓐ $\mathrm{23,658}$
2. ⓑ $\mathrm{3,978}$

Solution

ⓐ
Locate the hundreds place.
The digit to the right of the hundreds place is 5. Underline the digit to the right of the hundreds place.
Since 5 is greater than or equal to 5, round up by adding 1 to the digit in the hundreds place. Then replace all digits to the right of the hundreds place with zeros.	So 23,658 rounded to the nearest hundred is 23,700.

ⓑ
Locate the hundreds place.
Underline the digit to the right of the hundreds place.
The digit to the right of the hundreds place is 7. Since 7 is greater than or equal to 5, round up by adding 1 to the 9. Then place all digits to the right of the hundreds place with zeros.	So 3,978 rounded to the nearest hundred is 4,000.

Example 1.10

Round each number to the nearest thousand:

1. ⓐ $\mathrm{147,032}$
2. ⓑ $\mathrm{29,504}$

Solution

ⓐ
Locate the thousands place. Underline the digit to the right of the thousands place.
The digit to the right of the thousands place is 0. Since 0 is less than 5, we do not change the digit in the thousands place.
We then replace all digits to the right of the thousands pace with zeros.	So 147,032 rounded to the nearest thousand is 147,000.

ⓑ
Locate the thousands place.
Underline the digit to the right of the thousands place.
The digit to the right of the thousands place is 5. Since 5 is greater than or equal to 5, round up by adding 1 to the 9. Then replace all digits to the right of the thousands place with zeros.	So 29,504 rounded to the nearest thousand is 30,000.

Notice that in part ⓑ , when we add $1$ thousand to the $9$ thousands, the total is $10$ thousands. We regroup this as $1$ ten thousand and $0$ thousands. We add the $1$ ten thousand to the $2$ ten thousands and put a $0$ in the thousands place.