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.

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 2. How much money is in the wallet?

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 3. 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 4 shows the number $138$ modeled with $\text{base-10}$ blocks.

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

Solution

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 $215.$

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

• 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 $4.$

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.

Use Place Value to Write Whole Numbers

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

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 7. Suppose we want to round the number $76$ to the nearest ten. Is $76$ closer to $70$ or $80$ on the number line?

Now consider the number $72.$ Find $72$ in Figure 8.

How do we round $75$ to the nearest ten. Find $75$ in Figure 9.

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 $70.$

Round $843$ to the nearest ten.

Solution

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.

Round each number to the nearest hundred:

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

Solution

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.

Round each number to the nearest thousand:

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

Solution

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.

Use Addition Notation

A college student has a part-time job. Last week he worked $3$ hours on Monday and $4$ hours on Friday. To find the total number of hours he worked last week, he added $3$ and $4.$

The operation of addition combines numbers to get a sum. The notation we use to find the sum of $3$ and $4$ is:

We read this as three plus four and the result is the sum of three and four. The numbers $3$ and $4$ are called the addends. A math statement that includes numbers and operations is called an expression.

Model Addition of Whole Numbers

Addition is really just counting. We will model addition with $\text{base-10}$ blocks. Remember, a block represents $1$ and a rod represents $10.$ Let’s start by modeling the addition expression we just considered, $3+4.$

Each addend is less than $10,$ so we can use ones blocks.

We start by modeling the first number with 3 blocks.
Then we model the second number with 4 blocks.
Count the total number of blocks.

There are $7$ blocks in all. We use an equal sign $\text{(=)}$ to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. $3+4=7.$

Model the addition $2+6.$

Solution

Solution

$2+6$ means the sum of $2$ and $6$

Each addend is less than 10, so we can use ones blocks.

Model the first number with 2 blocks.
Model the second number with 6 blocks.
Count the total number of blocks	There are $8$ blocks in all, so $2+6=8.$

When the result is $10$ or more ones blocks, we will exchange the $10$ blocks for one rod.

Model the addition $5+8.$

Solution

Solution

$5+8$ means the sum of $5$ and $8.$

Each addend is less than 10, se we can use ones blocks.
Model the first number with 5 blocks.
Model the second number with 8 blocks.
Count the result. There are more than 10 blocks so we exchange 10 ones blocks for 1 tens rod.
Now we have 1 ten and 3 ones, which is 13.	5 + 8 = 13

Notice that we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. From now on, we will use the shorter version but keep in mind that they mean the same thing.

Next we will model adding two digit numbers.

Model the addition: $17+26.$

Solution

Solution

$17+26$ means the sum of 17 and 26.

Model the 17.	1 ten and 7 ones
Model the 26.	2 tens and 6 ones
Combine.	3 tens and 13 ones
Exchange 10 ones for 1 ten.	4 tens and 3 ones $40+3=43$
We have shown that $17+26=43$

Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.

Imagine filling in Table 6 by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.

Look at the pairs of sums.

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.

In the previous example, the sum of the ones and the sum of the tens were both less than $10.$ But what happens if the sum is $10$ or more? Let’s use our $\text{base-10}$ model to find out. Figure 1 shows the addition of $17$ and $26$ again.

When we add the ones, $7+6,$ we get $13$ ones. Because we have more than $10$ ones, we can exchange $10$ of the ones for $1$ ten. Now we have $4$ tens and $3$ ones. Without using the model, we show this as a small red $1$ above the digits in the tens place.

When the sum in a place value column is greater than $9,$ we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, $10$ ones for $1$ ten or $10$ tens for $1$ hundred.

Add: $324+586.$

Solution

Solution

Step-by-step guide to adding multi-digit numbers with visual examples.

Write the numbers so the digits line up vertically.
Add the digits in each place value. Add the ones: $4+6=10$ Write the $0$ in the ones place in the sum and carry the $1$ ten to the tens place.
Add the tens: $1+2+8=11$ Write the $1$ in the tens place in the sum and carry the $1$ hundred to the hundreds
Add the hundreds: $1+3+5=9$ Write the $9$ in the hundreds place.

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in Table 16.

Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.

Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.

Find the perimeter of the patio shown.

Solution

Solution

This table illustrates the step-by-step process of calculating the perimeter of an object, from defining the problem to providing the final solution.

We are asked to find the perimeter.
Write a phrase.	the sum of the sides
Translate to math notation.	$4+6+2+3+2+9$
Simplify by adding.	$26$
Write a sentence to answer the question.
We added feet, so the sum is $26$ feet.	The perimeter of the patio is $26$ feet.

Use Subtraction Notation

Suppose there are seven bananas in a bowl. Elana uses three of them to make a smoothie. How many bananas are left in the bowl? To answer the question, we subtract three from seven. When we subtract, we take one number away from another to find the difference. The notation we use to subtract $3$ from $7$ is

We read $7-3$ as seven minus three and the result is the difference of seven and three.

Model Subtraction of Whole Numbers

A model can help us visualize the process of subtraction much as it did with addition. Again, we will use $\text{base-10}$ blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered, $7-3.$

This table visually demonstrates the subtraction of 3 from 7 using blocks, illustrating each step from initial modeling to the final mathematical result.

We start by modeling the first number, 7.
Now take away the second number, 3. We'll circle 3 blocks to show that we are taking them away.
Count the number of blocks remaining.
There are 4 ones blocks left.	We have shown that $7-3=4$.

Model the subtraction: $8-2.$

Solution

Solution

$8-2$ means the difference of 8 and 2.
Model the first, 8.
Take away the second number, 2.
Count the number of blocks remaining.
There are 6 ones blocks left.	We have shown that $8-2=6$.

Model the subtraction: $13-8.$

Solution

Solution

Model the first number, 13. We use 1 ten and 3 ones.
Take away the second number, 8. However, there are not 8 ones, so we will exchange the 1 ten for 10 ones.
Now we can take away 8 ones.
Count the blocks remaining.
There are five ones left.	We have shown that $13-8=5$.

As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens.

Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.

We know $7-3=4$ because $4+3=7.$ Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.

To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.

When we modeled subtracting $26$ from $43,$ we exchanged $1$ ten for $10$ ones. When we do this without the model, we say we borrow $1$ from the tens place and add $10$ to the ones place.

Subtract: $43-26.$

Solution

Solution

A step-by-step visual guide demonstrating the process of subtracting numbers with borrowing, including textual instructions and corresponding illustrations.

Write the numbers so each place value lines up vertically.
Subtract the ones. We cannot subtract 6 from 3, so we borrow 1 ten. This makes 3 tens and 13 ones. We write these numbers above each place and cross out the original digits.
Now we can subtract the ones. $13-6=7.$ We write the 7 in the ones place in the difference.
Now we subtract the tens. $3-2=1.$ We write the 1 in the tens place in the difference.
Check by adding. Our answer is correct.

Subtract and then check by adding: $207-64.$

Solution

Solution

This table illustrates the step-by-step process of subtracting multi-digit numbers with borrowing, followed by a verification step.

Write the numbers so each place value lines up vertically.
Subtract the ones. $7-4=3.$ Write the 3 in the ones place in the difference.
Subtract the tens. We cannot subtract 6 from 0 so we borrow 1 hundred and add 10 tens to the 0 tens we had. This makes a total of 10 tens. We write 10 above the tens place and cross out the 0. Then we cross out the 2 in the hundreds place and write 1 above it.
Now we subtract the tens. $10-6=4.$ We write the 4 in the tens place in the difference.
Finally, subtract the hundreds. There is no digit in the hundreds place in the bottom number so we can imagine a 0 in that place. Since $1-0=\mathrm{1,}$ we write 1 in the hundreds place in the difference.
Check by adding. Our answer is correct.

Subtract and then check by adding: $910-586.$

Solution

Solution

Step-by-step guide demonstrating multi-digit subtraction with borrowing, showing instructional text alongside visual examples and a final check.

Write the numbers so each place value lines up vertically.
Subtract the ones. We cannot subtract 6 from 0, so we borrow 1 ten and add 10 ones to the 0 ones we had. This makes 10 ones. We write a 0 above the tens place and cross out the 1. We write the 10 above the ones place and cross out the 0. Now we can subtract the ones. $10-6=4.$
Write the 4 in the ones place of the difference.
Subtract the tens. We cannot subtract 8 from 0, so we borrow 1 hundred and add 10 tens to the 0 tens we had, which gives us 10 tens. Write 8 above the hundreds place and cross out the 9. Write 10 above the tens place.
Now we can subtract the tens. $10-8=2$.
Subtract the hundreds place. $8-5=3$ Write the 3 in the hundreds place in the difference.
Check by adding. Our answer is correct.

Subtract and then check by adding: $\mathrm{2,162}-479.$

Solution

Solution

Write the numbers so each place value lines up vertically.
Subtract the ones. Since we cannot subtract 9 from 2, borrow 1 ten and add 10 ones to the 2 ones to make 12 ones. Write 5 above the tens place and cross out the 6. Write 12 above the ones place and cross out the 2.
Now we can subtract the ones.	$12-9=3$
Write 3 in the ones place in the difference.
Subtract the tens. Since we cannot subtract 7 from 5, borrow 1 hundred and add 10 tens to the 5 tens to make 15 tens. Write 0 above the hundreds place and cross out the 1. Write 15 above the tens place.
Now we can subtract the tens.	$15-7=8$
Write 8 in the tens place in the difference.
Now we can subtract the hundreds.
Write 6 in the hundreds place in the difference.
Subtract the thousands. There is no digit in the thousands place of the bottom number, so we imagine a 0. $1-0=1.$ Write 1 in the thousands place of the difference.
Check by adding. $\begin{array}{}\\ \stackrel{1}{1},\stackrel{1}{6}\stackrel{1}{8}3\hfill \\ \underset{\text{\_\_\_\_\_\_}}{+479}\hfill \\ \mathrm{2,}162✓\hfill \end{array}$

Our answer is correct.

Translate Word Phrases to Math Notation

As with addition, word phrases can tell us to operate on two numbers using subtraction. To translate from a word phrase to math notation, we look for key words that indicate subtraction. Some of the words that indicate subtraction are listed in Table 12.

Subtract Whole Numbers in Applications

To solve applications with subtraction, we will use the same plan that we used with addition. First, we need to determine what we are asked to find. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question, using the appropriate units.

The temperature in Chicago one morning was $73$ degrees Fahrenheit. A cold front arrived and by noon the temperature was $27$ degrees Fahrenheit. What was the difference between the temperature in the morning and the temperature at noon?

Solution

Solution

We are asked to find the difference between the morning temperature and the noon temperature.

This table illustrates the step-by-step process of translating a word phrase into a subtraction problem and calculating its numerical difference.

Write a phrase.	the difference of 73 and 27
Translate to math notation. Difference tells us to subtract.	$73-27$
Then we do the subtraction.
Write a sentence to answer the question.	The difference in temperatures was 46 degrees Fahrenheit.

A washing machine is on sale for $\text{\$399}.$ Its regular price is $\text{\$588}.$ What is the difference between the regular price and the sale price?

Solution

Solution

We are asked to find the difference between the regular price and the sale price.

This table illustrates the step-by-step process of solving a subtraction word problem, from phrasing to the final answer.

Write a phrase.	the difference between 588 and 399
Translate to math notation.	$588-399$
Subtract.
Write a sentence to answer the question.	The difference between the regular price and the sale price is $189.

Use Multiplication Notation

Suppose you were asked to count all these pennies shown in Figure 1.

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number $3$ times.

Multiplication is a way to represent repeated addition. So instead of adding $8$ three times, we could write a multiplication expression.

We call each number being multiplied a factor and the result the product. We read $3\times 8$ as three times eight, and the result as the product of three and eight.

There are several symbols that represent multiplication. These include the symbol $\times$ as well as the dot, $·$, and parentheses $(\text{).}$

Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used $\text{base-10}$ blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.

Multiply Whole Numbers

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section.

Table 2 shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.

What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and $1$ is called the multiplicative identity.

Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that $8+9=17$ is the same as $9+8=17.$

Is this also true for multiplication? Let’s look at a few pairs of factors.

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

We start by multiplying $3$ by $7.$

We write the $1$ in the ones place of the product. We carry the $2$ tens by writing $2$ above the tens place.

Then we multiply the $3$ by the $2,$ and add the $2$ above the tens place to the product. So $3\times 2=6,$ and $6+2=8.$ Write the $8$ in the tens place of the product.

The product is $81.$

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.