1.1.2: Rounding Whole Numbers

Learning Objectives

• Learn the rules for rounding.
• Round whole numbers to specific place values, including tens, hundreds, and thousands.

In some situations, you don’t need an exact answer. In these cases, rounding the number to a specific place value is possible. For example, if you travelled 973 miles, you might want to round the distance to 1,000 miles, which is easier to think about. Rounding also comes in handy to see if a calculation is reasonable.

These are the rules for rounding whole numbers:

First, identify the digit with the place value to which you are rounding. You might circle or highlight the digit so you can focus on it better.

Then, determine the possible numbers that you would obtain by rounding. These possible numbers are close to the number that you’re rounding to, but have zeros in the digits to the right.

If you are rounding 186 to the nearest ten, then 180 and 190 are the two possible numbers to round to, as 186 is between 180 and 190. But how do you know whether to round to 180 and 190?

Usually, round a number to the number that is closest to the original number.

When a number is halfway between the two possible numbers, round up to the greater number.

Since 186 is between 180 and 190, and 186 is closer to 190, you round up to 190.

You can use a number line to help you round numbers.

Example

A camera is dropped out of a boat, and sinks to the bottom of a pond that is 37 feet deep. Round 37 to the nearest ten.

Solution

The digit you’re rounding to is the tens digit, 3.

37 is between 30 and 40.

37 is only 3 away from 40, but it’s 7 away from 30. So, 37 is closer to 40.

To the nearest ten, 37 rounds to 40.

Example

Round 33 to the nearest ten.

Solution

33 rounds to 30 because 33 is closer to 30.

To the nearest ten, 33 rounds to 30.

You can determine where to round without using a number line by looking at the digit to the right of the one you’re rounding to. If that digit is less than 5, round down. If it’s 5 or greater, round up. In the example above, you can see without a number line that 33 is rounded to 30 because the ones digit, 3, is less than 5.

Example

Round 77 to the nearest ten.

Solution

77 rounds to 80 because the ones digit, 7, is 5 or greater.

77 rounded to the nearest ten is 80.

Example

There are 576 jellybeans in a jar. Round this number to the nearest ten.

Solution

576 rounds to 580 because the ones digit, 6, is 5 or greater.

576 rounded to the nearest ten is 580.

In the previous examples, you rounded to the tens place. The rounded numbers had a 0 in the ones place. If you round to the nearest hundred, the rounded number will have zeros in the tens and ones places. The rounded number will resemble 100, 500, or 1, 200.

Example

A runner ran 1,539 meters, but describes the distance he ran with a rounded number. Round 1,539 to the nearest hundred.

Solution

1,539 rounds to 1,500 because the next digit is less than 5.

1,539 rounded to the nearest hundred is 1,500.

If you round to the nearest thousand, the rounded number will have zeros in the hundreds, tens, and ones places. The rounded number will resemble 1,000 , 2,000 , or 14,000.

Example

A plane’s altitude increased by 2,721 feet. Round this number to the nearest thousand.

Solution

2,721 rounds to 3,000 because the next digit, 7, is 5 or greater.

2,721 rounded to the nearest thousand is 3,000.

Now that you know how to round to the nearest ten, hundred, and thousand, try rounding to the nearest ten thousand.

Example

Round 326,749 to the nearest ten thousand.

Solution

326,749 rounds to 330,000 because the next digit, 6, is 5 or greater.

326,749 rounded to the nearest ten thousand is 330,000.

Exercise

A record number of 23,386 people voted in a city election. Round this number to the nearest hundred.

1. 23,300
2. 23,400
3. 23,000
4. 23,390
1. 23,300

Incorrect. The two possible numbers are 23,300 and 23,400, but 23,386 is closer to 23,400. The tens digit, 8, is 5 or greater, so you should round up. The correct answer is 23,400.

2. 23,400

Correct. The two possible numbers are 23,300 and 23,400, and 23,386 is closer to 23,400. The tens digit, 8, is 5 or greater, so you should round up.

3. 23,000

Incorrect. This number is rounded to the nearest thousand, not the nearest hundred. The correct answer is 23,400.

4. 23,390

Incorrect. This number is rounded to the nearest ten, not the nearest hundred. The correct answer is 23,400.

In situations when you don’t need an exact answer, you can round numbers. When you round numbers, you are always rounding to a particular place value, such as the nearest thousand or the nearest ten. Whether you round up or round down usually depends on which number is closest to your original number. When a number is halfway between the two possible numbers, round up to the larger number.

1.1 Introduction to 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 , … 1 , 2 , 3 , 4 , 5 , … 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 1 and continue.

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

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.

Example 1.1

Which of the following are ⓐ counting numbers? ⓑ whole numbers?

Solution

Which of the following are ⓐ counting numbers ⓑ whole numbers?

0 , 2 3 , 2 , 9 , 11.8 , 241 , 376 0 , 2 3 , 2 , 9 , 11.8 , 241 , 376

Which of the following are ⓐ counting numbers ⓑ whole numbers?

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 537 537 has a different value than the number 735 . 735 . Even though they use the same digits, their value is different because of the different placement of the 7 7 and the 5 . 5 .

Money gives us a familiar model of place value. Suppose a wallet contains three $100$100 bills, seven $10$10 bills, and four $1$1 bills. The amounts are summarized in Figure 1.3. 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 $374 .$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 10 ones, and the hundreds square is made of 10 10 tens, or 100 100 ones.

Example 1.2

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

Solution

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

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

Example 1.3

Solution

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

For each number, find the place value of digits listed: 27,493,615 27,493,615

For each number, find the place value of digits listed: 519,711,641,328 519,711,641,328

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.

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

How To

Name a whole number in words.

1. Step 1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.
2. Step 2. Use commas in the number to separate the periods.

Example 1.4

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

Name each number in words: 9,258,137,904,061 9,258,137,904,061

Name each number in words: 17,864,325,619,004 17,864,325,619,004

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 2014 was 327,577,529 . 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.

The population in a country is 316,128,839 . 316,128,839 . Name that number.

Use Place Value to Write Whole Numbers

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

How To

Use place value to write a whole number.

1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
3. Step 3. Name the number in each period and place the digits in the correct place value position.

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 53,401,742 . 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.

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.

Write each number in standard form:

fifty-three million, eight hundred nine thousand, fifty-one.

Write each number in standard form:

two billion, twenty-two million, seven hundred fourteen thousand, four hundred sixty-six.

Example 1.7

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 $77,000,000,000.$77,000,000,000.

Write each number in standard form:

The closest distance from Earth to Mars is about 34 34 million miles.

Write each number in standard form:

The total weight of an aircraft carrier is 204 204 million pounds.

Round Whole Numbers

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 76 to the nearest ten. Is 76 76 closer to 70 70 or 80 80 on the number line?

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, 80 . 80 . So, 75 75 rounded to the nearest ten is 80 . 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 , 5 , round down. If it is greater than or equal to 5 , 5 , round up.

How To

Round a whole number to a specific place value.

1. Step 1. Locate the given place value. All digits to the left of that place value do not change unless the digit immediately to the left is 9, in which case it may. (See Step 3.)
2. Step 2. Underline the digit to the right of the given place value.
3. Step 3. Determine if this digit is greater than or equal to 5 . 5 .
• Yes—add 1 1 to the digit in the given place value. If that digit is 9, replace it with 0 and add 1 to the digit immediately to its left. If that digit is also a 9, repeat.
• No—do not change the digit in the given place value.
4. Step 4. Replace all digits to the right of the given place value with zeros.

Example 1.8

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 to the nearest ten: 157 . 157 .

Round to the nearest ten: 884 . 884 .

Example 1.9

Round each number to the nearest hundred:

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 added 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 to the nearest hundred: 17,852 . 17,852 .

Round to the nearest hundred: 4,951 . 4,951 .

Example 1.10

Round each number to the nearest thousand:

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.

Round to the nearest thousand: 63,921 . 63,921 .

Round to the nearest thousand: 156,437 . 156,437 .

Section 1.1 Exercises

Practice Makes Perfect

Identify Counting Numbers and Whole Numbers

In the following exercises, determine which of the following numbers are ⓐ counting numbers ⓑ whole numbers.

Model Whole Numbers

In the following exercises, use place value notation to find the value of the number modeled by the base-10 base-10 blocks.

Identify the Place Value of a Digit

In the following exercises, find the place value of the given digits.

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

The height of Mount Ranier is 14,410 14,410 feet.

The height of Mount Adams is 12,276 12,276 feet.

The U.S. Census estimate of the population of Miami-Dade county was 2,617,176 . 2,617,176 .

The population of Chicago was 2,718,782 . 2,718,782 .

The population of China is expected to reach 1,377,583,156 1,377,583,156 in 2016 . 2016 .

The population of India is estimated at 1,267,401,849 1,267,401,849 as of July 1 , 2014 . 1 , 2014 .

Use Place Value to Write Whole Numbers

In the following exercises, write each number as a whole number using digits.

thirty-five thousand, nine hundred seventy-five

sixty-one thousand, four hundred fifteen

eleven million, forty-four thousand, one hundred sixty-seven

eighteen million, one hundred two thousand, seven hundred eighty-three

three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

The population of the world was estimated to be seven billion, one hundred seventy-three million people.

The age of the solar system is estimated to be four billion, five hundred sixty-eight million years.

Lake Tahoe has a capacity of thirty-nine trillion gallons of water.

The federal government budget was three trillion, five hundred billion dollars.

Round Whole Numbers

In the following exercises, round to the indicated place value.

Round to the nearest hundred:

Round to the nearest hundred:

Round to the nearest thousand:

Round to the nearest hundred:

Round to the nearest thousand:

Everyday Math

Buying a Car Jorge bought a car for $24,493 .$24,493 . Round the price to the nearest:

1. ⓐ ten dollars
2. ⓑ hundred dollars
3. ⓒ thousand dollars
4. ⓓ ten-thousand dollars

Remodeling a Kitchen Marissa’s kitchen remodeling cost $18,549 .$18,549 . Round the cost to the nearest:

1. ⓐ ten dollars
2. ⓑ hundred dollars
3. ⓒ thousand dollars
4. ⓓ ten-thousand dollars

Population The population of China was 1,355,692,544 1,355,692,544 in 2014 . 2014 . Round the population to the nearest:

Astronomy The average distance between Earth and the sun is 149,597,888 149,597,888 kilometers. Round the distance to the nearest:

Writing Exercises

In your own words, explain the difference between the counting numbers and the whole numbers.

Give an example from your everyday life where it helps to round numbers.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were.

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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Excel ROUND Function

The Excel ROUND function returns a number rounded to a given number of digits. The ROUND function can round to the right or left of the decimal point.

• number - The number to round.
• num_digits - The number of digits to which number should be rounded.

The ROUNDDOWN function rounds a number to a given number of places. The number of places to round to is controlled by the number of digits provided to ROUND, as seen in the table below:

The Excel ROUNDUP function returns a number rounded up to a given number of decimal places. Unlike standard rounding, where numbers less than 5 are rounded down, ROUNDUP rounds all numbers up.

The Excel ROUNDDOWN function returns a number rounded down to a given number of places. Unlike standard rounding, where only numbers less than 5 are rounded down, ROUNDDOWN rounds all numbers down.

The Excel MROUND function returns a number rounded to a given multiple. MROUND will round a number up or down, depending on the nearest multiple.

The Excel CEILING function rounds a given number up to the nearest specified multiple. CEILING works like the MROUND function, but CEILING always rounds up.

The Excel FLOOR function rounds a given number down to the nearest specified multiple. FLOOR works like the MROUND function, but FLOOR always rounds down.

The Excel INT function returns the integer part of a decimal number by rounding down to the integer. Note the INT function rounds down, so negative numbers become more negative. For example, while INT(10.8) returns 10, INT(-10.8) returns.

The Excel TRUNC function returns a truncated number based on an (optional) number of digits. For example, TRUNC(4.9) will return 4, and TRUNC(-3.5) will return -3. The TRUNC function does no rounding, it simply truncates as specified.

Round a number to a specified multiple

There may be times when you want to round to a multiple of a number that you specify. For example, suppose your company ships a product in crates of 18 items. You can use the MROUND function to find out how many crates you will need to ship 204 items. In this case, the answer is 12, because 204 divided by 18 is 11.333, and you will need to round up. The 12th crate will contain only 6 items.

There may also be times where you need to round a negative number to a negative multiple or a number that contains decimal places to a multiple that contains decimal places. You can also use the MROUND function in these cases.

Fraction Calculator

1 2/3 is a mixed number with a whole number and a fraction. We want to round to the nearest whole number without the fraction.

First, we convert the mixed fraction to an improper fraction or fraction like this:

Then, we divide the numerator by the denominator from the previous step:

Now we have one decimal number. Round using the following rules:

If the number to the right of the decimal point is .5 or higher, then add 1 to the left of the decimal point.

If the number to the right of the decimal point is less than .5, then leave the number to the left of the decimal point as is.

.66 is .5 or higher. Thus, we add 1 to the left of the decimal point. 1 2/3 rounded to the nearest whole number is therefore:

Mixed Number Rounded to the Nearest Whole Number
"What is 1 2/3 rounded to the nearest whole number?" is not the only question we can answer. Round another mixed number here.

What is 1 2/4 rounded to the nearest whole number?
Here is the next mixed number on our list that we have rounded to the nearest whole number.

Measuring in Inches

This is a complete lesson with instruction and exercises for fifth grade. It teaches students about measuring in inches, using the 1/16 parts of an inch.

1. Find the ½-inch mark, 1 ½ -inch mark, and 2 ½-inch mark on all of the rulers above.

2. Find the ¼-inch mark, the ¾-inch mark, the 1 ¼-inch mark, the 1 ¾-inch mark, the 2 ¼-inch mark,
the 2 ¾ - inch mark, and the 3 ¼-inch mark on the bottom three rulers above.

3. On the ruler that measures in 8th parts of an inch, find and label tick marks for these points: the
1/8-inch point, the 5/8-inch point, the 7/8-inch point, the 1 5/8-inch point, and the 2 3/8-inch point.

Also, find these same points on the ruler that measures in 16th parts of an inch.

4. Look at the ruler that measures in 16th parts of an inch. On that ruler find tick marks for these points:

• 3/16 inch
• 7/16 inch
• 11/16 inch
• 1 1/8 inches
• 2 3/8 inches
• 7/8 inch
• 1/4 inch
• 1 1/4 inches
• 2 3/4 inches

5. Measure the following colored lines with the rulers given. If the end of the line does not fall exactly
on a tick mark, then read the mark that is CLOSEST to the end of the line.

6. Measure the following lines using different rulers. Cut out the rulers from the bottom of this page.

Using the 1/4-inch ruler: __________ in.

Using the 1/8-inch ruler: __________ in.

Using the 1/4-inch ruler: __________ in.

Using the 1/8-inch ruler: __________ in.

Using the 1/4-inch ruler: __________ in.

Using the 1/8-inch ruler: __________ in.

Using the 1/4-inch ruler: __________ in.

Using the 1/8-inch ruler: __________ in.

Using the 1/4-inch ruler: __________ in.

Using the 1/8-inch ruler: __________ in.

Using the 1/4-inch ruler: __________ in.

Using the 1/8-inch ruler: __________ in.

You may cut out the following rulers:

7. Find six items in your home that you can measure with your ruler and measure them.

 a. _________________________ _______ in. b. _________________________ _______ in. c. _________________________ _______ in. d. _________________________ _______ in. e. _________________________ _______ in. f. _________________________ _______ in.

8. Carefully measure the sides of the
quadrilateral at the right, and
find its perimeter.

9. A small rectangular bulletin board
measures 15 3/4 in. by 9 1/8 in.
What is its perimeter?

10. Janet checked the amount of sugar in 10
different cookie recipes. The amounts were (in cups):

1 1/2 1 3/8 1 1 3/4 1 1/2 1 1/8 1 1/4 1 1/4 1 1/2 3/4

a. Make a line plot from this data (below) by drawing an X-mark
for each measurement above the number line.

b. If Janet made the recipe with the least amount of sugar
three times, how much sugar would she need?

c. If Janet made the recipe with the largest amount of sugar
three times, how much sugar would she need?

11. Make a line plot from these measurements (lengths of cockroaches, in inches, in Jake's collection):

1 1/4 1 1/8 1 1/8 1 1/2 1 1 1/8 1 3/8 1 3/4 1 3/8 7/8 1 1/4 2 1/8 1/2 1 1/4 1 1/4

This time, you will need to do the scaling on the number line.

b. What is the mode of this data set?

c. Jake took his five longest cockroaches, and placed them
end-to-end. How long a &ldquotrain&rdquo did they form?

12. Measure a bunch of pencils to the nearest 1/8 or 1/16 of an inch. Then make a line plot of

This lesson is taken from Maria Miller's book Math Mammoth Fractions 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

Math Mammoth Fractions 1

A self-teaching worktext for 5th grade that teaches fractions and their operations with visual models. The book covers fractions, mixed numbers, adding and subtracting like fractions, adding and subtracting mixed numbers, adding and subtracting unlike fractions, and comparing fractions.

First work out which number will be left when we finish.

• Rounding to tenths means to leave one number after the decimal point.
• Rounding to hundredths means to leave two numbers after the decimal point.
• etc.

3.1416 rounded to hundredths is 3.14

as the next digit (1) is less than 5

3.1416 rounded to thousandths is 3.142

as the next digit (6) is more than 5

1.2735 rounded to tenths is 1.3

as the next digit (7) is 5 or more

To round to "so many decimal places" count that many digits from the decimal point:

1.2735 rounded to 3 decimal places is 1.274

as the next digit (5) is 5 or more

Excel rounding by changing the cell format

If you want to round numbers in Excel solely for presentations purposes, you can change the cell's format by performing the following steps:

1. Select the cell with the number(s) you want to round.
2. Open the Format Cells dialog by pressing Ctrl + 1 or right click the cell(s) and choose Format Cells. from the context menu.
3. In the Format Cells window, switch to either Number or Currency tab, and type the number of decimal places you want to display in the Decimal paces box. A preview of the rounded number will immediately show up under Sample.
4. Click the OK button to save the changes and close the Format Cells dialog.

1.1 Introduction to Whole Numbers

A more thorough introduction to the topics covered in this section can be found in Prealgebra in the chapters Whole Numbers and The Language of Algebra.

As we begin our study of elementary algebra, we need to refresh some of our skills and vocabulary. This chapter will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

Use Place Value with Whole Numbers

The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting number s. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole number s.

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly.

We can visualize counting numbers and whole numbers on a number line (see Figure 1.2).

Manipulative Mathematics

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 1.3 shows the place values . The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

Example 1.1

In the number 63,407,218, find the place value of each digit:

Solution

Place the number in the place value chart:

ⓐ 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.

For the number 27,493,615, find the place value of each digit:

For the number 519,711,641,328, find the place value of each digit:

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 at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see Figure 1.4). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

How To

Name a Whole Number in Words.

1. Step 1. Start at the left and name the number in each period, followed by the period name.
2. Step 2. Put commas in the number to separate the periods.
3. Step 3. Do not name the ones period.

Example 1.2

Name the number 8,165,432,098,710 using words.

Solution

Name the number in each period, followed by the period name.

Put the commas in to separate the periods.

Name the number 9 , 258 , 137 , 904 , 061 9 , 258 , 137 , 904 , 061 using words.

Name the number 17 , 864 , 325 , 619 , 004 17 , 864 , 325 , 619 , 004 using words.

We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers, separating the periods with commas.

How To

Write a Whole Number Using Digits.

1. Step 1. Identify the words that indicate periods. (Remember, the ones period is never named.)
2. Step 2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
3. Step 3. Name the number in each period and place the digits in the correct place value position.

Example 1.3

Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine as a whole number using digits.

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.

Write the number two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one as a whole number using digits.

Write the number eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six as a whole number using digits.

In 2013, the U.S. Census Bureau estimated the population of the state of New York as 19,651,127. We could say the population of New York was approximately 20 million. In many cases, you don’t need the exact value an approximate number is good enough.

The process of approximating a number is called rounding . Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of New York is approximately 20 million means that we rounded to the millions place.

Example 1.4

How to Round Whole Numbers

Round 23,658 to the nearest hundred.

Solution

Round to the nearest hundred: 17,852 . 17,852 .

Round to the nearest hundred: 468,751 . 468,751 .

How To

Round Whole Numbers.

1. Step 1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
2. Step 2. Underline the digit to the right of the given place value.
3. Step 3. Is this digit greater than or equal to 5?
• Yes–add 1 1 to the digit in the given place value.
• No–do not change the digit in the given place value.
4. Step 4. Replace all digits to the right of the given place value with zeros.

Example 1.5

Solution

 Locate the hundreds place in 103,978. Underline the digit to the right of the hundreds place. Since 7 is greater than or equal to 5, add 1 to the 9. Replace all digits to the right of the hundreds place with zeros. So, 104,000 is 103,978 rounded to the nearest hundred.
 Locate the thousands place and underline the digit to the right of the thousands place. Since 9 is greater than or equal to 5, add 1 to the 3. Replace all digits to the right of the hundreds place with zeros. So, 104,000 is 103,978 rounded to the nearest thousand.
 Locate the ten thousands place and underline the digit to the right of the ten thousands place. Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros. So, 100,000 is 103,978 rounded to the nearest ten thousand.

Round 206,981 to the nearest: ⓐ hundred ⓑ thousand ⓒ ten thousand.

Round 784,951 to the nearest: ⓐ hundred ⓑ thousand ⓒ ten thousand.

Identify Multiples and Apply Divisibility Tests

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

Similarly, a multiple of 3 would be the product of a counting number and 3.

We could find the multiples of any number by continuing this process.

Manipulative Mathematics

Table 1.1 shows the multiples of 2 through 9 for the first 12 counting numbers.

Counting Number 1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108
Multiples of 10 10 20 30 40 50 60 70 80 90 100 110 120

Multiple of a Number

A number is a multiple of n if it is the product of a counting number and n.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, 15 ÷ 3 15 ÷ 3 is 5, so 15 is 5 · 3 . 5 · 3 .

Divisible by a Number

If a number m is a multiple of n, then m is divisible by n.

Look at the multiples of 5 in Table 1.1. They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1.1 that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:

Divisibility Tests

• 2 if the last digit is 0, 2, 4, 6, or 8.
• 3 if the sum of the digits is divisible by 3.
• 5 if the last digit is 5 or 0.
• 6 if it is divisible by both 2 and 3.
• 10 if it ends with 0.

Example 1.6

Is 5,625 divisible by 2? By 3? By 5? By 6? By 10?

Solution

 Is 5,625 divisible by 2? Does it end in 0,2,4,6, or 8? No. 5,625 is not divisible by 2. Is 5,625 divisible by 3? What is the sum of the digits? 5 + 6 + 2 + 5 = 18 5 + 6 + 2 + 5 = 18 Is the sum divisible by 3? Yes. 5,625 is divisble by 3. Is 5,625 divisible by 5 or 10? What is the last digit? It is 5. 5,625 is divisble by 5 but not by 10. Is 5,625 divisible by 6? Is it divisible by both 2 or 3? No, 5,625 is not divisible by 2, so 5,625 is not divisible by 6.

Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10.

Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10.

Find Prime Factorizations and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Factors

Some numbers, like 72, have many factors. Other numbers have only two factors.

Prime Number and Composite Number

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

Manipulative Mathematics

The counting numbers from 2 to 19 are listed in Figure 1.5, with their factors. Make sure to agree with the “prime” or “composite” label for each!

The prime number s less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

Example 1.7

Solution

Find the prime factorization of 80.

Find the prime factorization of 60.

How To

Find the Prime Factorization of a Composite Number.

1. Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
2. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
3. Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
4. Step 4. Write the composite number as the product of all the circled primes.

Example 1.8

Find the prime factorization of 252.

Solution

 Step 1. Find two factors whose product is 252. 12 and 21 are not prime. Break 12 and 21 into two more factors. Continue until all primes are factored. Step 2. Write 252 as the product of all the circled primes. 252 = 2 · 2 · 3 · 3 · 7 252 = 2 · 2 · 3 · 3 · 7

Find the prime factorization of 126.

Find the prime factorization of 294.

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominator s. Two methods are used most often to find the least common multiple and we will look at both of them.

The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18.

We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18.

How To

Find the Least Common Multiple by Listing Multiples.

1. Step 1. List several multiples of each number.
2. Step 2. Look for the smallest number that appears on both lists.
3. Step 3. This number is the LCM.

Example 1.9

Find the least common multiple of 15 and 20 by listing multiples.

Solution

 Make lists of the first few multiples of 15 and of 20, and use them to find the least common multiple. Look for the smallest number that appears in both lists. The first number to appear on both lists is 60, so 60 is the least common multiple of 15 and 20.

Notice that 120 is in both lists, too. It is a common multiple, but it is not the least common multiple.

Find the least common multiple by listing multiples: 9 and 12.

Find the least common multiple by listing multiples: 18 and 24.

Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.

Example 1.10

How to Find the Least Common Multiple Using the Prime Factors Method

Find the Least Common Multiple (LCM) of 12 and 18 using the prime factors method.

Solution

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

Find the LCM using the prime factors method: 9 and 12.

Find the LCM using the prime factors method: 18 and 24.

How To

Find the Least Common Multiple Using the Prime Factors Method.

1. Step 1. Write each number as a product of primes.
2. Step 2. List the primes of each number. Match primes vertically when possible.
3. Step 3. Bring down the columns.
4. Step 4. Multiply the factors.

Example 1.11

Find the Least Common Multiple (LCM) of 24 and 36 using the prime factors method.

Solution

 Find the primes of 24 and 36. Match primes vertically when possible. Bring down all columns. Multiply the factors. The LCM of 24 and 36 is 72.

Find the LCM using the prime factors method: 21 and 28.

Find the LCM using the prime factors method: 24 and 32.

Media

Access this online resource for additional instruction and practice with using whole numbers. You will need to enable Java in your web browser to use the application.

Section 1.1 Exercises

Practice Makes Perfect

Use Place Value with Whole Numbers

In the following exercises, find the place value of each digit in the given numbers.

7,284,915,860,132 ⓐ 7 ⓑ 4 ⓒ 5 ⓓ 3 ⓔ 0

In the following exercises, name each number using words.

In the following exercises, write each number as a whole number using digits.

thirty-five thousand, nine hundred seventy-five

sixty-one thousand, four hundred fifteen

eleven million, forty-four thousand, one hundred sixty-seven

eighteen million, one hundred two thousand, seven hundred eighty-three

three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

In the following, round to the indicated place value.

Round to the nearest hundred.

Round to the nearest hundred.

Round to the nearest hundred.

Round to the nearest hundred.

In the following exercises, round each number to the nearest ⓐ hundred, ⓑ thousand, ⓒ ten thousand.

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 5, 6, and 10.

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

In the following exercises, find the least common multiple of the each pair of numbers using the multiples method.

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

Everyday Math

Writing a Check Jorge bought a car for $24,493. He paid for the car with a check. Write the purchase price in words. Writing a Check Marissa’s kitchen remodeling cost$18,549. She wrote a check to the contractor. Write the amount paid in words.

Buying a Car Jorge bought a car for $24,493. Round the price to the nearest ⓐ ten ⓑ hundred ⓒ thousand and ⓓ ten-thousand. Remodeling a Kitchen Marissa’s kitchen remodeling cost$18,549, Round the cost to the nearest ⓐ ten ⓑ hundred ⓒ thousand and ⓓ ten-thousand.

Population The population of China was 1,339,724,852 on November 1, 2010. Round the population to the nearest ⓐ billion ⓑ hundred-million and ⓒ million.

Astronomy The average distance between Earth and the sun is 149,597,888 kilometers. Round the distance to the nearest ⓐ hundred-million ⓑ ten-million and ⓒ million.

Grocery Shopping Hot dogs are sold in packages of 10, but hot dog buns come in packs of eight. What is the smallest number that makes the hot dogs and buns come out even?

Grocery Shopping Paper plates are sold in packages of 12 and party cups come in packs of eight. What is the smallest number that makes the plates and cups come out even?

Writing Exercises

Give an everyday example where it helps to round numbers.

If a number is divisible by 2 and by 3 why is it also divisible by 6?

What is the difference between prime numbers and composite numbers?

Explain in your own words how to find the prime factorization of a composite number, using any method you prefer.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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What you're seeing here is actually the effect of two roundings. Numbers in ECMAScript are internally represented double-precision floating-point. When id is set to 714341252076979033 ( 0x9e9d9958274c359 in hex), it actually is assigned the nearest representable double-precision value, which is 714341252076979072 ( 0x9e9d9958274c380 ). When you print out the value, it is being rounded to 15 significant decimal digits, which gives 14341252076979100 .

You're overflowing the capacity of JavaScript's number type, see §8.5 of the spec for details. Those IDs will need to be strings.

IEEE-754 double-precision floating point (the kind of number JavaScript uses) can't precisely represent all numbers (of course). Famously, 0.1 + 0.2 == 0.3 is false. That can affect whole numbers just like it affects fractional numbers it starts once you get above 9,007,199,254,740,991 ( Number.MAX_SAFE_INTEGER ).

Beyond Number.MAX_SAFE_INTEGER + 1 ( 9007199254740992 ), the IEEE-754 floating-point format can no longer represent every consecutive integer. 9007199254740991 + 1 is 9007199254740992 , but 9007199254740992 + 1 is also 9007199254740992 because 9007199254740993 cannot be represented in the format. The next that can be is 9007199254740994 . Then 9007199254740995 can't be, but 9007199254740996 can.

The reason is we've run out of bits, so we no longer have a 1s bit the lowest-order bit now represents multiples of 2. Eventually, if we keep going, we lose that bit and only work in multiples of 4. And so on.

Your values are well above that threshold, and so they get rounded to the nearest representable value.

As of ES2020, you can use BigInt for integers that are arbitrarily large, but there is no JSON representation for them. You could use strings and a reviver function:

If you're curious about the bits, here's what happens: An IEEE-754 binary double-precision floating-point number has a sign bit, 11 bits of exponent (which defines the overall scale of the number, as a power of 2 [because this is a binary format]), and 52 bits of significand (but the format is so clever it gets 53 bits of precision out of those 52 bits). How the exponent is used is complicated (described here), but in very vague terms, if we add one to the exponent, the value of the significand is doubled, since the exponent is used for powers of 2 (again, caveat there, it's not direct, there's cleverness in there).

So let's look at the value 9007199254740991 (aka, Number.MAX_SAFE_INTEGER ):

That exponent value, 10000110011 , means that every time we add one to the significand, the number represented goes up by 1 (the whole number 1, we lost the ability to represent fractional numbers much earlier).

But now that significand is full. To go past that number, we have to increase the exponent, which means that if we add one to the significand, the value of the number represented goes up by 2, not 1 (because the exponent is applied to 2, the base of this binary floating point number):

Well, that's okay, because 9007199254740991 + 1 is 9007199254740992 anyway. But! We can't represent 9007199254740993 . We've run out of bits. If we add just 1 to the significand, it adds 2 to the value:

The format just cannot represent odd numbers anymore as we increase the value, the exponent is too big.

Eventually, we run out of significand bits again and have to increase the exponent, so we end up only being able to represent multiples of 4. Then multiples of 8. Then multiples of 16. And so on.