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Introduction to Whole Numbers

Use place value with whole numbers

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 numbers. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole numbers.

\(\\)Counting Numbers: 1, 2, 3, …

\(\\)Whole Numbers: 0, 1, 2, 3, …

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 ).

Our number system is called a place value system, because the value of a digit depends on its position in a number. 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

Try it.

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

ⓐ 7
ⓑ 0
ⓒ 1
ⓓ 6
ⓔ 3

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.

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 ). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Example

Try it.

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.

So, \(8,165,432,098,710\) is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

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 2 and a counting number.

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.

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

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 15 by 3, we get a counting number. In fact, \(15\div 3\) is 5, so 15 is \(5\cdot 3.\)

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

Example

Try it.

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?	\(5+6+2+5=18\)
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 and 3?	No, 5,625 is not divisible by 2, so 5,625 is not divisible by 6.

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.

Since \(8\cdot 9=72,\) we say that 8 and 9 are factors of 72. When we write \(72=8\cdot 9,\) we say we have factored 72.

Other ways to factor 72 are \(1\cdot 72,2\cdot 36,3\cdot 24,4\cdot 18,\ \text{and}\ 6\cdot 12.\) Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.

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

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

The prime numbers 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.

How to Find the Prime Factorization of a Composite Number

Try it.

Factor 48.

Solution

We say \(2\cdot 2\cdot 2\cdot 2\cdot 3\) is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!

If we first factored 48 in a different way, for example as \(6\cdot 8,\) the result would still be the same. Finish the prime factorization and verify this for yourself.

Example

Try it.

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.	\(252=2\cdot 2\cdot 3\cdot 3\cdot 7\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Place Value as in .
Name a Whole Number in Words
1. Start at the left and name the number in each period, followed by the period name.
2. Put commas in the number to separate the periods.
3. Do not name the ones period.
Write a Whole Number Using Digits
1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.
3. Name the number in each period and place the digits in the correct place value position.
Round Whole Numbers
1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
2. Underline the digit to the right of the given place value.
3. Is this digit greater than or equal to 5?
  - Yes—add 1 to the digit in the given place value.
  - No—do not change the digit in the given place value.
4. Replace all digits to the right of the given place value with zeros.
Divisibility Tests: A number is divisible by:
- 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.
Find the Prime Factorization of a Composite Number
1. Find two factors whose product is the given number, and use these numbers to create two branches.
2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
3. If a factor is not prime, write it as the product of two factors and continue the process.
4. Write the composite number as the product of all the circled primes.
Find the Least Common Multiple by Listing Multiples
1. List several multiples of each number.
2. Look for the smallest number that appears on both lists.
3. This number is the LCM.
Find the Least Common Multiple Using the Prime Factors Method
1. Write each number as a product of primes.
2. List the primes of each number. Match primes vertically when possible.
3. Bring down the columns.
4. Multiply the factors.

Introduction to Whole Numbers

Use Place Value with Whole Numbers

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

Try it.

51,493 ⓐ 1, ⓑ 4, ⓒ 9, ⓓ 5, ⓔ 3

Solution

ⓐ thousands ⓑ hundreds ⓒ tens ⓓ ten thousands ⓔ ones

Try it.

87,210 ⓐ 2 ⓑ 8 ⓒ 0 ⓓ 7 ⓔ 1

Try it.

164,285 ⓐ 5, ⓑ 6, ⓒ 1, ⓓ 8, ⓔ 2

Solution

ⓐ ones ⓑ ten thousands ⓒ hundred thousands ⓓ tens ⓔ hundreds

Try it.

395,076 ⓐ 5 ⓑ 3 ⓒ 7 ⓓ 0 ⓔ 9

Try it.

93,285,170 ⓐ 9 ⓑ 8 ⓒ 7 ⓓ 5 ⓔ 3

Solution

ⓐ ten millions ⓑ ten thousands ⓒ tens ⓓ thousands ⓔ millions

Try it.

36,084,215 ⓐ 8 ⓑ 6 ⓒ 5 ⓓ 4 ⓔ 3

Try it.

7,284,915,860,132 ⓐ 7 ⓑ 4 ⓒ 5 ⓓ 3 ⓔ 0

Solution

ⓐ trillions ⓑ billions ⓒ millions ⓓ tens ⓔ thousands

Try it.

2,850,361,159,433
ⓐ 9
ⓑ 8
ⓒ 6
ⓓ 4
ⓔ 2

In the following exercises, name each number using words.

Try it.

1,078

Solution

one thousand, seventy-eight

Try it.

5,902

Try it.

364,510

Solution

three hundred sixty-four thousand, five hundred ten

Try it.

146,023

Try it.

5,846,103

Solution

five million, eight hundred forty-six thousand, one hundred three

Try it.

1,458,398

Try it.

37,889,005

Solution

thirty-seven million, eight hundred eighty-nine thousand, five

Try it.

62,008,465

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

Try it.

four hundred twelve

Solution

412

Try it.

two hundred fifty-three

Try it.

thirty-five thousand, nine hundred seventy-five

Solution

35,975

Try it.

sixty-one thousand, four hundred fifteen

Try it.

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

Solution

11,044,167

Try it.

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

Try it.

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

Solution

3,226,512,017

Try it.

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

In the following, round to the indicated place value.

Try it.

Round to the nearest ten.

ⓐ 386 ⓑ 2,931

Solution

ⓐ 390 ⓑ 2,930

Try it.

Round to the nearest ten.

ⓐ 792 ⓑ 5,647

Try it.

Round to the nearest hundred.

ⓐ 13,748 ⓑ 391,794

Solution

ⓐ 13,700 ⓑ 391,800

Try it.

Round to the nearest hundred.

ⓐ 28,166 ⓑ 481,628

Try it.

Round to the nearest ten.

ⓐ 1,492 ⓑ 1,497

Solution

ⓐ 1,490 ⓑ 1,500

Try it.

Round to the nearest ten.

ⓐ 2,791 ⓑ 2,795

Try it.

Round to the nearest hundred.

ⓐ 63,994 ⓑ 63,940

Solution

ⓐ 64,000 ⓑ 63,900

Try it.

Round to the nearest hundred.

ⓐ 49,584 ⓑ 49,548

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

Try it.

392,546

Solution

ⓐ 392,500 ⓑ 393,000 ⓒ 390,000

Try it.

619,348

Try it.

2,586,991

Solution

ⓐ 2,587,000 ⓑ 2,587,000 ⓒ 2,590,000

Try it.

4,287,965

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.

Try it.

Solution

divisible by 2, 3, and 6

Try it.

9,696

Try it.

Solution

divisible by 3 and 5

Try it.

900

Solution

divisible by 2, 3, 5, 6, and 10

Try it.

800

Try it.

986

Solution

divisible by 2

Try it.

942

Try it.

350

Solution

divisible by 2, 5, and 10

Try it.

550

Try it.

22,335

Solution

divisible by 3 and 5

Try it.

39,075

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

Try it.

Solution

\(2\cdot 43\)

Try it.

132

Solution

\(2\cdot 2\cdot 3\cdot 11\)

Try it.

455

Try it.

693

Solution

\(3\cdot 3\cdot 7\cdot 11\)

Try it.

400

Try it.

432

Solution

\(2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3\)

Try it.

627

Try it.

2,160

Solution

\(2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3\cdot 5\)

Try it.

2,520

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

Try it.

8, 12

Solution

Try it.

4, 3

Try it.

12, 16

Solution

Try it.

30, 40

Try it.

20, 30

Solution

Try it.

44, 55

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

Try it.

8, 12

Solution

Try it.

12, 16

Try it.

28, 40

Solution

280

Try it.

84, 90

Try it.

55, 88

Solution

440

Try it.

60, 72

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

In the number 63,407,218, find the place value of each digit:
1. ⓐ 7
2. ⓑ 0
3. ⓒ 1
4. ⓓ 6
5. ⓔ 3
i

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:
ⓐ 2 ⓑ 1 ⓒ 4 ⓓ 7 ⓔ 5

i

ⓐ ten millions ⓑ tens ⓒ hundred thousands ⓓ millions ⓔ ones
For the number 519,711,641,328, find the place value of each digit:
ⓐ 9 ⓑ 4 ⓒ 2 ⓓ 6 ⓔ 7

i

ⓐ billions ⓑ ten thousands ⓒ tens ⓓ hundred thousands ⓔ hundred millions
Name the number 8,165,432,098,710 using words.

i

Name the number in each period, followed by the period name.
Put the commas in to separate the periods.
So, \(8,165,432,098,710\) is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.
Name the number \(9,258,137,904,061\) using words.

i

nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one
Name the number \(17,864,325,619,004\) using words.

i

seventeen trillion, eight hundred sixty-four billion, three hundred twenty-five million, six hundred nineteen thousand, four
Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine as a whole number using digits.

i

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 9,246,073,189.
Write the number two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one as a whole number using digits.

i

\(2,466,714,051\)
Write the number eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six as a whole number using digits.

i

\(11,921,830,106\)
Round 23,658 to the nearest hundred.
Round to the nearest hundred: \(17,852.\)

i

\(17,900\)
Round to the nearest hundred: \(468,751.\)

i

\(468,800\)

Round \(103,978\) to the nearest:

ⓐ hundred
ⓑ thousand
ⓒ ten thousand

ⓐ

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.

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

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?	\(5+6+2+5=18\)
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 and 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.

i

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

i

by 3 and 5
Factor 48.

i

We say \(2\cdot 2\cdot 2\cdot 2\cdot 3\) is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!
If we first factored 48 in a different way, for example as \(6\cdot 8,\) the result would still be the same. Finish the prime factorization and verify this for yourself.
Find the prime factorization of 80.

i

\(2\cdot 2\cdot 2\cdot 2\cdot 5\)
Find the prime factorization of 60.

i

\(2\cdot 2\cdot 3\cdot 5\)

Find the prime factorization of 252.

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.	\(252=2\cdot 2\cdot 3\cdot 3\cdot 7\)

Find the prime factorization of 126.

i

\(2\cdot 3\cdot 3\cdot 7\)
Find the prime factorization of 294.

i

\(2\cdot 3\cdot 7\cdot 7\)

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

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.

i

36
Find the least common multiple by listing multiples: 18 and 24.

i

72
Find the Least Common Multiple (LCM) of 12 and 18 using the prime factors method.
Find the LCM using the prime factors method: 9 and 12.

i

36
Find the LCM using the prime factors method: 18 and 24.

i

72
Find the Least Common Multiple (LCM) of 24 and 36 using the prime factors method.

i

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.

i

84
Find the LCM using the prime factors method: 24 and 32.

i

96

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Introduction to Whole Numbers

Use place value with whole numbers
Identify multiples and apply divisibility tests
Find prime factorizations and least common multiples
Start at the left and name the number in each period, followed by the period name.
Put commas in the number to separate the periods.
Do not name the ones period.
Identify the words that indicate periods. (Remember, the ones period is never named.)
Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Kuri Gukoresha

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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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.