Introduction to 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: $1, 2, 3, \dots$

Whole numbers: $0, 1, 2, 3, \dots$

The notation “$\dots$” 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, which gets larger from left to right and smaller from right to left:

While this number line shows only the whole numbers $0$ through $6$, the numbers keep going without end.

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

The table below shows the place values for the number $5{,}278{,}194$:

The digit $5$ is in the millions place. The digit $2$ is in the hundred-thousands place. The digit $7$ is in the ten-thousands place. The digit $8$ is in the thousands place. The digit $1$ is in the hundreds place. The digit $9$ is in the tens place. The digit $4$ is in the ones place.

Example. In the number $63{,}407{,}218$, find the place value of each digit: (a) $7$ (b) $0$ (c) $1$ (d) $6$ (e) $3$

Placing the number in the place value chart:

(a) The $7$ is in the thousands place. (b) The $0$ is in the ten-thousands place. (c) The $1$ is in the tens place. (d) The $6$ is in the ten-millions place. (e) 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. For example, $74{,}218{,}369$ is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Example. Name the number $8{,}165{,}432{,}098{,}710$ using words.

Naming the number in each period, followed by the period name: $8$ is eight trillion, $165$ is one hundred sixty-five billion, $432$ is four hundred thirty-two million, $098$ is ninety-eight thousand, and $710$ is seven hundred ten. Putting the commas in to separate the periods, we get: eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

We can also reverse the process, writing the digits from the name of a number. To do this, we first look for the clue words that indicate the periods. It helps to draw a blank for each place in a period and then fill in the blanks with the digits, separating the periods with commas.

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

Identifying the words that indicate periods, then drawing blanks for the number of places needed in each period (billions, millions, thousands, ones): the billions period needs only one place here (nine), the millions period needs three places ($246$), the thousands period needs three places ($073$), and the ones period needs three places ($189$). The number is $9{,}246{,}073{,}189$.

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. Round $23{,}658$ to the nearest hundred.

The hundreds place in $23{,}658$ holds the $6$. The digit to its right is $5$, which is greater than or equal to $5$, so we add $1$ to the $6$, making it $7$, and replace the remaining digits with zeros. So $23{,}658$ rounds to $23{,}700$.

Example. Round $103{,}978$ to the nearest (a) hundred (b) thousand (c) ten thousand.

(a) The hundreds place holds the $9$. The digit to its right is $7$, which is greater than or equal to $5$, so we add $1$ to the $9$ (which carries), giving $103{,}978$ rounded to the nearest hundred as $104{,}000$.

(b) The thousands place holds the $3$. The digit to its right is $9$, which is greater than or equal to $5$, so we add $1$ to the $3$, giving $4$. Replacing the rest with zeros, $103{,}978$ rounded to the nearest thousand is $104{,}000$.

(c) The ten-thousands place holds the $0$. The digit to its right is $3$, which is less than $5$, so we leave the $0$ as is. Replacing the rest with zeros, $103{,}978$ rounded to the nearest ten thousand is $100{,}000$.

In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are $a, b, c, m, n, x$, and $y$.

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.

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

Example. Is $5{,}625$ divisible by $2$? By $3$? By $5$? By $6$? By $10$?

Is $5{,}625$ divisible by $2$? It does not end in $0, 2, 4, 6$, or $8$, so $5{,}625$ is not divisible by $2$.

Is $5{,}625$ divisible by $3$? The sum of the digits is $5 + 6 + 2 + 5 = 18$, and $18$ is divisible by $3$, so $5{,}625$ is divisible by $3$.

Is $5{,}625$ divisible by $5$ or $10$? The last digit is $5$, so $5{,}625$ is divisible by $5$ but not by $10$.

Is $5{,}625$ divisible by $6$? It is not divisible by both $2$ and $3$ (it fails $2$), 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 $1 \cdot 72$, $2 \cdot 36$, $3 \cdot 24$, $4 \cdot 18$, and $6 \cdot 12$. Seventy-two has many factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36$, and $72$.

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

The counting numbers from $2$ to $19$, with their factors, show which are prime and which are composite:

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.

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 it 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. Factor $48$.

We start by finding two factors whose product is $48$ — say $2$ and $24$. Since $2$ is prime, that branch is complete. $24$ is not prime, so we break it into $4$ and $6$. Neither is prime, so we break $4$ into $2$ and $2$, and $6$ into $2$ and $3$. All of these are prime, so we circle them:

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.

Example. Find the prime factorization of $252$.

We find two factors of $252$ that are not prime — $12$ and $21$ — and break each into two more factors. Continuing until all branches end in a prime:

Writing $252$ as the product of all the circled primes:

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 denominators. 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 each:

Notice that some numbers appear in both lists. They are the common multiples of $12$ and $18$: $36, 72$, and $108$. Since $36$ is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

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

Listing the first few multiples of $15$ and of $20$:

The smallest number to appear on both lists is $60$, so $60$ is the least common multiple of $15$ and $20$:

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

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. Find the LCM of $12$ and $18$ using the prime factors method.

Writing each number as a product of primes and matching primes vertically when possible, then bringing down the columns:

Multiplying the factors: $\text{LCM}(12, 18) = 2 \cdot 2 \cdot 3 \cdot 3 = 36$.

Notice that the prime factors of $12$ ($2 \cdot 2 \cdot 3$) and the prime factors of $18$ ($2 \cdot 3 \cdot 3$) are included in the LCM ($2 \cdot 2 \cdot 3 \cdot 3$). By matching up the common primes, each common prime factor is used only once. This ensures that $36$ is the least common multiple.

Example. Find the LCM of $24$ and $36$ using the prime factors method.

Finding the primes of $24$ and $36$ and matching primes vertically when possible:

Multiplying the factors: $\text{LCM}(24, 36) = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 72$. The LCM of $24$ and $36$ is $72$.

Key terms

counting numbers — the numbers $1, 2, 3, \dots$, also called natural numbers. whole numbers — the counting numbers together with $0$. rounding — approximating a number to a specific place value. multiple of a number — a number that is the product of a counting number and $n$. divisible — $m$ is divisible by $n$ if $m$ is a multiple of $n$. factors — numbers whose product is a given number. prime number — a counting number greater than $1$ whose only factors are $1$ and itself. composite number — a counting number that is not prime. prime factorization — the product of prime numbers that equals a given number. least common multiple (LCM) — the smallest number that is a multiple of two given numbers.

This section is adapted from Elementary Algebra 2e, Section 1.1: Introduction to Whole Numbers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the place-value chart, factor-tree diagrams, and prime-factor column alignments as tables and accessible inline graphics; omitted the Be Prepared quiz, Manipulative Mathematics callouts, media links, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.