# Foundations

## Introduction to Whole Numbers

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 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.
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.
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.
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.
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. Further discussion of constants and variables appears later in this section.

### 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.
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,  is 5, so 15 is 
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:

### 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  we say that 8 and 9 are factors of 72. When we write  we say we have factored 72.
Other ways to factor 72 are  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.
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.
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 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.
The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18.
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.
Notice that the prime factors of 12  and the prime factors of 18  are included in the LCM  So 36 is the least common multiple of 12 and 18.
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.

### Key Concepts

1. Place Value as in .
2. Name a Whole Number in Words


3. Write a Whole Number Using Digits


4. Round Whole Numbers


5. Divisibility Tests: A number is divisible by:


6. Find the Prime Factorization of a Composite Number


7. Find the Least Common Multiple by Listing Multiples


8. Find the Least Common Multiple Using the Prime Factors Method



### Practice Makes Perfect

Use Place Value with Whole Numbers
In the following exercises, find the place value of each digit in the given numbers.
In the following exercises, name each number using words.
In the following exercises, write each number as a whole number using digits.
In the following, round to the indicated place value.
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 Exercises


### 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.