Lesson 1.1: Introduction to Whole Numbers

New Concept

Welcome to algebra's foundation! We'll start with whole numbers, the basic counting blocks ($0, 1, 2, \ldots$). You'll learn to use place value, find factors and multiples, and apply divisibility tests to build your number sense.

What’s next

Next, you'll tackle interactive examples and practice cards on place value, prime factorization, and finding the least common multiple. Let's begin!

Property

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: $1, 2, 3, \ldots$
Whole Numbers: $0, 1, 2, 3, \ldots$
The notation “$\ldots$” is called ellipsis and means “and so on,” or that the pattern continues endlessly.

Examples

• The first five counting numbers are $1, 2, 3, 4, 5$.
• The first five whole numbers are $0, 1, 2, 3, 4$.
• The numbers $15, 250,$ and $3001$ are all both counting numbers and whole numbers.

Explanation

Counting numbers are for counting objects, like $1, 2, 3$ apples. Whole numbers add zero to this group. The ellipsis (...) is a shortcut to show that the list of numbers goes on forever in the same pattern.

Property

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.

Examples

• In the number $72,845,193$, the digit $8$ is in the hundred thousands place, the $2$ is in the millions place, and the $9$ is in the tens place.
• The number $4,321,987$ is named four million, three hundred twenty-one thousand, nine hundred eighty-seven.
• Sixty-three million, four hundred five thousand, two hundred ten is written as the whole number $63,405,210$.

Explanation

The place value system gives each digit a specific job based on where it sits in the number. A $7$ in the tens place is worth $70$, but in the millions place, it's worth $7,000,000$.

Property

To round a whole number:

1. Locate the given place value.
2. Underline the digit to the right of the given place value.
3. If this digit is greater than or equal to $5$, add $1$ to the digit in the given place value. If it is less than $5$, do not change the digit in the given place value.
4. Replace all digits to the right of the given place value with zeros.

Examples

• Round $38,461$ to the nearest hundred. The digit to the right is $6$, so we round up the $4$ to $5$, giving $38,500$.
• Round $572,399$ to the nearest thousand. The digit to the right is $3$, so we keep the $2$ as is, giving $572,000$.
• Round $49,815$ to the nearest thousand. The digit to the right is $8$, so we round up the $9$. This carries over, resulting in $50,000$.

Explanation

Rounding simplifies numbers to make them easier to work with. It's perfect for getting a quick estimate when you don't need an exact answer. Think of it as finding the closest 'neat' number.

Property

To find the prime factorization of a composite number:

1. Find two factors whose product is the given number. Use these numbers to create two branches.
2. If a factor is prime, that branch is complete. Circle the prime.
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.

Examples

• The prime factorization of $45$ is found by breaking it into $5 \cdot 9$. Since $5$ is prime and $9 = 3 \cdot 3$, the factorization is $3 \cdot 3 \cdot 5$.
• The prime factorization of $72$ can be found from $8 \cdot 9$. Then $8 = 2 \cdot 2 \cdot 2$ and $9 = 3 \cdot 3$. The result is $2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$.
• The prime factorization of $100$ comes from $10 \cdot 10$. Each $10$ is $2 \cdot 5$. So, the final factorization is $2 \cdot 2 \cdot 5 \cdot 5$.

Explanation

Every whole number greater than $1$ is either a prime number or can be broken down into a unique set of prime factors. This is like finding the secret recipe or building blocks for that specific number.

Property

To find the Least Common Multiple (LCM) 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 number from each column.
4. Multiply the factors.

Examples

• Find the LCM of $9$ and $12$. The prime factors are $9 = 3 \cdot 3$ and $12 = 2 \cdot 2 \cdot 3$. The LCM is $2 \cdot 2 \cdot 3 \cdot 3 = 36$.
• Find the LCM of $15$ and $25$. The prime factors are $15 = 3 \cdot 5$ and $25 = 5 \cdot 5$. The LCM is $3 \cdot 5 \cdot 5 = 75$.
• Find the LCM of $40$ and $50$. The prime factors are $40 = 2 \cdot 2 \cdot 2 \cdot 5$ and $50 = 2 \cdot 5 \cdot 5$. The LCM is $2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 = 200$.

Explanation

The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. It's great for solving problems where events repeat in cycles, like planning schedules or buying supplies.