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Here are a few recommended readings before getting started with this lesson.
Numbers are used to counting things in everyday life. The most basic numbers for counting are natural and whole numbers.
Natural numbers, also known as counting numbers, are the base numbers for counting things and objects. A natural number contains no fraction, decimal part, or negative sign. A set of natural numbers is represented by the letter $N$ and contains the numbers from $1$ to infinity.
$N={1,2,3,4,5,…}$
Whole numbers represented by the symbol $W$ are a set consisting of all natural numbers and $0.$
$W={0,1,2,3…}$
The dots indicate that the set continues to increase indefinitely one step at a time. These are some things to consider about whole numbers and natural numbers.
Natural Numbers | Whole Numbers |
---|---|
$N={1,2,3,4,5,…}$ | $W={0,1,2,3…}$ |
The smallest natural number is $1.$ | The smallest whole number is $0.$ |
All natural numbers are whole numbers | Not all whole numbers are natural numbers. |
Example: Marky's favorite toys are three car figurines, two puzzles, and a guitar. | Example: The school principal has one notebook and ten pencils, yet he has zero erasers. |
A number line is a horizontal straight line used to represent numbers visually. Numbers are placed along the line at equal intervals jointly with a vertical segment at each interval.
The arrows in the line indicate that the line extends infinitely on the left- and right-hand sides. A number on the left-hand side of the line is always less than a number on the right-hand side.
Natural numbers can be represented in a number line that starts from $1.$ Then $2,$ $3,$ $4,$ and so on, are marked at equal distances along the line to the right of $1.$
A number line can also be used to represent whole numbers. The line begins at $0$ and the rest of the whole numbers are drawn along the line to the right.
Find the whole number represented by the point on the number line.
The product of two or more numbers can be divided evenly by each number in the multiplication expression. This is true every time multiplication is performed. The concepts of factors and multiples are important characteristics to understand when performing such multiplication and division. Consider their definitions.
A number $a$ is a factor of another number $b$ if $a$ divides $b$ evenly and there is no remainder. This also means that the resulting number $c$ is such that the product of $a$ and $c$ equals $b.$
The numbers $a$ and $c$ are factors of $b$ because of the Commutative Property of Multiplication. For this reason, each multiplicand in a multiplication expression is called a factor. A number can have many factors.
Number | Factors |
---|---|
$48$ | $1,$ $2,$ $3,$ $4,$ $6,$ $8,$ $12,$ $16,$ $24,$ $48$ |
$21$ | $1,$ $3,$ $7,$ $21$ |
$9$ | $1,$ $3,$ $9$ |
Dominika loves Halloween. She plans to give candy to her friends at a Halloween party. Her parents give her a bag containing $20$ pieces of candy. She wants to divide the candy into equal amounts.
Find the following information to help Dominika have a great Halloween party.
Factors of $20$ | |
---|---|
$20=1⋅20$ | |
$20=2⋅10$ | |
$20=4⋅5$ | |
$20=5⋅4$ |
Multiples of $20$ |
---|
$20⋅1=20$ |
$20⋅2=40$ |
$20⋅3=60$ |
$20⋅4=80$ |
$20⋅5=100$ |
$20⋅6=120$ |
The sixth multiple of $20$ is $120.$ This means that Dominika's parents should buy $6$ bags of candy. That is because $20⋅6=120.$ Great! The candy is ready and Dominika can now have fun at the Halloween party.
Oh no! Dominika forgot to arrange balloons around her house for the party. She has $100$ balloons. She wants to create arrangements of $6$ balloons without having any balloons leftover. She also has $12$ boxes of $5$ spider decorations to decorate the walls.
Help Dominika complete her Halloween decorations!
Multiples of $5$ | $5,$ $10,$ $15,$ $20,$ $25,$ $30,$ $35,$ $40,$ $45,$ $50,$ $55,$ $60$ |
---|
There are infinitely many whole numbers. Yet, they can be classified into two groups depending on the number of factors they have. Consider, for example, the factors of $11$ and $9.$
Number | Factors | Number of Factors |
---|---|---|
$11$ | $1$ and $11$ | $2$ |
$9$ | $1,$ $3,$ and $9$ | $3$ |
The number $11$ belongs to a set called prime numbers, while $9$ belongs to a group called composite numbers.
A prime number is a whole number greater than $1$ that is only divisible by $1$ and the number itself. In other words, the only factors of a prime number are $1$ and itself.
Number | Factors | Prime? |
---|---|---|
$5$ | $1$ and $5$ | $✓$ |
$6$ | $1,$ $2,$ $3,$ and $6$ | $×$ |
$7$ | $1$ and $7$ | $✓$ |
A composite number is a whole number with more than two factors. This means that there is at least another number different than $1$ and the number itself that divides this number evenly.
Number | Factors | Composite? |
---|---|---|
$4$ | $1,$ $2,$ and $4$ | $✓$ |
$6$ | $1,$ $2,$ $3,$ and $6$ | $✓$ |
$7$ | $1$ and $7$ | $×$ |
The set of composite numbers contains infinitely many numbers. These are some facts about prime and composite numbers.
Prime | Composite |
---|---|
A prime number has only two factors. | A composite number has more than two factors. |
The only even prime number is $2.$ | Every even number greater than $2$ is composite. |
Any whole number greater than $1$ is either prime or composite. | |
The number $1$ is neither prime nor composite. |
The number $1$ is a unique number because it is neither prime nor composite. Why is this?
Number | Prime Factorization |
---|---|
$2$ | $2=2$ |
$3$ | $3=3$ |
$4$ | $4=2⋅2$ |
$5$ | $5=5$ |
$6$ | $6=2⋅3$ |
$12$ | $6=2⋅2⋅3$ |
A composite number is defined as having more than two factors. This means that $1$ cannot be composite because its only factor is itself.
Now that it is known that $1$ is neither prime nor composite let's summarize these facts.
$1$ is not a prime number.
$1$ is not a composite number.
The only factor of $1$ is itself.