Natural and whole numbers

Natural and whole numbers

Natural numbers are also known as counting numbers. These are the numbers we use when we count things starting from one. Counting always starts with $1$ and continues as: $1, 2, 3, 4, 5, \ldots$

The three dots ($\ldots$) mean that the numbers go on forever without end. This is what we mean by saying they continue to infinity. Therefore, natural numbers are the numbers:

$\mathbb{N} = \{1, 2, 3, 4, 5, \ldots\}$

Natural numbers do not include zero.

Natural numbers are denoted by the letter $\mathbb{N}$. They can be represented on a horizontal line called a number line as shown below:

1 ——— 2 ——— 3 ——— 4 ——— 5 ——— ⋯

Whole numbers are like natural numbers but include the number zero. They are written as:

$\mathbb{W} = \{0, 1, 2, 3, 4, 5, \ldots\}$

Whole numbers are denoted by the letter $\mathbb{W}$. They can also be represented on a number line as shown below:

0 ——— 1 ——— 2 ——— 3 ——— 4 ——— ⋯

The difference between natural and whole numbers

Natural numbers are used for counting objects or representing the position of an object in a sequence. They start from one and go on to infinity. That is why they are sometimes referred to as counting numbers.

Whole numbers include all natural numbers and the number zero. The only number that is a whole number but not a natural number is zero.

Symbolically, we write:

$\mathbb{N} = \{1, 2, 3, 4, 5, \ldots\}$

$\mathbb{W} = \{0, 1, 2, 3, 4, 5, \ldots\}$

Counting numbers (natural numbers) can be classified into different types such as:

1. Perfect numbers
2. Composite numbers
3. Co-prime (relatively prime) numbers
4. Prime numbers
5. Even numbers
6. Odd numbers

Even, odd, and prime numbers

Even numbers are natural numbers that are divisible by 2. That means if a number $n$ satisfies $n \div 2 = \text{whole number}$, then $n$ is an even number.

Examples: $2, 4, 6, 8, 10, 12, \ldots$

Odd numbers are natural numbers that are not divisible by 2. That means if a number $n \div 2$ gives a remainder of 1, then $n$ is an odd number.

Examples: $1, 3, 5, 7, 9, 11, 13, \ldots$

Prime numbers are natural numbers greater than 1 that are divisible by only 1 and themselves. That means they have exactly two positive divisors.

Examples: $2, 3, 5, 7, 11, 13, 17, \ldots$