An Introduction To Natural Numbers

What is a Natural Number?

Natural numbers, also known as counting numbers, are the positive whole numbers you use to count with: 1, 2, 3, 4, 5, ...

Examples: Are the following natural numbers?

36: This is a natural number since it is a counting number.
-17: This is not a natural number since it is negative.
0: This is somewhat debatable: some mathematicians like to include 0 as a natural number but others do not and only count the positive whole numbers as natural numbers.

Different Types of Natural Numbers

Natural numbers can be split up into different categories or types depending on their properties. Let`s look at a few of them here:
Even numbers: An even number is a number that is exactly divisible by 2 (it doesn`t leave any remainder when it is divided by 2). We can quickly spot even numbers as they end in 0, 2, 4, 6 or 8. Some examples are 2, 68 and 10,838,596.

Odd numbers: An odd number is a number that is not exactly divisible by 2 (it leaves a remainder of 1 when it is divided by 2). We can also quickly spot odd numbers as they end in 1, 3, 5, 7 or 9. Some examples are 1, 13 and 163,627.

Prime numbers: A prime number is a number greater than 1 which has only two positive factors: 1 and itself. Note that 1 is not included as a prime number as its only factor is itself, and prime numbers are required to have two different factors (1 and itself). Some examples are 5, 31 and 103.

Factors and Composite Numbers

We briefly mentioned factors in the last section, but what exactly are they?
A factor of a number is any number that divides that number exactly (it leaves no remainder after division).

Examples:

• 3 is a factor of 12, since 3 divides 12 exactly: 12 = 3 x 4.
• 5 is a factor of 35, since 5 divides 35 exactly: 35 = 5 x 7.
  
• 8 is a factor of 72, since 8 divides 72 exactly: 72 = 8 x 9.
• 2 is not a factor of 11, since 2 does not divide 11 exactly: 11 = 2 x 5 + 1 (a remainder of 1 is left after division).
• 6 is not a factor of 38, since 6 does not divide 38 exactly: 38 = 6 x 6 + 2 (a remainder of 2 is left after division).
• 7 is not a factor of 46, since 7 does not divide 46 exactly: 46 = 7 x 6 + 4 (a remainder of 4 is left after division).
  

With our knowledge of factors, we can now have a look at another type of natural number:
Composite numbers: A composite number is a number with positive factors other than 1 and itself. It can be made by multiplying other numbers together. Some examples are 6 (6 has factors 1, 2, 3, and 6), 28 (28 has factors 1, 2, 4, 7, 14 and 28) and 45 (45 has factors 1, 3, 5, 9, 15 and 45).

Interesting Natural Number Facts

Here are some interesting facts about the natural numbers:

• 2 is the only even prime number: all the rest are odd.

• Any natural number greater than 1 is either prime or composite: 2 is prime, 3 is prime, 4 is composite, 5 is prime, 6 is composite, 7 is prime, 8 is composite, 9 is composite, ...

• Any natural number greater than 1 can be expressed as a product of prime factors. For prime numbers, the product consists of just the number itself. For example, 12 = 2 x 2 x 3, 17 = 17 (it is prime), 28 = 2 x 2 x 7 and 45 = 3 x 3 x 5. For this reason, the prime numbers are often described as the building blocks of the natural numbers.

• There are an infinite number of prime numbers: no matter how large a number you think of, you can always find a prime number greater than it!