### Natural Numbers:

The numbers used for **counting** (“there are six coins on the table”) and **ordering** (“this is the third largest city in the country”) are called Natural Numbers.

When we count objects in a group of objects, we start counting from one and then go onto two, three, four etc. This is a natural way of counting objects. Hence, 1, 2, 3, 4, … are called natural numbers.

**Notation:**

**‘N’** is used to refer to the set of all natural numbers. This set is **countably infinite**: it is infinite but countable by definition.

However, fractional numbers like \(\frac{7}{3} \), \(\frac{67}{8} \), \(\frac{548}{2151} \) and decimal numbers like 7.27, 74.52, 845.021 are not natural numbers.

If we add 1 to the first natural number 1, then we get 2, the second natural number. By adding 1 to 2, we get 3, the third natural number. In fact, by adding 1 to any natural number, we get the next natural number. For example, 1000 is the natural number next to 999, 10001 is the natural number next to 10000 and so on.

If we think of any natural number, there is always a natural number next to it.

1 is the first natural number and there is no last natural number.

**Properties of Natural Numbers:**

1) A natural number can be used to express the size of a finite set.

2) The first and the smallest natural number is 1.

3) Every natural number (except 1) can he obtained by adding 1 to the previous natural number.

4) For the natural number 1, there is no ‘previous’ natural number (Though 1 = 0 + 1, but 0 is not a natural number).

5) There is no last or greatest natural number.

6) We cannot complete the counting of all natural numbers. We express this fact by saying that there are infinitely many natural numbers.

0 represents the absence of item being referred to or nothingness or emptiness. Therefore, 0 is not regarded as a natural number.

**Algebraic Properties of Natural Numbers:**

The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:

**Property 1)** **Closure under addition and multiplication:** For all natural numbers a and b, both a + b and a × b are natural numbers.

**Property 2)** **Associativity under addition and multiplication:** For all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

**Property 3) Commutativity**For all natural numbers a and b, a + b = b + a and a × b = b × a.

**under addition and multiplication**:**Property 4)**

**Existence of identity elements:**For every natural number a, a + 0 = a and a × 1 = a.

Natural number addition identity is zero and Natural number multiplication identity is one.

**Property 5)** **Distributivity of multiplication over addition:** For all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c)

**Property 6)** **No zero divisors:** If a and b are natural numbers such that a × b = 0, then a = 0 or b = 0.