A natural number is an integer greater than 0. Natural numbers consist of positive numbers such as 1,2,3.

The set of natural numbers is denoted by English Capital N.That is $N=[1,2,2,4,5,6,7......]$.

It should be noted that negative numbers such as $-1,-2,-3$ are not included in natural numbers.

Also, zero is not wholly considered a natural number because it has neither positive nor negative value.

Because natural numbers are positive numbers, they are sometimes referred to as positive integers, counting numbers.

## Properties of Natural Numbers

1. **Summation property**: This states that a natural number will always result from the addition of two or more natural numbers.

Consider the natural number 6,3,2,1, for instance. They can be added together to get 12, which is a natural number.

2. **Multiplicative property**: Just like adding natural numbers will produce natural numbers, multiplication of two or more natural numbers will produce natural numbers.

For instance, multiplying 3,5 and 10 will result in 150, a natural number.

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3. **Subtractive property**: The subtraction of any two natural numbers may or may not result in a natural number.

For example, $10-4$ will result in 6, but $4-10$ will result in -6, which is an integer, not a natural number.

Therefore, we can conclude that the subtraction of two natural numbers will result in a non-natural number if a bigger number is subtracted from a lesser number.

In the same vein, subtraction of two natural numbers will result in a natural number if a lesser number is substrates from a bigger number.

4. **Division property**: The division of two natural numbers may or may not result in natural numbers.

For example, $12÷4$ will yield 3, which is a natural number.

However, $10÷4$ will produce 2.5, which is not a natural number.

For clarity, 2.5 is a rational number.

5. **Square property**: The square of any natural number will always produce a natural number also.

That is if $B\in N$, then $B^2\in N$.

For example, $3^2$ will produce 9, just like $5^2$ will produce 25.

6. **Additive commutative property**: The order of two or more natural number does not affect their addition.

That is, a+b+c will yield the same result as b+c+a, which will also yield the same result as c+a+b.

For example $3+2+1=6$, just like $1+2+3$ and $3+2+1$ is 6.

Thus, we see that the order in which the operands do not affect the additive result.

7. **Multiplicative commutative property:** When natural numbers are multiplied, the order in which the operands are taken has no bearing on the outcome.

For example, $3×6×7$ will give the same answer as $6×3×7$.

8. **Square root property**: The square root of a natural number may or may not produce a natural number.

As an illustration, the square root of 9 is 3, whereas the square root of 3 is a non-natural integer, 1.7321.

**Observations: if** a natural number is a perfect square, its square root will also be a natural number.

On the other hand, if the natural number is not a perfect square, then its square root will result in an irrational number.

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9. **Non-commutative division property**: The division of two natural numbers is not commutative, so the order of the two natural numbers matters.

For example, $10÷5=2$ but $5÷10=0.5$. That is $10÷5≠5÷10$

As can be seen, the division of natural numbers is non-commutative

10. **Non-commutative subtractive property**: The order of two natural numbers affects their subtraction.

In other words, the subtractive result depends on the order in which the operands are taken.

For example, $5-6$ will produce $-1$ but 6-5 will produce $1$.

To repeat, Natural numbers are positive numbers that are greater than zero.

There are ten properties of natural numbers, namely; summation property, multiplicative property, subtractive property, division property, square property, commutative summation property, multiplicative commutative property, square root property, non-commutative division property, and non-commutative subtraction property.

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