# Natural Numbers

## Natural Numbers: Understanding and Applying Their Unique Properties

Natural numbers, also known as counting numbers, are a set of positive whole numbers starting from 1. They are represented on a number line, providing a clear visual representation of their order and value. While negative numbers are not considered natural numbers, they are closely related to other number classes, including whole numbers, integers, rational and irrational numbers, and real numbers.

To get a better understanding of natural numbers, let's take a closer look at their defining properties.

**Number Classes:** *natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.*

Natural numbers possess four key properties that make them unique and useful in mathematical operations. These properties are known as the closure property, associative property, commutative property, and distributive property.

**Closure Property:** This property states that when two natural numbers are added or multiplied together, the result will always be another natural number. For example, 4+5=9 and 3x6=18.

**Associative Property:** This property means that when three or more natural numbers are added or multiplied, the result will remain the same regardless of how they are grouped. For instance, (4+3)+2=9 and 4+(3+2)=9.

**Commutative Property:** This property states that the order of the numbers being added or multiplied does not affect the result. For example, 6+2=8 and 2+6=8.

**Distributive Property:** This property allows for the simplification of multiplying three natural numbers with brackets by multiplying them separately. For example, 5(2+3)=5(2)+5(3)=25.

The symbol **N** is commonly used to represent a set of natural numbers.

## Calculating the Sum of Natural Numbers

The list of natural numbers follows an arithmetic sequence, which means that there is a simple formula to calculate the sum of any sequence of natural numbers.

The formula is as follows:

*Sum of n terms = (n/2)(n+1)*

In this formula, *n* represents the number of terms, and the sequence always begins with 1. It is important to note that the sum of all natural numbers is infinite.

The symbol **Σ** is used to represent the sum of terms in a sequence.

Let's put this formula to use by finding the sum of the first 50 and 100 natural numbers.

**Example:** Find the sum of the first 50 natural numbers.

Using the formula, we can determine that *n*=50.

Substituting this value into the formula, we get:

Sum = (50/2)(50+1) = 1275

Therefore, the sum of the first 50 natural numbers is 1275.

We can apply the same formula to find the sum of the first 100 natural numbers.

**Example:** Find the sum of the first 100 natural numbers.

Using the formula, we can identify that *n*=100.

Substituting this value into the formula, we get:

Sum = (100/2)(100+1) = 5050

Therefore, the sum of the first 100 natural numbers is 5050.

Looking to expand your knowledge on arithmetic sequences and series? Consider exploring StudySmarter for additional resources.

## Key Takeaways

- Natural numbers are positive whole numbers starting from 1 and are not considered negative numbers.
- The difference between natural and whole numbers is the inclusion of 0 in whole numbers.
- Natural numbers follow an arithmetic sequence, with consecutive numbers following one another, such as 1, 2, 3, 4, 5, 6, etc.
- Natural numbers possess four key properties: closure, associative, commutative, and distributive.
- The symbol
**N**is used to represent a set of natural numbers. - The summation of natural numbers can be calculated using the formula
*(n/2)(n+1)*. - Keep in mind, the sum of all natural numbers is infinite!