 # Natural Numbers  View Notes

## Natural Numbers, The Definition of Natural Numbers and Types of Natural Numbers

Natural numbers are an important part of the number system, including all the positive integers from 1 to infinity, used for counting purposes. Natural numbers come under real numbers and include the positive integers 1, 2, 3, 4, 5, 6, 7, 8... and so on.

### Define a Natural Number

Natural numbers are the positive integers, including numbers from 1 to infinity. Natural numbers are countable numbers and are preferable for calculations. 1 is the smallest natural number and the sum of natural numbers from 1 to 100 is (n (n+1) /2).

### Whole Numbers and Natural Numbers

Natural numbers and whole numbers are different from each other in the matter of including zero. Whole numbers include zero, but all natural numbers are the positive numbers excluding zero.

Every natural number is a whole number, but every whole number is not a natural number.

### Set of Natural Numbers in Maths

N is the natural numbers’ set representation and represents the following:

Statement:

N = Set of numbers starting from 1 and lasting till infinity.

Roster Form:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10... and so on}

Set Builder Form:

N = {x: x is a number starting from 1}

## Properties of the Natural Number

Natural numbers follow four main properties, which are as follows:

1. Closure Property

2. Commutative Property

3. Associative Property

4. Distributive Property

### Closure Property

A natural number is closed under addition and multiplication. This means that adding or multiplying two natural numbers results in a natural number. However, for subtraction and division, natural numbers do not follow closure property.

When a and b are two natural numbers, a+b is also a natural number. For example, 2+3=5, 6+7=13, and similarly, all the resultants are natural numbers.

Subtraction

For two natural numbers a and b, a-b might not result in a natural number. E.g. 6-5 = 1 but 5-6=-1.

Multiplication

When a and b are two natural numbers, a*b is also a natural number. Example, 3*5 =15, and similarly all resultants from multiplication are natural numbers.

Division

For the two rational numbers a and b, the division might or might not result in a natural number. E.g. 10/2=5 but 10/3=3.33.

### Associative Property

Natural numbers follow associative property for addition and multiplication. For three rational numbers, say, a, b and c, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c. Whereas, natural numbers do not follow associative property for multiplication and division.

For natural numbers a, b and c, addition is associative, i.e. a + (b + c) = (a + b) + c. For example, (15 +3) +1 = 19 = 15 + (3 + 1)

Multiplication

For natural numbers a, b and c, multiplication is associative, which means, a * (b * c) = (a * b) * c. Example: (3 * 1) * 15 = 45 = 3 * (1 * 15).

Subtraction

For three natural numbers a, b, and c, subtraction is not associative, meaning, a – (b – c) is not equal to (a –b) – c. For example: (2 – 15) – 1 = -14 but 2 – (15 – 1) = -12.

Division

For three natural numbers a, b, and c, division is not associative, i.e. a / (b / c) is not equal to (a / b ) / c. Example: 2 / (3 / 6) = 4 but (2 / 3) / 6 = 0.11

### Commutative Property

For any two given natural numbers a and b, addition and multiplication are commutative, i.e. a+b = b+a and a*b = b*a. However, division and subtraction are not commutative for the natural number (s), i.e. a-b is not equal to b-a and a/b is not similar to b/a.

### Distributive Property

For the given three natural numbers a, b and c, multiplication is distributive over addition and subtraction. This means that a * (b + c) = ab + ac and a * (b – c) = ab – ac.

1. What are the natural numbers? Explain the difference between natural numbers and whole numbers? State whether 0 is a natural number or not.

Natural Numbers are the non-negative, positive set of integers, starting from 1 and ending at infinity. These are countable numbers and are used for counting the physical objects.

N = {1, 2, 3, 4, 5, 6, 7... and so on till infinity}

Whole numbers are different from natural numbers because natural numbers lie between 1 and infinity, whereas, whole numbers lie between 0 and infinity. All-natural numbers are whole numbers, but whole numbers are not natural numbers.

Zero is not a natural number as a set of natural numbers only includes positive numbers starting from 1. Zero is included in the collection of whole numbers as they start from 0, 1, 2.... and so on.

2. State the properties of natural numbers.

Natural numbers follow the below-mentioned properties:

1. Closure Property – Natural numbers follow closure property for addition and multiplication but not for division and subtraction.

2. Commutative Property – All the natural numbers follow commutative property only for addition and subtraction.

3. Associative Property – The set of natural numbers is associative under addition and subtraction but not under multiplication and division.

4. Distributive Property – For given natural numbers, multiplication is distributive under addition and subtraction.

Addition and multiplication are the operations for which the set of natural numbers follow most of the properties. However, that is not true for division and subtraction.

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