Natural Numbers
Natural numbers are a part of the number system, including all the positive integers from 1 till infinity. Natural numbers are also called counting numbers because they do not include zero or negative numbers. They are a part of real numbers including just the positive integers, but not zero, fractions, decimals, and negative numbers.
Introduction to Natural Numbers
We see numbers everywhere around us, for counting objects, for representing or exchanging money, for measuring the temperature, telling the time, etc. These numbers that are used for counting objects are called “natural numbers”. For example, while counting objects, we say 5 cups, 6 books, 1 bottle, etc.
What are Natural Numbers?
Natural numbers refer to a set of all the whole numbers excluding 0. These numbers are significantly used in our day-to-day activities and speech.
Natural Numbers Definition
Natural numbers are the numbers that are used for counting and are a part of real numbers. The set of natural numbers include only the positive integers, i.e., 1, 2, 3, 4, 5, 6, ……….∞.
Examples of Natural Numbers
Natural numbers, also known as non-negative integers, include the positive integers (also known as non-negative integers) and a few examples include 23, 56, 78, 999, 100202, and so on.
Set of Natural Numbers
A set is a collection of elements (numbers in this context). The set of natural numbers in Mathematics is written as {1,2,3,...}. The set of natural numbers is denoted by the symbol, N. N = {1,2,3,4,5,...}
| Statement Form | N = Set of all numbers starting from 1. |
| Roaster Form | N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ………………………………} |
| Set Builder Form | N = {x : x is an integer starting from 1} |
Smallest Natural Number
The smallest natural number is 1. We know that the smallest element in N is 1 and that for every element in N, we can talk about the next element in terms of 1 and N (which is 1 more than that element). For example, two is one more than one, three is one more than two, and so on.
Natural Numbers from 1 to 100
The natural numbers from 1 to 100 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99 and 100.
Is 0 a Natural Number?
No, 0 is NOT a natural number because natural numbers are counting numbers. For counting any number of objects, we start counting from 1 and not from 0.
Odd Natural Numbers
The odd natural numbers are the numbers that are odd and belong to the set N. So the set of odd natural numbers is {1,3,5,7,...}.
Even Natural Numbers
The even natural numbers are the numbers that are even and belong to the set N. So the set of even natural numbers is {2,4,6,8,...}.
Natural Numbers and Whole Numbers
The set of whole numbers is the same as the set of natural numbers, except that it includes an additional number which is 0. The set of whole numbers in Mathematics is written as {0,1,2,3,...}. It is denoted by the letter, W. W = {0,1,2,3,4…}
From the above definitions, we can understand that every natural number is a whole number. Also, every whole number other than 0 is a natural number. We can say that the set of natural numbers is a subset of the set of whole numbers.
Difference Between Natural Numbers and Whole Numbers
Natural numbers are all positive numbers like 1, 2, 3, 4, and so on. They are the numbers you usually count and they continue till infinity. Whole numbers are all natural numbers including 0, for example, 0, 1, 2, 3, 4, and so on. Integers include all whole numbers and their negative counterpart. e.g, -4, -3, -2, -1, 0,1, 2, 3, 4 and so on. The following table shows the difference between a natural number and a whole number.
| Natural Number | Whole Number |
|---|---|
| The set of natural numbers is N= {1,2,3,...} | The set of whole numbers is W={0,1,2,3,...} |
| The smallest natural number is 1. | The smallest whole number is 0. |
| All natural numbers are whole numbers, but all whole numbers are not natural numbers. | Each whole number is a natural number, except zero. |
Natural Numbers on Number Line
The set of natural numbers and whole numbers can be shown on the number line as given below. All the positive integers or the integers on the right-hand side of 0, represent the natural numbers, whereas, all the positive integers along with zero, represent the whole numbers.
Properties of Natural Numbers
The four operations, addition, subtraction, multiplication, and division, on natural numbers, lead to four main properties of natural numbers as shown below:
- Closure Property
- Associative Property
- Commutative Property
- Distributive Property
1. Closure Property:
The sum and product of two natural numbers is always a natural number.
- Closure Property of Addition: a+b=c ⇒ 1+2=3, 7+8=15. This shows that the sum of natural numbers is always a natural number.
- Closure Property of Multiplication: a×b=c ⇒ 2×3=6, 7×8=56, etc. This shows that the product of natural numbers is always a natural number.
So, the set of natural numbers, N is closed under addition and multiplication but this is not the case in subtraction and division.
2. Associative Property:
The sum or product of any three natural numbers remains the same even if the grouping of numbers is changed.
- Associative Property of Addition: a+(b+c)=(a+b)+c ⇒ 2+(3+1)=2+4=6 and the same result is obtained in (2+3)+1=5+1=6.
- Associative Property of Multiplication: a×(b×c)=(a×b)×c ⇒ 2×(3×1)=2×3=6= and the same result is obtained in (a×b)×c=(2×3)×1=6×1=6.
So, the set of natural numbers, N is associative under addition and multiplication but this does not happen in the case of subtraction and division.
3. Commutative Property:
The sum or product of two natural numbers remains the same even after interchanging the order of the numbers. The commutative property of N states that: For all a,b∈N: a+b=b+a and a×b=b×a.
- Commutative Property of Addition: a+b=b+a ⇒ 8+9=17 and b+a=9+8=17.
- Commutative Property of Multiplication: a×b=b×a ⇒ 8×9=72 and 9×8=72.
So, the set of natural numbers, N is commutative under addition and multiplication but not in the case of subtraction and division.
Let us summarise these three properties of natural numbers in a table. So, the set of natural numbers, N is commutative under addition and multiplication.
| Operation | Closure Property | Associative Property | Commutative Property |
|---|---|---|---|
| Addition | yes | yes | yes |
| Subtraction | no | no | no |
| Multiplication | yes | yes | yes |
| Division | no | no | no |
4. Distributive Property:
- The distributive property of multiplication over addition is a×(b+c)=a×b+a×c
- The distributive property of multiplication over subtraction is a×(b−c)=a×b−a×c
To learn more about the properties of natural numbers, click here.
Important Points
- 0 is not a natural number, it is a whole number.
- Negative numbers, fractions, and decimals are neither natural numbers nor whole numbers.
- N is closed, associative, and commutative under both addition and multiplication (but not under subtraction and division).
FAQs on Natural Numbers
Is the Number 0 a Natural Number?
No, 0 is not a natural number. Natural numbers start from 1 and can be listed as 1, 2, 3, 4, 5, and so on.
What is an Example of a Natural Number?
Natural numbers can be listed as 1, 2, 3, 4, 5, and so on. So, one example can be 5.
Is 23 a Natural Number?
Yes, 23 is a natural number because it is a positive number that is used in counting.
Why are Natural Numbers Called Natural?
Natural numbers are called natural because they are used for counting naturally. The set of natural numbers is the most basic system of numbers because it is intuitive, or natural, and hence the name. We use natural numbers in our everyday life, in counting discrete objects, that is, objects which can be counted.
What are the First Five Natural Numbers?
Natural numbers are the numbers that are used for counting and are a part of real numbers. The first five natural numbers are 1, 2, 3, 4, and 5.
How to Find the Sum of n Natural Numbers?
To find the sum of 'n' natural numbers, we use the formula: Sum= n(n + 1)/2, where 'n' represents the number of terms. For example, if we want to find the sum of the first six natural numbers: 1, 2, 3, 4, 5, 6, we will replace 'n' with 6 (the total number of terms) and solve the formula. Sum= n(n + 1)/2. 6(6+1)/2 = 42/2 = 21. We get 21 as the answer.
Why are All Natural Numbers Integers?
An integer is a number from the set of negative and positive numbers, including zero, and the positive numbers come under the category of natural numbers. Thus, all natural numbers are integers.