Introduction
Let us first start with the meaning 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.
Numbers can be found everywhere around us, used for counting objects, representing or transferring money, calculating temperature, telling time, and so on. "Natural Numbers" refer to the Numbers used to count objects. When counting objects, we might say 5 glasses, 6 books, 1 bottle, and so on.
The number system includes all positive integers from 1 to infinity, which is known as Natural Numbers. Natural Numbers are sometimes known as counting numbers because they do not include zero or negative numbers. They are only positive integers, not zeros, fractions, decimals, or negative Numbers, and they are part of the real Number system.
Natural Numbers
A set of all whole numbers except 0 is referred to as Natural Numbers. These figures play a significant role in our day-to-day activities and communication.
Natural Numbers are those that can be counted and are a portion of real Numbers. The set of Natural Numbers contains only positive integers such as 1, 2, 3, 4, 5, 6, and so on.
Natural Numbers refer to non-negative integers (all positive integers). Examples can be 39, 696, 63, 05110, and so on.
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
The term "Set" refers to a group of items (Numbers in this context). In mathematics, the Set of Natural Numbers is written as 1,2,3,... The Set of Natural Numbers is symbolised by the symbol N. N = 1,2,3,4,5 and so on. In mathematics, the Set of Natural Numbers is written as 1,2,3,...
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:
Closure Property
Commutative Property
Associative Property
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.
Addition
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. \[\frac{10}{2} =5\] but \[\frac{10}{3} = 3.33.\].
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.
Addition
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. \[\frac{a}{(b/c)}\] is not equal to \[\frac{(a/b)}{c}\] . Example: \[\frac{2}{(3/6)} = 4\] but \[\frac{(2/3)}{6} = 0.11\]
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 \[\frac{a}{b}\] is not similar to \[\frac{b}{a}\].
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.
Smallest Natural Number
1 is the Smallest Natural Number. We know that the Smallest element in N is 1 and that for each element in N, we may talk about the next element in terms of 1 and N. (which is 1 more than that element). 2 is one greater than one, 3 is one greater than two, and so on.
FAQs on Natural Numbers
1. What is the distinction between Natural and Whole numbers?
Natural numbers, such as 1, 2, 3, 4, and so on, are all positive numbers. They're the numbers you're used to counting, and they never seem to stop. The whole numbers, on the other hand, are all natural numbers, including zero, and include numbers like 0, 1, 2, 3, 4, and so on. Integers are all whole numbers and their negative equivalents. Examples are -4, -3, -2, -1, 0,1, 2, 3, 4, and so on. The distinction between a whole number and a natural number is-
Natural Numbers-
N=1,2,3,... is the set of natural numbers.
1 is the smallest natural number.
Whole Numbers-
W=0,1,2,3,... is the set of whole numbers.
0 is the smallest whole number.
2. What are the properties of Natural Numbers?
The four operations on natural numbers, addition, subtraction, multiplication, and division, result in four main features of natural numbers, which are listed below:
Closure Property
Associative Property
Commutative Property
Distributive Property
1. Closure Property:
A natural number will always be the sum and product of two natural numbers only.
Addition Closure Property: a+b=c 1+2=3, 7+8=15. As this shows, the sum of natural numbers is always a natural number.
Multiplication Closure Property: ab=c 23=6, 788=56, etc. As it can be seen, the product of two natural numbers is always a natural number.
In addition and multiplication, the set of natural numbers, N, is closed, but this is not the case in subtraction and division.
2. Associative Property:
Even if the order of the numbers is modified, the sum or product of any three natural numbers remains the same.
Associative Property of Addition: a+(b+c)=(a+b)+c ⇒ 2+(3+1)=2+4=6 and the result is obtained in (2+3)+1=5+1=6.
Multiplication Associative Property: a(bc)=(ab)c 2(31)=23=6=, and the same result is achieved in (ab)c=(23)1=61=6.
So, when it comes to addition and multiplication, the set of natural numbers N is associative, but not when it comes to subtraction and division.
3. Commutative Property:
Even if the sequence of the numbers is changed, the sum or product of two natural numbers remains the same. The commutative property of N asserts that a+b=b+a and ab=ba for all a, bN.
Commutative properties of addition: a+b=b+a 8+9=17 and b+a=9+8=17
Multiplication Commutative Property: ab=ba 89=72 and 98=72.
The set N of natural numbers is commutative in addition and multiplication, but not in subtraction and division.
4. Distributive Property:
Multiplication over addition has the distributive property a(b+c)=ab+ac.
Multiplication over subtraction has the distributive property a(bc)=abac.
3. What is a generalisation on Natural numbers?
The two uses of counting and ordering result in two major generalisations of natural numbers: cardinal numbers and ordinal numbers.
A natural number is not just a number, they can also be used to represent the size of a finite set. Natural numbers are cardinal numbers, which is a measure of set size that can be used to infinite sets. This concept of "size" is based on mappings between sets, with two sets having the same size if there is a bijection between them. The set of natural numbers, as well as any bijective image of it, is said to be countably infinite and have aleph-null (0) cardinality.
Natural numbers are also used as ordinal numbers in linguistics: "first," "second," "third," and so on. They can be assigned to the elements of a completely ordered finite set as well as any well-ordered countably infinite set in this fashion. The ordinal numbers can be obtained by generalising this assignment to extending well-orderings with a cardinality beyond countability. In a sense that differs from cardinality, an ordinal number can be used to define the notion of "size" for a well-ordered set: if two well-ordered sets have the same ordinal number (rather than a bijection! ), they have the same ordinal number. The first ordinal number that isn't a natural number is; it's also the first ordinal number in the set of natural numbers.
4. What are different types of numbers?
The types are-
1. Natural Numbers (Numbers)
The numbers 1, 2, 3, 4, and so on are natural numbers, often known as counting numbers. They're the positive numbers we employ to keep track of our possessions. The number zero is not regarded as a "natural number."
2. Whole Numbers
The numbers 0, 1, 2, 3, 4, and so on are whole numbers (the natural numbers and zero). "Whole numbers" do not include negative numbers. All-natural numbers are integers, but not all integers are natural numbers because zero is both an integer and a natural number.
3. Integers
All the whole numbers and their opposites are represented by the integers..., -4, -3, -2, -1, 0, 1, 2, 3, 4,... (the positive whole numbers, the negative whole numbers, and zero). Integers are integers, but fractions and decimals are not. All integers -5, for instance, is an integer but not a whole or natural number are integers (and all-natural numbers are integers), but not all integers or natural numbers are integers.
4. Rational Numbers
All integers, as well as all fractions, terminating decimals, and repeating decimals, are included in the rational numbers. Every rational number can be expressed as a fraction a/b, with a and b being integers. Although all rational numbers are natural numbers, whole numbers, and integers, not all rational numbers are natural numbers, whole numbers, or integers.
5. What are number lines and how do they present natural numbers on them?
A number line is a representation of a straight line that has been graduated. The integers are represented on the number line as marked points that are uniformly spaced. The line contains all real numbers in both directions, as well as numbers between the integers that are not marked. It is used to assist in the teaching of basic addition and subtraction with negative integers. A horizontal number line is the most common representation. Positive numbers are always to the right of zero, negative numbers are always to the left of zero, and the arrows at both ends of the line indicate that, by convention, the line extends infinitely in the positive and negative directions.
On the number line, show the set of natural numbers and whole numbers as shown below. Natural numbers are represented by all positive integers or integers to the right of 0, whereas integers are represented by all positive integers plus zero.