Contents
Natural numbers
Introduction
Natural numbers are fundamental concept in mathematics, representing positive integers that are used for counting and ordering. They are an essential component of many mathematical concepts and operations, making them crucial topic for understanding mathematics. This article will provide an overview of natural numbers, including their definition, history, set notation, types, comparison with whole numbers, properties, operations, applications, and frequently asked questions.
A. Overview of Natural Numbers
Natural numbers are positive integers that are used for counting and ordering. They are denoted by symbol “N,” which stands for set of all natural numbers. Natural numbers have rich history that dates back to ancient civilizations, where they were used for basic arithmetic and counting. Today, natural numbers are fundamental concept in mathematics, used in wide range of mathematical operations and applications.
B. Importance and Relevance of Natural Numbers in Mathematics
Natural numbers are crucial in mathematics because they form foundation for many mathematical concepts and operations. For example, they are used in basic arithmetic, algebra, calculus, number theory, and geometry. Natural numbers also play vital role in real-world applications, such as science, engineering, and economics. Understanding properties and rules of natural numbers is essential for solving mathematical problems and making informed decisions.
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What are Natural Numbers?
A. Natural Numbers Definition:
Natural numbers are set of integers which are positive that start from 1 and continue indefinitely without any upper bound. In mathematical notation, natural number represented as
![\[N{\rm{ }} = {\rm{ }}\left\{ {1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5,{\rm{ }}...} \right\}.\]](https://quicklatex.com/cache3/9c/ql_0c992fd76f1fded636803c516f353f9c_l3.png "Rendered by QuickLaTeX.com")
This definition is widely accepted and used in mathematics, and it forms basis for many other mathematical concepts and systems. Natural numbers are used for counting objects and to describe quantities in many real-world applications, and their properties can be seen in depth in number theory and other areas of mathematics.
B. History of Natural Numbers:
The concept of natural numbers dates back to ancient civilizations, where they were used for basic arithmetic and counting. Babylonians, Egyptians, and Greeks all used natural numbers in their mathematical systems. However, concept of zero, which is essential for modern mathematics, was not introduced until much later.
In medieval period, natural numbers were developed further, with introduction of Hindu-Arabic numerals and development of algebra. Negative numbers concept was also introduced, which expanded range of numbers that could be used in mathematical operations.
Today, natural numbers are fundamental concept in mathematics, forming basis for many mathematical operations and concepts.
Examples of Numbers that are Natural
A. Examples of Numbers that are Natural in Everyday Life
Numbers that are natural can be seen in many everyday situations, such as counting money, telling time, and measuring distances. For example, if you have three apples, you are using natural number 3. If you need to be at meeting at 2:30 pm, you are using natural numbers 2 and 30. If you need to travel 10 miles to get to work, you are using natural number 10.
B. Natural Numbers in Mathematical Operations and Equations
Natural numbers are used in many mathematical operations and equations, such as subtraction, addition, division, multiplication, For example, if you add natural numbers 3 and 5, you get natural number 8. If you subtract natural number 2 from natural number 7, you get natural number 5. If you multiply natural numbers 4 and 6, you get natural number 24. If you divide natural number 10 by natural number 2, you get natural
Set of Numbers that are Natural
Set of numbers that are natural are is known as numbers used for counting, which is fundamental concept in mathematics. It is defined as set of positive integers that are used for counting and ordering. Set of numbers that are natural is infinite set, as there is no largest natural number. In this section, we will discuss different forms of representing set of natural numbers, including statement form, roaster form, set builder form, smallest natural number, and set of numbers that are natural is from 1 to 100.
Statement Form:
The statement form of set of numbers that are natural is “set of integers that are positive.” This form defines set numbers that are natural in terms of its properties.
Roaster Form:
The roaster form of natural numbers set is
![\[\left\{ {1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}...} \right\}\]](https://quicklatex.com/cache3/bd/ql_9518a49f9787ea4e279c33b77d92f2bd_l3.png "Rendered by QuickLaTeX.com")
Set Builder Form:
The set builder form of numbers that are natural is
![\[\left\{ {x|x{\rm{ }}is{\rm{ }}a{\rm{ }}positive{\rm{ }}integer} \right\}\]](https://quicklatex.com/cache3/d9/ql_ba9da5aeaecb91e24ca86153011132d9_l3.png "Rendered by QuickLaTeX.com")
Smallest Natural Number:
Smallest natural number is 1. This is because set of natural numbers begins with 1 and has no negative numbers.
Natural Numbers from 1 to 100:
Set of natural numbers from 1 to 100 is
![\[\left\{ {1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}...,{\rm{ }}100} \right\}\;\]](https://quicklatex.com/cache3/67/ql_7249407c23af03b3585f398e1c6ce867_l3.png "Rendered by QuickLaTeX.com")
In summary, numbers that are natural is of an infinite set of positive integers that is used for counting and ordering. It can be represented in different forms, including statement form, roster form, and set builder form. Smallest natural number is 1, and set of natural numbers from 1 to 100 is
Is 0 Natural Number?
The question of whether 0 is natural number is somewhat controversial. Traditionally, natural numbers are defined as positive integers used for counting and ordering, which would exclude 0. However, in some contexts, 0 is included as natural number.
In modern mathematics, term “whole numbers” is often used to refer to set of non-negative integers, including 0. This is broader definition than traditional definition of natural numbers, which excludes 0. Therefore, whether or not 0 is considered natural number may depend on context in which it is being used.
Types of Natural Numbers
A. Odd Numbers that are Natural: Integers which is not divisible by 2
Odd natural numbers are integers which is not divisible by 2. Examples of odd numbers that are natural includes 1,3,5,7,and 9 and so on. Odd natural numbers can be expressed in form
![\[2n{\rm{ }} + {\rm{ }}1\]](https://quicklatex.com/cache3/99/ql_8f455753520ebd522917e05f77026e99_l3.png "Rendered by QuickLaTeX.com")
B. Even Natural Numbers: Integers which are divisible by 2
Even numbers that are natural integers that is divisible by 2. Examples of even numbers that are natural include 2, 4, 6, 8, and 10 and so on. Even natural numbers can be expressed in form
![\[2n\]](https://quicklatex.com/cache3/4d/ql_9b21599b9bd1ba767fc111563ea15f4d_l3.png "Rendered by QuickLaTeX.com")
Natural Numbers and Whole Numbers
A. Definition of Whole Numbers
Whole numbers are set of numbers that has all natural numbers and zero. Whole numbers are also used for counting and ordering, but they include 0, which is not natural number.
B. Difference between Numbers That Are Natural and Numbers That Are Whole
The main difference found in numbers that are natural and numbers that are whole is that numbers that are whole include 0, while numbers that are natural does not. Both sets of numbers are used for counting and ordering, but inclusion of 0 in whole numbers expands their range of applications beyond those of natural numbers.
Every Natural Number is Whole Number. True or False?
True. Every natural number is also whole number, since whole numbers include all numbers that are natural and 0
Numbers that are Natural on Number Line
A. Representation of Numbers that are Natural on Number Line
Numbers that are natural can be shown on number line by placing number 1 at beginning of line, and then continuing to right in increments of 1 for each successive natural number. Number line can be used to visualize ordering and distance between natural numbers.
B. Comparison of Natural Numbers on Number Line
Natural numbers can be compared on number line by their position relative to each other. Numbers to right are greater than numbers to left, and distance between numbers represents difference between them
.
Properties of Natural Numbers
A. Closure Property: Addition and multiplication of natural numbers gives natural number
The closure property of natural numbers states that result of adding or multiplying any two numbers that are natural is always a natural number. Example 3 + 2 = 5, and
![\[2{\rm{ }} \times {\rm{ }}3{\rm{ }} = {\rm{ }}6\]](https://quicklatex.com/cache3/1f/ql_5818d54399e04bbddafa86ca3bae531f_l3.png "Rendered by QuickLaTeX.com")
B. Associative Property: order of operations does not change result
The associative property of natural numbers states that way in which numbers are grouped when adding or multiplying does not affect result. Example
![\[\left( {2 + 3} \right) + 4 = 2 + \left( {3 + 4} \right)\]](https://quicklatex.com/cache3/da/ql_87fe99ea01ca6cd3b4ceb64fdf2fc9da_l3.png "Rendered by QuickLaTeX.com")
![\[\left( {2 \times 3} \right) \times 4 = 2 \times \left( {3 \times 4} \right)\]](https://quicklatex.com/cache3/c2/ql_8d44fd2f208f4fcded40b59a031cfcc2_l3.png "Rendered by QuickLaTeX.com")
C. Commutative Property: commutative property is fundamental property that applies to certain operations, such as addition and multiplication, and states that order in which terms or factors are combined does not affect result.
For the Multiplication and Addition of numbers that are natural, commutative property can be stated as follows:
Commutative addition: For any two numbers that are natural a and b,
![\[a + b = b + a\]](https://quicklatex.com/cache3/ff/ql_936eb2d446950af236438dc91bd083ff_l3.png "Rendered by QuickLaTeX.com")
Commutative multiplication: For any two numbers that are natural a and b,
![\[a \times b{\rm{ }} = {\rm{ }}b \times a\]](https://quicklatex.com/cache3/04/ql_d0c32ef0baee9b1cd77d29eb67fa7d04_l3.png "Rendered by QuickLaTeX.com")
For example, commutative property of addition tells us that
![\[2 + 3 = 3 + 2\]](https://quicklatex.com/cache3/b1/ql_19abe9515d9e3da7be38ac82fbdc92b1_l3.png "Rendered by QuickLaTeX.com")
D. Distributive Property: This property states that multiplication distributes over addition. If a, b, and c are numbers that are natural, then
![\[a{\rm{ }} \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)\]](https://quicklatex.com/cache3/64/ql_2317c687ec149ce5cc8c447fbbfea564_l3.png "Rendered by QuickLaTeX.com")
E. Inverse element: Every natural number has unique additive inverse, which is negative number. If additive inverse of 5 is -5, and additive inverse of 10 is -10.
F. Principle of well-ordering: This principle states that every non-empty set of numbers that are natural contains least element. For example, set
![\[\left\{ {1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4} \right\}\]](https://quicklatex.com/cache3/ad/ql_444faba5e0ca8a7ddfe177278fe0baad_l3.png "Rendered by QuickLaTeX.com")
Operations with Natural Numbers
Basic mathematical operations:
Basic mathematical operations with natural numbers include addition, subtraction, multiplication, and division. Addition and multiplication are commutative and associative, meaning that order of numbers being added or multiplied does not change result, and order of operations does not change result.
Advanced mathematical operations:
Advanced mathematical operations with natural numbers include exponents, factorials, and prime factorization. Exponents represent repeated multiplication of number by itself, while factorials represent product of all positive integers up to given number. Prime factorization is process of breaking down a number into its prime factors, which are factors are prime numbers.
Applications of Natural Numbers
Natural numbers have wide range of real world applications in fields such as science, economics, engineering and computer science. For example, natural numbers are used in physics to represent quantities such as distance, time, and mass, and in chemistry to represent number of atoms or molecules in sample.
In economics, natural numbers are used to represent quantities such as prices, quantities of goods, and number of people employed in company. In computer science, natural numbers are used to represent data sizes, memory addresses, and other important concepts.
Conclusion
Natural numbers are fundamental concept in mathematics, and they form basis for many other mathematical ideas and systems. They are defined as positive integers, starting from 1 and continuing infinitely. Numbers that are natural are closed under addition, subtraction, and multiplication, and have several important properties, such as associative, commutative, and distributive laws.
The study of numbers that are natural and their properties is called number theory, and it has many practical applications, such as in cryptography, computer science, and physics. Study of prime numbers, for example, it is central topic in number theory and has important applications in cryptography.
One interesting aspect of natural numbers is their infinite nature. They have no upper bound, meaning that there is always larger natural number that can be found. This has important implications for mathematical analysis and proof, as it allows for creation of infinite series and limits.
Another important concept related to natural numbers is notion of divisibility. Two natural numbers are said to be divisible if one divides other without leaving remainder. Divisibility has many applications, such as in identification of prime numbers and simplification of fractions.
The study of natural numbers also includes concepts such as factors, multiples, and divisors, which are all related to idea of divisibility. These concepts are important in fields such as number theory, algebra, and geometry.
Finally, natural numbers are also used to describe and solve many real-world problems, such as counting objects or measuring time. They are fundamental part of our everyday lives, and their study has important practical applications in fields such as finance, economics, and engineering.
In conclusion, study of natural numbers is fascinating and important area of mathematics that has wide-ranging applications in many fields. Their infinite nature and various properties make them rich area of study, and their practical applications make them an essential tool for solving real-world problems From simple counting of objects to complex calculations used in science and technology, natural numbers are an essential tool for understanding and describing world around us.
Solved examples:
1. What is sum of first 50 natural numbers?
Ans: sum of first n natural numbers is given by formula:
![\[\frac{{n(n + 1)}}{2}\]](https://quicklatex.com/cache3/9d/ql_a07276758b0b3fde198cf27685998e9d_l3.png "Rendered by QuickLaTeX.com")
Substituting n=50, we get:
![\[\frac{{50(50 + 1)}}{2} = 25 \times 51 = 1275\]](https://quicklatex.com/cache3/7d/ql_075ca2c7db3c9db790096847d4bc2a7d_l3.png "Rendered by QuickLaTeX.com")
Therefore, sum of first 50 natural numbers is 1275.
2. What is product of first 10 odd natural numbers?
Ans: first 10 odd natural numbers are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
Their product is given by multiplying all numbers together:
![\[1 \times 3 \times 5 \times 7 \times 9 \times 11 \times 13 \times 15 \times 17 \times 19{\rm{ }} = {\rm{ }}654,729,075\]](https://quicklatex.com/cache3/55/ql_28231611924c16171cee12660e3efe55_l3.png "Rendered by QuickLaTeX.com")
Therefore, product of first 10 odd natural numbers is 654,729,075.
3. What is difference between sum of first 20 even natural numbers and sum of first 20 odd natural numbers?
Ans: sum of first n even natural numbers is given by formula:
![\[n(n + 1)\]](https://quicklatex.com/cache3/df/ql_73d69261a8fa5f1ec18e7ec85e6c37df_l3.png "Rendered by QuickLaTeX.com")
The sum of first n odd natural numbers is also given by formula:
![\[{n^2}\]](https://quicklatex.com/cache3/fc/ql_141e8033ba105a496f1b122f1e4bbafc_l3.png "Rendered by QuickLaTeX.com")
Substituting n=20, we get:
Sum of first 20 even numbers =
![\[20(20 + 1) = 420\]](https://quicklatex.com/cache3/b9/ql_a29cc8deb8374d31c0cde5b76cc140b9_l3.png "Rendered by QuickLaTeX.com")
Sum of first 20 odd numbers =
![\[{20^2} = 400\]](https://quicklatex.com/cache3/07/ql_4d305f100852eccddee06a7f5d5fab07_l3.png "Rendered by QuickLaTeX.com")
The difference between two sums is:
![\[420 - 400 = 20\]](https://quicklatex.com/cache3/b9/ql_67351effb2e29641353284ad2a20cfb9_l3.png "Rendered by QuickLaTeX.com")
Therefore, difference between sum of first 20 even natural numbers and sum of first 20 odd natural numbers is 20.
4. What is value of 2 to power of 10?
Ans: 2 to power of 10 can be calculated using formula:
![\[{2^n}\]](https://quicklatex.com/cache3/f0/ql_870c9dccc650bc7f8e7665bd11097cf0_l3.png "Rendered by QuickLaTeX.com")
Substituting n=10, we get:
![\[{2^{10}} = 1024\]](https://quicklatex.com/cache3/43/ql_c52bfadddb03fb5efdd02b01eae8aa43_l3.png "Rendered by QuickLaTeX.com")
Therefore, value of 2 to power of 10 is 1024.
5. What is factorial of 7?
Ans: factorial of positive integer n is defined as product of all positive integers from 1 to n.
Therefore, factorial of 7 is:
![\[7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1{\rm{ }} = {\rm{ }}5040\]](https://quicklatex.com/cache3/67/ql_6c1cd09f78acc30a07e1e3fa37df2067_l3.png "Rendered by QuickLaTeX.com")
Therefore, factorial of 7 is 5040.
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FAQs
1. Is zero a natural number?
Ans: No, zero is not a natural number. Natural numbers are defined as positive integers used for counting and ordering.
2. Can natural numbers be negative?
Ans: No, natural numbers are defined as positive integers only. Negative integers are not considered natural numbers.
3. What is the difference between natural numbers and whole numbers?
Ans: A set of whole numbers includes zero, whereas a set of natural numbers does not. Therefore, natural numbers are a subset of whole numbers.
4. What is the well-ordering principle of natural numbers?
Ans: The well-ordering principle states that every non-empty set of natural numbers contains the least element. This principle is used to prove many mathematical theorems.
5. What are some real-world applications of natural numbers?
Ans: Natural numbers are used in many real-world applications such as counting items, measuring time, and calculating distances. They are also used in fields such as science, engineering, and economics.
Reference:
Rosen, K. H. (2012). Discrete Mathematics and Its Applications (7th ed.). McGraw-Hill
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