पूर्ण संख्याएँ - Revision history

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2024-11-05T02:50:06Z

added internal links

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	== पूर्ण संख्याएँ क्या हैं? ==		== पूर्ण संख्याएँ क्या हैं? ==
	शून्य (<math>0</math>) सहित प्राकृतिक संख्याओं को '''पूर्ण संख्याएँ''' कहा जाता है। हमे ज्ञात है कि प्राकृतिक संख्याएँ <math>1,2,3,4</math> इत्यादि से प्रारंभ होने वाली गिनती संख्याओं के एक समूह को संदर्भित करती हैं। सरल शब्दों में, पूर्ण संख्याएँ भिन्नों, दशमलवों या यहाँ तक कि ऋणात्मक पूर्णांकों के बिना संख्याओं का एक समूह होती हैं। यह धनात्मक पूर्णांकों और शून्य का एक संग्रह है। या हम कह सकते हैं कि पूर्ण संख्याएँ गैर-ऋणात्मक पूर्णांकों का एक समूह हैं। प्राकृतिक और पूर्ण संख्याओं के बीच प्राथमिक अंतर पूर्ण संख्या समुच्चय में शून्य की उपस्थिति है।		शून्य (<math>0</math>) सहित प्राकृतिक संख्याओं को '''पूर्ण संख्याएँ''' कहा जाता है। हमे ज्ञात है कि [[प्राकृतिक संख्याएँ]] <math>1,2,3,4</math> इत्यादि से प्रारंभ होने वाली गिनती संख्याओं के एक समूह को संदर्भित करती हैं। सरल शब्दों में, पूर्ण संख्याएँ भिन्नों, दशमलवों या यहाँ तक कि ऋणात्मक पूर्णांकों के बिना संख्याओं का एक समूह होती हैं। यह धनात्मक पूर्णांकों और शून्य का एक संग्रह है। या हम कह सकते हैं कि पूर्ण संख्याएँ गैर-ऋणात्मक पूर्णांकों का एक समूह हैं। प्राकृतिक और पूर्ण संख्याओं के बीच प्राथमिक अंतर पूर्ण संख्या समुच्चय में शून्य की उपस्थिति है।

	== परिभाषा ==		== परिभाषा ==
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	* शून्य सहित सभी धनात्मक पूर्णांक पूर्ण संख्याएँ होती हैं।		* शून्य सहित सभी धनात्मक पूर्णांक पूर्ण संख्याएँ होती हैं।
	* सभी पूर्ण संख्याएँ वास्तविक संख्याएँ होती हैं।		* सभी पूर्ण संख्याएँ वास्तविक संख्याएँ होती हैं।
	* प्रत्येक पूर्ण संख्या एक परिमेय संख्या होती है।		* प्रत्येक पूर्ण संख्या एक [[परिमेय संख्याएँ\|परिमेय]] संख्या होती है।

	=== लघुतम पूर्ण संख्या ===		=== लघुतम पूर्ण संख्या ===

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	<math>W = \{0,1,2,3,4,....\}</math>		<math>W = \{0,1,2,3,4,....\}</math>

	~~Here are some facts about whole numbers~~:		पूर्ण संख्याओं के बारे में कुछ तथ्य इस प्रकार हैं:

	* ~~All natural numbers are whole numbers.~~		* सभी प्राकृतिक संख्याएँ पूर्ण संख्याएँ होती हैं।
	* ~~All counting numbers are whole numbers.~~		* सभी गिनती वाली संख्याएँ पूर्ण संख्याएँ होती हैं।
	* ~~All positive integers including zero are whole numbers.~~		* शून्य सहित सभी धनात्मक पूर्णांक पूर्ण संख्याएँ होती हैं।
	* ~~All whole numbers are real numbers.~~		* सभी पूर्ण संख्याएँ वास्तविक संख्याएँ होती हैं।
	* ~~Every whole number is a rational number.~~		* प्रत्येक पूर्ण संख्या एक परिमेय संख्या होती है।

	=== लघुतम पूर्ण संख्या ===		=== लघुतम पूर्ण संख्या ===

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Mani at 17:01, 9 October 2024

2024-10-09T17:01:23Z

← Older revision		Revision as of 22:31, 9 October 2024
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			== What are Whole Numbers? ==
			Natural numbers along with zero (<math>0</math>) are referred to as '''whole numbers'''. We know that natural numbers refer to a set of counting numbers starting from <math>1,2,3,4</math> and so on. In simple words, whole numbers are a set of numbers without fractions, decimals, or even negative integers. It is a collection of positive integers and zero. Or we can say that whole numbers are the set of non-negative integers. The primary difference between natural and whole numbers is the presence of zero in the whole numbers set.

			== Definition ==
			Whole Numbers are those counting positive numbers that start from <math>0</math> and go on till infinity. For example, <math>0,1,2,3,4,....</math>, and so on.

			=== Set of Whole Numbers ===
			The set of whole numbers includes the set of natural numbers along with the number <math>0</math>. The set of whole numbers in mathematics is given as <math>\{0,1,2,3,4,....\}</math> which is denoted by the symbol '''W.'''

			<math>W = \{0,1,2,3,4,....\}</math>

			Here are some facts about whole numbers:

			* All natural numbers are whole numbers.
			* All counting numbers are whole numbers.
			* All positive integers including zero are whole numbers.
			* All whole numbers are real numbers.
			* Every whole number is a rational number.

			=== Smallest Whole Number ===
			<math>0</math> is the smallest whole number because whole numbers start from <math>0</math> (from the definition of whole numbers). Zero is a number that lies between the positive and negative numbers on a number line. Although zero carries no value, it is used as a placeholder. So, zero is neither a positive number nor a negative number.

			=== List of Whole Numbers ===
			The list of whole numbers starts from <math>0,1,2,3,4</math> and goes on till infinity.

			== Natural Numbers and Whole Numbers ==
			From the above definitions, we can understand that every whole number other than <math>0</math> is a natural number. Also, every natural number is a whole number. So, the set of natural numbers is a part of the set of whole numbers or a subset of whole numbers.

			=== Difference Between Whole numbers and Natural Numbers ===

			{\| class="wikitable"
			!Whole Number
			!Natural Number
			\|-
			\|The set of whole numbers is, <math>W = \{0,1,2,3,4,....\}</math>.
			\|The set of natural numbers is, <math>N = \{1,2,3,4,....\}</math>.
			\|-
			\|The smallest whole number is <math>0</math>.
			\|The smallest natural number is <math>1</math>
			\|-
			\|Every natural number is a whole number.
			\|Every whole number is a natural number, except <math>0</math>.
			\|}

			== Whole Numbers on Number Line ==
			The set of natural numbers and the set of 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 <math>0</math>represent the natural numbers, whereas all the positive integers along with zero, altogether represent the whole numbers. Both sets of numbers can be represented on the number line as follows:
			[[File:Number Line.jpg\|alt=Fig.1 Number Line\|none\|thumb\|Fig.1 Number Line]]

			== Properties of Whole Numbers ==
			The basic operations on whole numbers: addition, subtraction, multiplication, and division, lead to four main properties of whole numbers that are listed below:

			* Closure Property
			* Associative Property
			* Commutative Property
			* Distributive Property

			'''Closure Property'''

			The sum and product of two whole numbers is always a whole number. For example, <math>7+3=10</math> (whole number), <math>7 \times 2 =14</math> (whole number).

			'''Associative Property'''

			The sum or product of any three whole numbers remains the same even if the grouping of numbers is changed. For example, when we add the following numbers we get the same sum: <math>10+(7+12)=(10+7)+12=29</math>. Similarly, when we multiply the following numbers we get the same product no matter how the numbers are grouped: <math>3 \times (2 \times 4)=(3 \times 2) \times 4=24</math>.

			'''Commutative Property'''

			The sum and the product of two whole numbers remain the same even after interchanging the order of the numbers. This property states that a change in the order of addition does not change the value of the sum. Let '<math>a</math>' and '<math>b</math>' be two whole numbers. According to the commutative property <math>a+b=b+a</math>. For example, <math>a=10</math> and <math>b=19</math> ⇒ <math>10+19=19+10=29</math>. This property also holds true for multiplication, but not for subtraction and division. For example: <math>7 \times 9 = 9 \times 7 =63</math>.

			'''Additive Identity'''

			When a whole number is added to <math>0</math>, its value remains unchanged, i.e., if <math>a</math> is a whole number then <math>a+0=0+a=a</math>. For example, <math>3+0=0+3=3</math>.

			'''Multiplicative Identity'''

			When a whole number is multiplied by <math>1</math>, its value remains unchanged, i.e., if <math>a</math> is a whole number then <math>a \times 1 = 1 \times a = a</math>. For example. <math>4 \times 1 = 1 \times 4 = 4</math>.

			'''Distributive Property'''

			This property states that the multiplication of a whole number is distributed over the sum or difference of the whole numbers. It means that when two numbers, for example, <math>a</math> and <math>b</math> are multiplied with the same number <math>c</math>, and are then added, then the sum of <math>a</math> and <math>b</math> can be multiplied by <math>c</math> to get the same answer. This property can be represented as: <math>a \times (b+c)= (a \times b) + (a \times c)</math>. Let <math>a=10</math>, <math>b=20</math> and <math>c=7</math> ⇒ <math>10 \times (20+7)= (10 \times 20) + (10 \times 7)=270</math>. The same property is true for subtraction as well. For example, we have <math>a \times (b-c)= (a \times b) - (a \times c)</math>. Let <math>a=10</math>, <math>b=20</math> and <math>c=7</math> ⇒ <math>10 \times (20-7)= (10 \times 20) - (10 \times 7)=130</math>.

			'''Multiplication by Zero'''

			When a whole number is multiplied by <math>0</math>, the result is always <math>0</math>, i.e., <math>a \times 0 = 0 \times a = 0</math>. For example, <math>4 \times 0 = 0 \times 4 = 0</math>

			'''Division by Zero'''

			The division of a whole number by <math>0</math> is not defined, i.e., if <math>a</math> is a whole number then <math>\frac{a}{0}</math> is not defined.

			'''Important Points related to Whole numbers'''

			* <math>0</math> is a whole number and not a natural number.
			* The first five whole numbers are <math>0,1,2,3,4</math>
			* The smallest whole number is <math>0</math>.
			* Whole numbers do not include negative numbers, fractions, and decimals.


	पूर्ण संख्याएँ भिन्न रहित संख्याएँ हैं और यह धनात्मक पूर्णांकों और शून्य का संग्रह है। पूर्ण संख्याओं के समुच्चय को W से दर्शाया जाता है। W= {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , 9 , 10 ................}		पूर्ण संख्याएँ भिन्न रहित संख्याएँ हैं और यह धनात्मक पूर्णांकों और शून्य का संग्रह है। पूर्ण संख्याओं के समुच्चय को W से दर्शाया जाता है। W= {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , 9 , 10 ................}
	{\| class="wikitable"		{\| class="wikitable"

Ramamurthy at 15:04, 26 September 2024

2024-09-26T15:04:48Z

← Older revision		Revision as of 20:34, 26 September 2024
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	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
	[[Category:~~कक्षा-9~~]][[Category:~~गणित~~]]		[[Category:गणित]]
			[[Category:कक्षा-9]]

Mani: New Mathematics Class 9 Hindi Page Created

2024-05-04T13:53:12Z

New Mathematics Class 9 Hindi Page Created

← Older revision		Revision as of 19:23, 4 May 2024
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	Whole Numbers are the numbers without fractions and it is a collection of positive integers and zero. The set of whole numbers is denoted by W. W= {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , 9 , 10 ................}		पूर्ण संख्याएँ भिन्न रहित संख्याएँ हैं और यह धनात्मक पूर्णांकों और शून्य का संग्रह है। पूर्ण संख्याओं के समुच्चय को W से दर्शाया जाता है। W= {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , 9 , 10 ................}
	{\| class="wikitable"		{\| class="wikitable"
	\|+~~Difference between Whole Numbers and Natural Numbers~~		\|+पूर्ण संख्याओं और प्राकृतिक संख्याओं के बीच अंतर
	!~~Whole Number~~		!पूर्ण संख्याएँ
	!~~Natural Number~~		!प्राकृतिक संख्याएँ
	\|-		\|-
	\|~~The set of whole numbers is~~ W={0,1,2,3.....}		\|पूर्ण संख्याओं का समुच्चय W={0,1,2,3......} है
	\|~~The set of Natural numbers is~~ N ={1,2,3.....}		\|प्राकृतिक संख्याओं का समुच्चय N ={1,2,3......} है
	\|-		\|-
	\|~~The smallest whole number is~~ 0.		\|सबसे छोटी पूर्ण संख्या 0 होती है.
	\|~~The smallest natural number is~~ 1		\|सबसे छोटी प्राकृतिक संख्या 1 है
	\|-		\|-
	\|~~Every [[Natural Numbers\|Natural number]] is a whole number.~~		\|प्रत्येक प्राकृत संख्या एक पूर्ण संख्या होती है।
	\|~~Every Whole number is a natural number except~~ 0.		\|0 को छोड़कर प्रत्येक पूर्ण संख्या एक प्राकृतिक संख्या होती है।
	\|}		\|}

	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
	[[Category:कक्षा-9]][[Category:गणित]]		[[Category:कक्षा-9]][[Category:गणित]]

Mani at 13:49, 4 May 2024

2024-05-04T13:49:07Z

← Older revision		Revision as of 19:19, 4 May 2024
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			Whole Numbers are the numbers without fractions and it is a collection of positive integers and zero. The set of whole numbers is denoted by W. W= {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , 9 , 10 ................}
			{\| class="wikitable"
			\|+Difference between Whole Numbers and Natural Numbers
			!Whole Number
			!Natural Number
			\|-
			\|The set of whole numbers is W={0,1,2,3.....}
			\|The set of Natural numbers is N ={1,2,3.....}
			\|-
			\|The smallest whole number is 0.
			\|The smallest natural number is 1
			\|-
			\|Every [[Natural Numbers\|Natural number]] is a whole number.
			\|Every Whole number is a natural number except 0.
			\|}

	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
	[[Category:कक्षा-9]][[Category:गणित]]		[[Category:कक्षा-9]][[Category:गणित]]
	~~Whole Numbers~~

Mani at 06:12, 16 August 2023

2023-08-16T06:12:52Z

← Older revision		Revision as of 11:42, 16 August 2023
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	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
	[[Category:कक्षा-9]][[Category:गणित]]		[[Category:कक्षा-9]][[Category:गणित]]
			Whole Numbers

Ramamurthy: Category updated

2023-08-04T12:53:52Z

Category updated

← Older revision		Revision as of 18:23, 4 August 2023
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	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
	~~[[Category:अंकगणित]]~~		[[Category:कक्षा-9]][[Category:गणित]]
	~~[[Category:वास्तविक संख्याएँ]]~~[[Category:कक्षा-9]][[Category:गणित]]