वास्तविक संख्याओं पर संक्रियाएँ - Revision history

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	== वास्तविक संख्याओं पर संक्रियाओं का नियम ==		== वास्तविक संख्याओं पर संक्रियाओं का नियम ==

	* एक परिमेय संख्या और अपरिमेय संख्या का योग या अंतर अपरिमेय होता है।		* एक [[परिमेय संख्याएँ\|परिमेय संख्या]] और अपरिमेय संख्या का योग या अंतर अपरिमेय होता है।
	* अपरिमेय संख्या के साथ एक गैर-शून्य परिमेय संख्या का गुणनफल या भागफल अपरिमेय संख्या होती है।		* [[अपरिमेय संख्याएँ\|अपरिमेय संख्या]] के साथ एक गैर-शून्य परिमेय संख्या का गुणनफल या भागफल अपरिमेय संख्या होती है।
	* जब दो अपरिमेय संख्याओं को जोड़ा, घटाया, गुणा या विभाजित किया जाता है, तो परिणाम एक परिमेय या अपरिमेय संख्या हो सकती है।		* जब दो अपरिमेय संख्याओं को जोड़ा, घटाया, गुणा या विभाजित किया जाता है, तो परिणाम एक परिमेय या अपरिमेय संख्या हो सकती है।

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Mani at 10:48, 2 November 2024

2024-11-02T10:48:56Z

← Older revision		Revision as of 16:18, 2 November 2024
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	Here we will be learning operations on [[Real Numbers]].		Here we will be learning operations on [[Real Numbers]].
			== वास्तविक संख्याओं पर संक्रियाओं का नियम ==

	~~== Operations on Real Numbers Rules ==~~		* एक परिमेय संख्या और अपरिमेय संख्या का योग या अंतर अपरिमेय होता है।
			* अपरिमेय संख्या के साथ एक गैर-शून्य परिमेय संख्या का गुणनफल या भागफल अपरिमेय संख्या होती है।
	* ~~The sum or difference of a rational number and an irrational number is irrational.~~		* जब दो अपरिमेय संख्याओं को जोड़ा, घटाया, गुणा या विभाजित किया जाता है, तो परिणाम एक परिमेय या अपरिमेय संख्या हो सकती है।

	* ~~The product or quotient of a non~~-~~zero rational number with an irrational number is irrational number.~~
	* ~~When two irrational numbers are added~~, ~~subtracted, multiplied or divided, the result may be a rational or an irrational number.~~

	~~If ''a'' and ''b'' are positive real numbers~~, ~~then we have~~,

			यदि <math>a </math> और <math>b</math> धनात्मक वास्तविक संख्याएँ हैं, तो हमारे पास है,
	* <math>\sqrt{ab}=\sqrt{a}\sqrt{b}</math>		* <math>\sqrt{ab}=\sqrt{a}\sqrt{b}</math>
	* <math>\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}</math>		* <math>\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}</math>
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	* <math>(\sqrt{a}+\sqrt{b})(\sqrt{c} +\sqrt{d})=\sqrt{ac}+\sqrt{ad}+\sqrt{bc}+\sqrt{bd}</math>		* <math>(\sqrt{a}+\sqrt{b})(\sqrt{c} +\sqrt{d})=\sqrt{ac}+\sqrt{ad}+\sqrt{bc}+\sqrt{bd}</math>
	* <math>(\sqrt{a} +\sqrt{b})^2=a+2\sqrt{ab}+b</math>		* <math>(\sqrt{a} +\sqrt{b})^2=a+2\sqrt{ab}+b</math>


	=== What are Mathematical Operations? ===		=== What are Mathematical Operations? ===

Mani at 10:47, 2 November 2024

2024-11-02T10:47:07Z

← Older revision		Revision as of 16:17, 2 November 2024
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			Here we will be learning operations on [[Real Numbers]].

			== Operations on Real Numbers Rules ==

			* The sum or difference of a rational number and an irrational number is irrational.

			* The product or quotient of a non-zero rational number with an irrational number is irrational number.
			* When two irrational numbers are added, subtracted, multiplied or divided, the result may be a rational or an irrational number.

			If ''a'' and ''b'' are positive real numbers, then we have,

			* <math>\sqrt{ab}=\sqrt{a}\sqrt{b}</math>
			* <math>\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}</math>
			* <math>(\sqrt{a} +\sqrt{b})(\sqrt{a} -\sqrt{b})=a-b</math>

			* <math>(a+\sqrt{b})(a -\sqrt{b})=a^2-b</math>
			* <math>(\sqrt{a}+\sqrt{b})(\sqrt{c} +\sqrt{d})=\sqrt{ac}+\sqrt{ad}+\sqrt{bc}+\sqrt{bd}</math>
			* <math>(\sqrt{a} +\sqrt{b})^2=a+2\sqrt{ab}+b</math>

			=== What are Mathematical Operations? ===
			The four basic Mathematical operations are addition (<math>+</math>), subtraction (<math>-</math>), multiplication (<math>\times</math>), and division (<math>/</math>).

			== Operations on Two Rational Numbers ==
			These are some of the operations:

			=== Addition of Two Rational Numbers ===
			When two rational numbers are added, the result is a rational number. For example, <math>0.24+0.68=0.92</math>. <math>0.92</math> can be written as <math>\frac{92}{100}</math>, which is a ratio or the <math>\frac{p}{q}</math> form.

			=== Subtraction of Two Rational Numbers ===
			When two rational numbers are subtracted, the result is a rational number. For example, <math>0.93-0.22=0.71</math> which can be written as <math>\frac{71}{100}</math>.

			=== Multiplication of Two Rational Numbers ===
			When two rational numbers are multiplied, the result is a rational number. For example, <math>0.5</math> multiplied by <math>185</math> is <math>92.5</math>, which can be written as <math>\frac{925}{10}</math>.

			=== Division of Two Rational Numbers ===
			When a rational number is divided by another rational number, the result is a rational number. For example, <math>0.352</math> divided by <math>0.6</math> is <math>0.58</math>, which can be written as <math>\frac{58}{100}</math>.

			== Operations on two Irrational Numbers ==

			=== Addition of Two Irrational Numbers ===
			When two irrational numbers are added, the result can be an irrational or a rational number. For example, <math>\sqrt{3}</math> added to <math>\sqrt{3}</math> is <math>2\sqrt{3}</math> which can which is a rational number. However, when <math>2\sqrt{5}</math> is added to <math>5\sqrt{3}</math>, we get a non-terminating and non-recurring decimal, an irrational number. It is written as <math>2\sqrt{5}+5\sqrt{3}</math>.

			=== Subtraction of Two Irrational Numbers ===
			Similarly, when two irrational numbers are subtracted, the result can be an irrational or a rational number. <math>\sqrt{2}</math> is subtracted from <math>\sqrt{2}</math>, the answer is <math>0</math>. When <math>4\sqrt{5}</math> is subtracted from <math>5\sqrt{3}</math>, we get <math>5\sqrt{3}-4\sqrt{5}</math>.

			=== Multiplication of Two Irrational Numbers ===
			The product of two irrational numbers can be an irrational number or a rational number. For example, when <math>\sqrt{2}</math> is multiplied by <math>\sqrt{2}</math>, we get <math>2</math> which is a rational number. However, when <math>\sqrt{2}</math> is multiplied by <math>\sqrt{3}</math>, we get <math>\sqrt{6}</math> which is an irrational number.

			=== Division of Two Irrational Numbers ===
			Similar to multiplication, we can get either an irrational number or a rational number as a result when an irrational number is divided by another. For example, when <math>\sqrt{2}</math> is divided by <math>\sqrt{2}</math>, we get <math>1</math> which is a rational number. But when <math>\sqrt{2}</math> is divided by <math>\sqrt{3}</math>, we get <math>\frac{\sqrt{2}}{\sqrt{3}}</math>, which is an irrational number.

			== Operations on a Rational and an Irrational Number ==

			=== Addition of an Irrational and a Rational Number ===
			The sum of a rational and an irrational number is always irrational. For example, when <math>2</math> is added to <math>5\sqrt{3}</math>, we get <math>2+5\sqrt{3}</math>, which is a rational number.

			=== Subtraction of an Irrational and a Rational Number ===
			The difference between a rational and an irrational number is always irrational. For example, when we subtract <math>5\sqrt{3}</math> from <math>2</math>, we get <math>2-5\sqrt{3}</math>, which is irrational.

			=== Multiplication of an Irrational and a Rational Number ===
			The product of a rational and an irrational number might be rational or irrational. For example, when <math>2</math> is multiplied by <math>\sqrt{2}</math>, we get <math>2\sqrt{2}</math> which is an irrational number, but when <math>\sqrt{12}</math> is multiplied by <math>\sqrt{3}</math>, we get <math>\sqrt{36}</math>, or <math>6</math>, which is a rational number.

			=== Division of an Irrational Number with a Rational Number ===
			When a rational number is divided by an irrational number or vice versa, the quotient is always an irrational number. For example, when <math>8</math> is divided by <math>\sqrt{2}</math>, we get <math>\frac{8}{\sqrt{2}}</math>, which is an irrational number. The answer can be further simplified to <math>4\sqrt{2}</math> which is also an irrational number.

			== Examples ==
			1.<math>(\sqrt{11} +\sqrt{7})(\sqrt{11} -\sqrt{b})</math>

			<math>11-7=4</math>



			2.<math>(\sqrt{3} +\sqrt{7})^2 </math>

			<math>(\sqrt{3})^2 +2(\sqrt{3})(\sqrt{7})+(\sqrt{7})^2 </math>

			<math>3 +2(\sqrt{21})+7 </math>

			<math>10 +2(\sqrt{21}) </math>



			3. <math>(5+\sqrt{7})(2 +\sqrt{5})</math>

			<math>10 + 5\sqrt{5}+2\sqrt{7}+\sqrt{35}</math>



			4.<math>(\sqrt{7} +\sqrt{5})(\sqrt{7} -\sqrt{5})</math>

			<math>(\sqrt{7})^2 - (\sqrt{5})^2</math>

			<math>7-5=2</math>





	यहां हम वास्तविक संख्याओं पर संक्रियाओं की विधि को सीखेंगे।		यहां हम वास्तविक संख्याओं पर संक्रियाओं की विधि को सीखेंगे।

Ramamurthy: /* उदाहरण */

2024-09-26T15:13:28Z

उदाहरण

← Older revision		Revision as of 20:43, 26 September 2024
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	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
			[[Category:गणित]]
	[[Category:कक्षा-9]]		[[Category:कक्षा-9]]
	~~[[Category:गणित]]~~

Mani: added content

2024-04-29T03:13:34Z

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← Older revision		Revision as of 08:43, 29 April 2024
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	यहां हम वास्तविक संख्याओं पर ~~संक्रियाएँ~~ सीखेंगे।		यहां हम वास्तविक संख्याओं पर संक्रियाओं की विधि को सीखेंगे।

	== वास्तविक संख्याओं पर संक्रियाओं का नियम ==		== वास्तविक संख्याओं पर संक्रियाओं का नियम ==

Mani: added content

2024-04-29T03:12:31Z

added content

← Older revision		Revision as of 08:42, 29 April 2024
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	== वास्तविक संख्याओं पर संक्रियाओं का नियम ==		== वास्तविक संख्याओं पर संक्रियाओं का नियम ==

	* ~~The sum or difference of a rational number and an irrational number is irrational.~~		* एक परिमेय संख्या और अपरिमेय संख्या का योग या अंतर अपरिमेय होता है।
			* अपरिमेय संख्या के साथ एक गैर-शून्य परिमेय संख्या का गुणनफल या भागफल अपरिमेय संख्या होती है।
	* ~~The product or quotient of a non~~-~~zero rational number with an irrational number is irrational number.~~		* जब दो अपरिमेय संख्याओं को जोड़ा, घटाया, गुणा या विभाजित किया जाता है, तो परिणाम एक परिमेय या अपरिमेय संख्या हो सकती है।
	* ~~When two irrational numbers are added~~, ~~subtracted, multiplied or divided, the result may be a rational or an irrational number.~~

	~~If ''a'' and ''b'' are positive real numbers~~, ~~then we have~~,

			यदि <math>a </math> और <math>b</math> धनात्मक वास्तविक संख्याएँ हैं, तो हमारे पास है,
	* <math>\sqrt{ab}=\sqrt{a}\sqrt{b}</math>		* <math>\sqrt{ab}=\sqrt{a}\sqrt{b}</math>
	* <math>\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}</math>		* <math>\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}</math>
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	<math>7-5=2</math>		<math>7-5=2</math>

	~~वास्तविक संख्याओं पर संक्रियाएँ~~
	[[Category:संख्या पद्धति]]		[[Category:संख्या पद्धति]]
	[[Category:कक्षा-9]]		[[Category:कक्षा-9]]
	[[Category:गणित]]		[[Category:गणित]]

Mani: New Mathematics Class 9 Hindi Page Created

2024-04-29T03:06:31Z

New Mathematics Class 9 Hindi Page Created

← Older revision		Revision as of 08:36, 29 April 2024
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	~~Here we will be learning operations on [[Real Numbers]].~~		यहां हम वास्तविक संख्याओं पर संक्रियाएँ सीखेंगे।

	== ~~Operations on Real Numbers Rules~~ ==		== वास्तविक संख्याओं पर संक्रियाओं का नियम ==

	* The sum or difference of a rational number and an irrational number is irrational.		* The sum or difference of a rational number and an irrational number is irrational.
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	* <math>(\sqrt{a} +\sqrt{b})^2=a+2\sqrt{ab}+b</math>		* <math>(\sqrt{a} +\sqrt{b})^2=a+2\sqrt{ab}+b</math>

	== ~~Examples~~ ==		== उदाहरण ==
	1.<math>(\sqrt{11} +\sqrt{7})(\sqrt{11} -\sqrt{b})</math>		1.<math>(\sqrt{11} +\sqrt{7})(\sqrt{11} -\sqrt{b})</math>

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	<math>7-5=2</math>		<math>7-5=2</math>

	[[Category:संख्या पद्धति]][[Category:कक्षा-9]][[Category:गणित]]		वास्तविक संख्याओं पर संक्रियाएँ
	~~Operations on Real Numbers~~		[[Category:संख्या पद्धति]]
			[[Category:कक्षा-9]]
			[[Category:गणित]]

Mani at 03:03, 29 April 2024

2024-04-29T03:03:08Z

← Older revision		Revision as of 08:33, 29 April 2024
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			Here we will be learning operations on [[Real Numbers]].

			== Operations on Real Numbers Rules ==

			* The sum or difference of a rational number and an irrational number is irrational.

			* The product or quotient of a non-zero rational number with an irrational number is irrational number.
			* When two irrational numbers are added, subtracted, multiplied or divided, the result may be a rational or an irrational number.

			If ''a'' and ''b'' are positive real numbers, then we have,

			* <math>\sqrt{ab}=\sqrt{a}\sqrt{b}</math>
			* <math>\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}</math>
			* <math>(\sqrt{a} +\sqrt{b})(\sqrt{a} -\sqrt{b})=a-b</math>

			* <math>(a+\sqrt{b})(a -\sqrt{b})=a^2-b</math>
			* <math>(\sqrt{a}+\sqrt{b})(\sqrt{c} +\sqrt{d})=\sqrt{ac}+\sqrt{ad}+\sqrt{bc}+\sqrt{bd}</math>
			* <math>(\sqrt{a} +\sqrt{b})^2=a+2\sqrt{ab}+b</math>

			== Examples ==
			1.<math>(\sqrt{11} +\sqrt{7})(\sqrt{11} -\sqrt{b})</math>

			<math>11-7=4</math>


			2.<math>(\sqrt{3} +\sqrt{7})^2 </math>

			<math>(\sqrt{3})^2 +2(\sqrt{3})(\sqrt{7})+(\sqrt{7})^2 </math>

			<math>3 +2(\sqrt{21})+7 </math>

			<math>10 +2(\sqrt{21}) </math>


			3. <math>(5+\sqrt{7})(2 +\sqrt{5})</math>

			<math>10 + 5\sqrt{5}+2\sqrt{7}+\sqrt{35}</math>


			4.<math>(\sqrt{7} +\sqrt{5})(\sqrt{7} -\sqrt{5})</math>

			<math>(\sqrt{7})^2 - (\sqrt{5})^2</math>

			<math>7-5=2</math>

	[[Category:संख्या पद्धति]][[Category:कक्षा-9]][[Category:गणित]]		[[Category:संख्या पद्धति]][[Category:कक्षा-9]][[Category:गणित]]
	Operations on Real Numbers		Operations on Real Numbers