"गणितसारसंग्रह": अवतरणों में अंतर
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पंक्ति 53:
:“A mathematician is to be known by eight qualities: conciseness, inference, <br /> confutation, vigour in work and progress, comprehension, concentration of mind and <br /> by the ability of finding solutions and uncovering quantities by investigation.”
==कला-सवर्न-व्यवहार ( Rules for decomposing fractions) ==
गणितसारसंग्रह में [[भिन्न|भिन्नों]] को इकाई भिन्न|इकाई भिन्नों]] के योग के रूप में व्यक्त करने की विधियाँ दी हुईं हैं। <ref name=k497>{{Harvnb|Kusuba|2004|pp=497–516}}</ref> ये विधियाँ वैदिक काल में प्रयुक्त इकाई भिन्नों तथा [[शूल्ब सूत्र]] का अनुसरण करतीं हैं जिसमें √2 का मान <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math> दिया गया है।<ref name=k497/>
In the ''Gaṇita-sāra-saṅgraha'' (GSS), the second section of the chapter on arithmetic is named ''kalā-savarṇa-vyavahāra'' (lit. "the operation of the reduction of fractions"). In this, the ''bhāgajāti'' section (verses 55–98) gives rules for the following:<ref name=k497/>
* To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):<ref name=k497/>
{{quote|rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /<br/>
dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //}}
{{quote|When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].}}
:: <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>
* To express 1 as the sum of an odd number of unit fractions (GSS ''kalāsavarṇa'' 77):<ref name=k497/>
:: <math>1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2} </math>
* To express a unit fraction <math>1/q</math> as the sum of ''n'' other fractions with given numerators <math>a_1, a_2, \dots, a_n</math> (GSS ''kalāsavarṇa'' 78, examples in 79):
:: <math>\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}</math>
* To express any fraction <math>p/q</math> as a sum of unit fractions (GSS ''kalāsavarṇa'' 80, examples in 81):<ref name=k497/>
: Choose an integer ''i'' such that <math>\tfrac{q+i}{p}</math> is an integer ''r'', then write
:: <math> \frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q} </math>
: and repeat the process for the second term, recursively. (Note that if ''i'' is always chosen to be the ''smallest'' such integer, this is identical to the [[greedy algorithm for Egyptian fractions]].)
* To express a unit fraction as the sum of two other unit fractions (GSS ''kalāsavarṇa'' 85, example in 86):<ref name=k497/>
:: <math>\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}</math> where <math>p</math> is to be chosen such that <math>\frac{p\cdot n}{n-1}</math> is an integer (for which <math>p</math> must be a multiple of <math>n-1</math>).
:: <math>\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}</math>
* To express a fraction <math>p/q</math> as the sum of two other fractions with given numerators <math>a</math> and <math>b</math> (GSS ''kalāsavarṇa'' 87, example in 88):<ref name=k497/>
:: <math>\frac{p}{q} = \frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}} + \frac{b}{\frac{ai+b}{p} \cdot \frac{q}{i} \cdot{i}}</math> where <math>i</math> is to be chosen such that <math>p</math> divides <math>ai + b</math>
कुछ और नियम १४वीं शताब्दी में [[नारायण पण्डित]] द्वारा [[गणित कौमुदी]] में दिये गये हैं। <ref name=k497/>
==सन्दर्भ==
{{टिप्पणीसूची}}
== बाहरी कड़ियाँ ==
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