# ganitsaarasangrah

ganitsaarasangrah: bhaarateeya ganitjnya mahaaveeraachaarya dvaara sanskrut bhaasha mein rachit ek ganit granth hai.

## sanrachana

2. parikrmavyavahaar: (Arithmetical operations)
3. kaalaasavarnavyavahaar: (Fractions)
4. prakeernakavyavahaar: (Miscellaneous problems)
5. trairaashikvyavahaar: (Rule of three)
6. mishrakavyavahaar: (Mixed problems)
7. kshetraganitvyavahaar: (Measurement of Areas)
8. khaatavyavahaar: (calculations regarding excavations)
9. chhaayaavyavahaar: (Calculations relating to shadows)

## ganitshaastraprashansa

ganitsaarasangrah: ke 'sanjnyaaadhikaar:' mein mangalaacharan ke pashchaat mahaaveeraachaarya ne bade hi maarmik dhang se ganit ki prashansha ki hai.

laukike vaidike vaapi tatha saamayikeऽpi y:.
vyaapaarastatr sarvatr sankhyaanamupayujyate.
kaamatantreऽrthashaastre ch gaandharve naatakeऽpi va.
kalaaguneshu sarveshu prastutan ganitan param.
triprashne chandravruttau ch sarvatraaङageekrutan hi tat.
dveepasaagarashailaanaan sankhyaavyaasaparikship:.
naarakaanaan ch sarveshaan shreneebandhendrakotkaraa:.
bahubhirpralaapai: kin trailokye sacharaachare.
yatkinyachidvastu tat sarvan ganiten bina n hi.
laukike vaidike vaapi tatha saamayikeऽpi y:.
vyaapaarastatr sarvatr sankhyaanamupayujyate.

arth: laukik, vaidik tatha saamayik mein jo vyaapaar hai vahaaain sarvatr sankhya ka hi upayog hota hai. kaamashaastr, arthashaastr, gandharvashaastr, gaayan, naatyashaastr, paakashaastr, aayurved, chhand, alankaar, kaavya, tark, vyaakaran aadi mein tatha kalaaon mein samast gunon mein ganit atyant upayogi hai. soorya aadi grahon ki gati gyaat karne mein, desh aur kaal ko gyaat karne mein sarvatr ganit angeekrut hai. dveepon, samoohon aur parvaton ki sankhya, vyaas aur paridhi, lok, antarlok, svarg aur narak ke nivaasi, sab shreneebaddh bhavanon, sabha evam mandiron ke nirmaan ganit ki sahaayata se hi jaane jaate hain. adhik kehne se kya prayojan? teenon lokon mein jo bhi vastueain hain unka astitv ganit ke bina naheen ho sakta.

## ganak ke gun

ganitsaarasangrah ke sanjnyaaadhikaar ke ant mein mahaaveeraachaarya ne ganakon (ganitjnyaon) ke 8 gun ginaae hain-

ath ganakagunaniroopanam

vyaktikraaङakavishishtair ganakoऽshtaabhir gunair jnyaey:.
(laghukaran, uh, apoh, anaalasya, grahan, dhaaran, upaaya, vyaktikraankavishisht - in aath gunon se ganak ko jaana jaata hai.)
“aA mathematician is to be known by eight qualities: conciseness, inference,
confutation, vigour in work and progress, comprehension, concentration of mind and
by the ability of finding solutions and uncovering quantities by investigation.”

## kala-savarn-vyavahaar ( Rules for decomposing fractions)

ganitsaarasangrah mein bhinnon ko ikaai bhinnon ke yog ke roop mein vyakt karne ki vidhiyaaain di huin hain.  ye vidhiyaaain vaidik kaal mein prayukt ikaai bhinnon tatha shulbasootr ka anusaran karateen hain jismein √a2 ka maan $1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}$ diya gaya hai.

In the Gaṇaita-sāara-saṅaagraha (GSS), the second section of the chapter on arithmetic is named kalā-savarṇaaa-vyavahāaara (lit. "the operation of the reduction of fractions"). In this, the bhāaagajāaati section (verses 55–98) gives rules for the following:

• To express 1 as the sum of n unit fractions (GSS kalāaasavarṇaaa 75, examples in 76):
rūaapāaaṃaaśaaakarāaaśaaīaanāaaṃ rūaapāaadyāaas triguṇaaitā harāaaḥ kramaśaaaḥ /

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].
$1={\frac {1}{1\cdot 2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n-2}}}+{\frac {1}{{\frac {2}{3}}\cdot 3^{n-1}}}$ • To express 1 as the sum of an odd number of unit fractions (GSS kalāaasavarṇaaa 77):
$1={\frac {1}{2\cdot 3\cdot 1/2}}+{\frac {1}{3\cdot 4\cdot 1/2}}+\dots +{\frac {1}{(2n-1)\cdot 2n\cdot 1/2}}+{\frac {1}{2n\cdot 1/2}}$ • To express a unit fraction $1/q$ as the sum of n other fractions with given numerators $a_{1},a_{2},\dots ,a_{n}$ (GSS kalāaasavarṇaaa 78, examples in 79):
${\frac {1}{q}}={\frac {a_{1}}{q(q+a_{1})}}+{\frac {a_{2}}{(q+a_{1})(q+a_{1}+a_{2})}}+\dots +{\frac {a_{n-1}}{q+a_{1}+\dots +a_{n-2})(q+a_{1}+\dots +a_{n-1})}}+{\frac {a_{n}}{a_{n}(q+a_{1}+\dots +a_{n-1})}}$ • To express any fraction $p/q$ as a sum of unit fractions (GSS kalāaasavarṇaaa 80, examples in 81):
Choose an integer i such that ${\tfrac {q+i}{p}}$ is an integer r, then write
${\frac {p}{q}}={\frac {1}{r}}+{\frac {i}{r\cdot q}}$ 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 kalāaasavarṇaaa 85, example in 86):
${\frac {1}{n}}={\frac {1}{p\cdot n}}+{\frac {1}{\frac {p\cdot n}{n-1}}}$ where $p$ is to be chosen such that ${\frac {p\cdot n}{n-1}}$ is an integer (for which $p$ must be a multiple of $n-1$ ).
${\frac {1}{a\cdot b}}={\frac {1}{a(a+b)}}+{\frac {1}{b(a+b)}}$ • To express a fraction $p/q$ as the sum of two other fractions with given numerators $a$ and $b$ (GSS kalāaasavarṇaaa 87, example in 88):
${\frac {p}{q}}={\frac {a}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}}}+{\frac {b}{{\frac {ai+b}{p}}\cdot {\frac {q}{i}}\cdot {i}}}$ where $i$ is to be chosen such that $p$ divides $ai+b$ kuchh aur niyam 14veen shataabdi mein Narayan pandit dvaara ganit kaumudi mein diye gaye hain. 

## sandarbh

1. Kusuba 2004, prushth 497–516