CONTASS3333S

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____By_cross_–_multiplication,__ ___________________________________n_(_ax_+_b)_=_m_(cx_+_d)_ ___________________________________nax_+_nb_=_mcx_+_md_ ___________________________________nax_-_mcx_=_md_–_nb__ ___________________________________x(_na_–_mc_)_=_md_–_nb_ _ ____________________________________________md_-_nb_ _______________________________x___=_____________ ____________________________________________na_-_mc._

____Now_look_at_the_problem_once_a ain__

_____________________________________ax_+_b_________m_ ______________________________________________=_____ _____________________________________cx+_d__________n_

para!artya_ i!es_________md_-_nb,_na_-_mc____and_ _______________________________________________________________________ _______________________________________________________________x___=___ _______________________________________________________________________

"xample_#$ ____________%x_+_#__________#%_ _______________________________________=_______ _______________________________&x_+_%___________#'_

_____________________md_-_nb___________#%_(%)_-_#'(#)___________%'_-_#'__

___________x__=______________=___________________=______________=______ _____________________na-_mc___________#'_(%)_-_#%(&)____________* _-_* _ _______________________________________________________________________

"xample_ $ ___________&x_+_*______________ _ _______________________________________=_______ _____________________________%x_+_#% __________ _ _______________________________________________________________________

____________________________( )_(#% )_-( )(*)_

_______________x___=____________________ ______________________________( )(&)_-_( )(%)_

__________________________('# )_-_& ____________('#_-_ ) ___________# ___________________=_________________=_________________=_____________=_ _____________________________% _–_ #_______________% _–_ #___________

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_

/ype_(i!)_$ 0onsider_the_problems_o1_the_type_____m_____________n_____ _______________________________________________________________________ _______________________________________________________________________

____________/ake_2.0.3_and_proceed.__

__________________________m(x+b)_+_n_(x+a)__ _____________________________________________=____ _ __________________________(x_+_a)_(x_+b)_

_________________________mx_+_mb_+_nx_+_na__ ____________________________________________=___ _ ____________________________x_+_a)(x_+_b)_

___________________(m_+_n)x_+_mb_+_na____=___ _____(m_+_n)x_=_-_mb_-_na__

_________________________________-mb_-_na_ ___________________x__=_______________ __________________________________(m_+_n)_

____/hus_the_problem________m______________n___ ___________________________________________+___________=___ ,____by_para ___________________________________x_+_a___________x_+_b_

____ i!es_directly_ ______________________________________-mb_-_na__ ___________________________x___=______________ _______________________________________(m_+_n)_

_

"xample_#_$ ________%____________&__ __________________________________+__________=___ _ ___________________________x_+_&_______x_–_4_

____ i!es____________-mb_-_na_ ___________________x__=________________Note_that_m_=_%,_n_=_&,_a_=_&,_b ______________________________(m_+_n)_ _ ___________________________-(%)(-4)_–_(&)_(&)__________# _-_#4________ _ _____________________=_____________________=____________=_____ __________________________________(_%_+_&)___________________ ________

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"xample_ _$ _ ________________________________*______________4__ _____________________________________+___________=___ _ _____________________________x_+_#_________x_–_ #_ _ ________ i!es_____________-(*)_(- #)_-_(4)_(#)__________# *_-_4_________''__ __________________x___=______________________=______________=_______=__ _____________________________________*_+_4________________________##___

_ _5_._6ol!e_the_1ollowin _problems_usin _the_sutra_7ara!artya_–_yo8ayet._ _ ____#)_%x_+_*_=_*x_–_%_____________4)_(x_+_#)_(_x_+_ )_=_(_x_–_%)_(x_–_ _ ____ )_( x %)_+_#=x_-_#________ )_(x_–_ )_(x_–_')=_(x_–_%)_(x_–_ )_ __________ ____%)_ x_+_ __________*_______________ )_(x_+_ )_(x_+_')=_(x_+_%_)_(x_+ ________________=______ _________%x-_*___________ _ _ ____&)_x_+_#_ _%_ __________________=__#_ ___________%x_-_#__ _ ____*)_____*_______________ __

________________+____________=___ _ _________x_+_%__________x_–_& __ _

_

_

55)__ ________#.6how_that_1or_the_type_o1_e9uations__ _

__________________m______________n________________p___ _______________________+____________+_________=___ ,_____the_solution_is_ _______________x_+_a__________x_+_b_________x_+_c_

______________________________-_mbc_–_nca_–_pab__ _______________x___=________________________________,_i1_m_+_n_+_p_= ._ ___________________________m(b_+_c)_+_n(c+a)_+_p(a_+_b)_

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__ ._:pply_the_abo!e_1ormula_to_set_the_solution_1or_the_problem__ _ _______________7roblem_________%______________ ________________*__ ___________________________________________+_________-____________=___ ___________________________________x_+_&_________x_+_4________x_+_* _

_

some_more_simple_solutions_$ _ _ _______________m_____________n____________m_+_n_ ____________________+___________=__________ ______________x_+_a_______x_+_b__________x_+_c__

Now_this_can_be_written_as,__ _ ________________m_____________n_____________m_______________n_ ____________________+___________=____________+_________ _____________x_+_a________x_+_b__________x_+_c________x_+_c__

____________m____________m_____________n________________n_ ____________________-__________=_____________-_________ _____________x_+_a________x_+_c_________x_+_c_________x_+_b__

________m(x_+c)_–_m(x_+_a)__________n(x_+_b)_–_n(x_+_c)_ ________________________________=_____________________

______________(x_+_a)_(x_+_c)________________(x_+_c)_(x_+_b)_ _

_

________mx_+_mc_–_mx_–_ma__________nx_+_nb_–_nx_–_nc_ ________________________________=____________________ ________________(x_+_a)_(x_+_c)_____________(x_+c_)_(x_+_b)_

________________m_(c_–_a)________________________n_(b_–c)_

_________________________________=_______________ ___________________x_+a____________________________x_+_b_

_______m_(c_-_a).x_+_m_(c_-_a).b____=____n_(b_-_c)._x_+_n(b_-_c).a_ ____________x_;_m(c_-_a)-_n(b_-_c)_<_____=____na(b_-_c)_–_mb_(c_-_a)__ ________or_x_;_m(c_-_a)_+_n(c_-_b)_<____=____na(b_-_c)_+_mb_(a_-_c)__

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_____

/hus____________________________mb(a_-_c)_+_na_(b_-_c)_ _______________________x___=_________________________ ________________________________________m(c-a)_+_n(c-b)._

____By_para!artya_rule_we_can_easily_remember_the_1ormula.__

"xample_#_$ _____sol!e__%_

_

*._6unyam_6amya_6amuccaye _

/he_6utra_ 6unyam_6amyasamuccaye _says_the_ 6amuccaya_is_the_same,_that_6amuccaya_is_>ero. _i.e.,_it_should_be_e9uated_to_?ero._/he_term_ 6amuccaya _has_se!eral_meanin s_under_di11erent_contexts.__

i)_@e_interpret,_ 6amuccaya _as_a_term_which_occurs_as_a_common_1actor_in_allthe_terms_concerned_and_proceed_as_1ollows._

"xample_#$ _/he_e9uation_ x_+_%x_=_&x_+_*x_has_the_same_1actor_A_x_A_in_aits_terms._ ence_by_the_sutra_it_is_?ero,i.e.,_x_=_ ._

Ctherwise_we_ha!e_to_work_like_this$__ _ ___________________ x_+_%x_=_&x_+_*x_ ________________________# x_=_'x_ _________________# x_–_'x_=_ _ ___________________________x_=_ _

____/his_is_applicable_not_only_1or_AxD_but_also_any_such_unknown_9uantity_as_1ollows._

"xample_ $______*(x+#)_=_%(x+#)_

No_need_to_proceed_in_the_usual_procedure_like__ _ _______________________*x_+__*_=_%x_+_%_ _______________________*x_–_%x_=__%_–_*_ ___________________ x_=_- _____or_____x_=_- _E_ _=_-#_

____6imply_think_o1_the_contextual_meanin _o1_ 6amuccaya __

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__

___Now_6amuccaya_is_____(_x_+_#)_ ___________________________________x_+_#_=_ _____ i!es_____x_=_-#__

ii)_Now_we_interpret_ 6amuccaya _as_product_o1_independent_terms_in_expressions_like_(x+a)_(x+b)_

"xample_%$ _____(_x_+_%_)_(_x_+_&)_=_(_x_–_ )_(_x_–_4_)_ _ _______________ ere_6amuccaya_is_____%_x_&_=_# _=_- _x_-4_ _______________6ince_it_is_same_,___we_deri!e_x_=_ _

___/his_example,_we_ha!e_already_dealt_in_type_(_ii_)_o1_7ara!artya_in_sol!in _simple_e9uations._

iii)_@e_interpret_ _6amuccaya_ as_the_sum_o1_the_denominators_o1_two_1ractionsha!in _the_same_numerical_numerator._

0onsider_the_example. _ _ ______________________________#____________#_ __________________________________+_________=___ _ ___________________________%x- ________ x-#_

____________1or_this_we_proceed_by_takin 2.0.3._

_ ___________________( x-#)+(%x– )_ __________________________________=___ _ ___________________(%x– )( x–#)_ _ _________________________*x–%_ ________________________________=___ _ ___________________(%x– )( x–#)_ _ _______________________*x_–_%_=_ ________*x_=_%_ _

_______________________________%_ _______________________x__=____ _______________________________*_

____5nstead_o1_this,_we_can_directly_put_the_6amuccaya_i.e.,_sum_o1_the_denominators__ _______________i.e.,_%x_–_ _+_ x_-_#_=_*x_-_%_=_ _

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_______________ i!in _*x_=_%_________x_=_%_ _*_

____5t_is_true_and_applicable_1or_all_problems_o1_the_type_ _ ___________________m___________m_ ______________________+__________=___ _ ________________ax+b_______cx+d_

____6amuccaya_is_ax+b+cx+d_and_solution_is_(_m_F_ _)_

_ _ _______________________-_(_b_+_d_)_ ________________x_=_____________ __________________________(_a_+_c_)_

____iii)_@e_now_interpret_ 6amuccaya _as_combination_or_total._ _ _________51_the_sum_o1_the_numerators_and_the_sum_o1_the_denominators_be_thesame,_then_that_sum_=_ ._ _ ____0onsider_examples_o1_type __ _ _______________________ax+_b_________ax_+_c_ ______________________________=__________ _______________________ax+_c_________ax_+_b_

________5n_this_case,__(ax+b)_(ax+b)_=(ax+c)_(ax+c)_ _______________________ax _+_ abx_+_b _=_ax _+_ acx_+c _ _________________________________ abx_–_ acx_=_c _–_b __ _____________________________x_(_ ab_–_ ac_)_=_c _–_b _ _ _______________________c–b ________(c+b)(c-b)______-(c+b)_ _____________x_=____________=_____________=_________ ______________________ a(b-c)________ a(b-c)____________ a_

____________:s_per_6amuccaya_(ax+b)_+_(ax+c)_=_ _

____________________________________________________ ax+b+c_=_ __ ________________________________________________________ ax_=_-b-c__ _ ____________________________________-(c+b)_ _____________________________x__=_________ ________________________________________ a____________ ence_the_statement

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_

"xample_&$ _ _____________________________%x+_&________%x_+_*_ ____________________________________=__________ _____________________________%x+_*________%x_+_&_

________6ince_N# _+_N _=_%x_+_&_+_%x_+_*_=_4x_+_'_,_ _____________:nd__G# _+_G _=_%x_+_&_+_%x_+_*_=_4x_+_'_ _____________@e_ha!eN# _+_N _=_G# _+_G _=_4x_+_'_ ___________________ ence_1rom_6unya_6amuccaya_we_ et_4x_+_'_=_ _

_____________________4x_=_-'_ _ _

________________________________-'________-%_ _________________________x_=_______=______ ________________________________4__________ _

"xample_*$ _ ________________________*x_+ _______*x_+_# _ _______________________________=__________ ________________________*x+# _______*x_+_ _

_______ ence_N# _+_N _=_*x_+_ _+_*x_+_# _=_# x_+_#'_

____________:nd____G# _+_G _=_*x_+_# _+_*x_+_ _=_# x_+_#'_ _____________________N# _+_N _=_G# _+_G _ i!es_# x_+_#'_=_ _ ___________________________________# x_=_-#'_ _ ________________________________________-#'_ _________________________________x__=_______ __________________________________________# _

0onsider_the_examples_o1_the_type,_where_N# _+_N _=_H_(G# _+_G _),_where_H_is_a_numerical_constant,_then_also_by_remo!in _the_numerical_constant_H,_we_can_proceed_as_abo!e._

"xample_4$ _ _ ________________________ x_+_%_______x__+_#_ _______________________________=__________ _______________________&x_+_*________ x_+_%_

________ ere_N# _+_N _=_ x_+_%_+_x_+_#_=_%x_+_&_

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___________________G# _+_G _=_&x_+_*_+_ x_+_%_=_4x_+_ _ ____________________________________________________________=_ _(_%x_+

________Iemo!in _the_numerical_1actor_ ,_we_ et_%x_+_&_on_both_sides._ _ ____________%x_+_&_=_ ____%x_=_-&_____x_=_-_&_ _%._

!)_ 6amuccaya _with_the_same_meanin _as_abo!e,_i.e.,_case_(i!),_we_sol!e_the_problems_leadin _to_9uadratic_e9uations._5n_this_context,_we_take_the_problems_as_1ollowsJ_

_______51_N# _+_N _=_G# _+_G _and_also_the_di11erences_ _______________N# _K_G# _=_N _K_G _then_both_the_thin s_are_e9uated_to_?ero,_the_solution_ i!es_the_two_!alues_1or_x._

"xample_ $ _ _ _____________________%x_+_ _____ x_+_*_ ____________________________=________ _____________________ x_+_*_____%x_+_ _

____5n_the_con!entional_text_book_method,_we_work_as_1ollows_$_ _ _______________________%x_+_ ____________ x_+_*_ __________________________________=___________ _______________________ x_+_*____________%x_+_ _

_ ___________________(_%x_+_ _)_(_%x_+_ _)_=_(_ x_+_*_)_(_ x_+_*_)_ ___________________________'x _+_# x_+_&_=_&x _+_ x_+_ *_ ____________________'x _+_# x_+_&_-_&x _-_ x_–_ *_=_ _ ___________________________________________*x _–_ x_–_ #_=_ _ ___________________________________*x _–_#*x_+_ x_–_ #_=_ _ _____________________________*x_(_x_–_%_)_+_ _(_x_–_%_)_=_ _ _____________________________________(x_–_%_)_(_*x_+_ _)_=_ _ ____________________________________x_–_%_=_ _or_*x_+_ _=_ _ _________________________________________x_=_%_or_-_ _ _*_

____Now_A6amuccayaD_sutra_comes_to_help_us_in_a_beauti1ul_way_as_1ollows_$_ _ ___________________Cbser!e_N# _+_N _=_%x_+_ _+_ x_+_*_=_*x_+_ _ _______________________________G# _+_G _=_ x_+_*_+_%x_+_ _=_*x_+_ _ _ ________Lurther__N# _K_G# _=_(_%x_+_ _)_–_(_ x_+_*_)_=_x_–_%_ ___________________N _K_G _=_(_ x_+_*)_–_(_%x_+_ _)_=_-_x_+_%_=_-_(_x_–_%_)__ _

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________ ence__*x_+_ _=_ _,_x_–_%_=_ _ _____________________*x_=_- _,_x_=_%_ ___________________________i.e.,_x_=_- _ _*_,_x_=_%_ _ ________Note_that_all_these_can_be_easily_calculated_by_mere_obser!ation.__

"xample_ $ __ _ ___________________________%x_+_&_______*x_+_4_ ___________________________________=________ ___________________________4x_+_ _______ x_+_%_ _ ________Cbser!e_that__ ___________________N# _+_N _=_%x_+_&_+_*x_+_4_=_ x_+_# _ ______________andG# _+_G _=_4x_+_ _+_ x_+_%_=_ x_+_# _ _ ________Lurther_____N# _KG# _=_(%x_+_&)_–_(4x_+_ )__ __________________________________=__%x_+_&_–_4x_–_ __ __________________________________=_-%x_–_%__=_-%_(_x_+_#_)_ _____________________N _K_G __=_(*x_+_4)_–_( x_+_%)_=_%x_+_%_=_%(_x_+_#)__

____By_A6unyam_6amuccayeD_we_ha!e_ _ ___________________ x_+_# _=_ ____________%(_x_+_#_)_=_ _ __________________________ x_=_-# ______________x_+_#_=_ _ ____________________________x_=_-_# _ _ _____________x_=_-#_

______________________________=_-_*_ _&_!i)A6amuccayaD_with_the_same_sense_but_with_a_di11erent_context_and_application_._ _ ___ _ ___"xample_'$ _ _ ________________________#__________#____________#___________#_ ___________________________+_________=________+_______ _____________________x_-_&______x_–_4______x_-_ ______x_-_ _

________Msually_we_proceed_as_1ollows._ _ ________________x–4+x-&________________x– +x- __ ____________________________=________________ _______________(x–&)_(x–4)____________(x– )_(x- )_ _ __________________

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____________________ x-# ________________ x-# _ _____________________________=_______________ _______________x–# x+ &___________x–# x+#4_ _ ______________(_ x_–_# _)_(_x _–_# x_+_#4_)_=_(_ x_–_# _)_(_x _–_# x_+_ &)_ ___________ x%– x +% x–# x+# x–#4 _=_ x%– x +& x–# x+# x- & _ _______________ x% _–_% x _+_#% x_–_#4 _=_ x% _–_% x _+_#& x_–_ & _ ________________________________#% x_–_#4 _=_#& x_–_ & _ ______________________________#% x_–_#& x_=_#4 _–_ & _ _______________________________________–_#4x_=_-_ _ __________________________________x_=_-_ _ _-_#4_=_*_

Now_A6amuccayaD_sutra,_tell_us_that,_i1_other_elements_bein _e9ual,_the_sum-total_o1_the_denominators_on_the_2. .6._and_their_total_on_the_I. .6._be_the_same,_that_total_is_?ero.__

____Now_G# _+_G _=_x_–_&_+_x_–_4_=_ x_–_# ,_and__ _______________G% _+_G& _=_x_–_ _+_x_–_ _=_ x_–_# _

By_6amuccaya,_ x_–_# _ i!es_ x_=_# _ _ __________________________# _ ___________________x__=______=__*_ ___________________________ _

"xample_# $ _

_ _______________#________#___________#___________#_ __________________+_______=_________+________ _____________x_- _____x_–_'______x_-*______x_–_# _ _ ____________G# _+G _=_x_–_ _+_x_–_'_=_ x_–_# ,_and__ ____________G% _+G& _=_x_–_*_+_x_–# _=_ x_–_# _ ___________________Now_ x_–_# _=_ _ i!es_ x_=_# __ _ ___________________________________# _ ___________________________x__=______=__ _

____________________________________ _"xample_##$ _ _ _______________#___________#___________#____________#_ __________________-_________=_________-________ _____________x_+ _____x_+_# ______x_+4______x_+_'_ _

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_____/his_is_not_in_the_expected_1orm._But_a_little_work_re ardin _transposition_makes_the_abo!e_as_1ollows._ _ _______________#__________#___________#___________#_ __________________+________=________+________ _____________x_+ _____x_+_'______x_+4______x_+_# _ _ _____Now_A6amuccayaD_sutra_applies__ _ ____________G# _+G _=_x_+_ _+_x_+_'_=_ x_+_#4,_and__ ____________G% _+G& _=_x_+_4_+_x_+_# _=_ x_+_#4_

___6olution_is_ i!en_by_ x_+_#4_=_ _i.e.,_ _x_=_-_#4._ ____________________________________________________________x_=_-_#4_ _

_6ol!e_the_1ollowin _problems_usin _6unyam_6amya-6amuccaye_process._ _ ________#._____ _(_x_+_ _)_+_%_(_x_+_ _)_=_4_(_x_+_ _)_+_*_(_x_+_ _)_ _ ________ ._____(_x_+_4_)_(_x_+_%_)_=_(_x_–_'_)_(_x_–_ )_ _ ________%._____(_x_-_#_)_(_x_+_#&_)_=_(_x_+_ _)_(_x_–_ )_ _

_ _____________________#________________#_ ________&._______________+__________=___ _ _________________&_x_-_%__________x_–_ _ _ _ ______________________&________________&_ ________*._______________+____________=___ _ _________________%x_+_#_________*x_+_ _ _ _

___________________ x_+_##_________ x+*_ ________4._________________=_________ ____________________ x+_*__________ x+##_ _ _ ___________________%x_+_&____________x_+_#_ ________ .________________=_________ ___________________4x_+_ ___________ x_+_%_

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_ _ ___________________&x_-_%_____________x+_&_ ________ .________________=_________ ____________________ x+_%___________%x_-_ _ _ _ _________ _________________#______________#______________#_____________#_ ________'.__________+_________=_________+________ ________________x_-_ _______x_-_*_________x_-_%________x_-_& _ _ _ ___________________#_____________#______________#_______________#_ ________# .__________-__________=__________-________ ___________________x_-_ _______x_-_4_________x_-_# _______x_-_' _

_

6unyam_6amya_6amuccaye_in_0ertain_0ubes$ _

0onsider_the_problem_(_x_–_&_)% _+_(_x_–_4_)% _=_ _(_x_–_*)%._Lor_the_solution_by_the_traditional_method_we_1ollow_the_steps_as_ i!en_below$_ _ ________________(_x_–_&_)% _+_(_x_–_4_)% _=_ _(_x_–_*_)% _ _________x% _–_# x _+_& x_–_4&_+_x% _–_# x _+_# x_–_ #4_

_______________________________________________=_ _(_x%

_–_#*x _+_ *x_–_# *_)_ ____________ x% _–_% x _+_#*4x_–_ _=_ x% _–_% x _+_#* x_–_ * _ _______________________________#*4x_–_ _=_#* x_–_ * _ ______________________________#*4x_–_#* x_=_ _–_ * _ ___________________________________________4x_=_% _ _____________________________________________x_=_% _ _4_=_*_

_But_once_a ain_obser!e_the_problem_in_the_!edic_sense_

_@e_ha!e_(_x_–_&_)_+_(_x_–_4_)_=_ x_–_# ._/akin _out_the_numerical_1actor_ ,we_ha!e_(_x_–_*_)_=_ ,_which_is_the_1actor_under_the_cube_on_I. .6._5n_such_a_

case_O6unyam_samya_6amuccayeP_1ormula_ i!es_that_x_–_*_=_ ._ ence_x_=_*_/hink_o1_sol!in _the_problem_(x– &') % _+_(x+ & )% _=_ (x–#)% _

/he_traditional_method_will_be_horrible_e!en_to_think_o1._

But_(_x_–_ &'_)_+_(_x_+_ & _)_=_ x_–_ _=_ _(_x_–_#_)._:nd_x_–_#._on_I. .6.

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cube,_it_is_enou h_to_state_that_x_–_#_=_ _by_the_AsutraD._

x_=_#_is_the_solution._No_cubin _or_any_other_mathematical_operations._

:l ebraic_7roo1_$ _ _ ________0onsider_(_x_–_ a)% _+_(_x_–_ b_)% _=_ _(_x_–_a_–_b_)% _it_is_clear_that__ _______________________________x_–_ a_+_x_–_ b_=_ x_–_ a_–_ b_ ____________________________________________________=_ _(_x_–_a_–_b_)__

________Now_the_expression,_ _ _________x% _-4x a_+_# xa _–_ a% _+_x% _–_4xb_+# xb _–_ b% __=__ ________________ (x%–%xa–%xb+%xa +%xb +4axb–a %–%ab–%ab–b%) _ _ _____________=_ x%–4x a–4x b+4xa +4xb +# xab– a%–4a b–4ab – b% __

__cancel_the_common_terms_on_both_sides__ _ _______# xa+# xb – a%– b% _=_4xa+4xb +# xab– a%–4a b–4ab – b% _ ____________4xa _+_4xb _–_# xab_=_4a% _+_4b% _–4a b_–_4ab _ _____________4x_(_a _+_b _–_ ab_)_=_4_;_a% _+_b% _–_ab_(_a_+_b_)<_ _________________x_(_a_–_b_) _=_;_(_a_+_b_)_(_a _+_b _–ab_)_–_(_a_+_b_)ab<_ _________________________________=_(_a_+_b_)_(_a _+_b _–_ ab_)_ _________________________________=_(_a_+_b_)_(_a_–_b_) ___ _

____________________________ __x_=_a_+_b_ _

6ol!e_the_1ollowin _usin _O6unyam_6amuccayeP_process_$_ _ ____________#.____(_x_–_%_)% _+_(_x_–_'_)% _=_ _(_x_–_4_)% _ _ ____________ .____(_x_+_&_)% _+_(_x_–_# _)% _=_ _(_x_–_%_)% _ _ ____________%.____(_x_+_a_+_b_–_c_)% _+_(_x_+_b_+_c_–_a_)% _=_ _(_x_+_b_)% _

_

_

_

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63_ _

_

"xample_$ _ _ _______________________(x_+_ )%_________x_+_#_ ________________________________=_________ _______________________(x_+_%)%_________x_+_&_

____with_the_text_book_procedures_we_proceed_as_1ollows_ _ _______________x%_+_4x _+_# x_+ _________x_+_#_ __________________________________=_________ _______________x%_+_'x _+_ x_+ _______x_+_&_

____Now_by_cross_multiplication,_ _ ____(_x_+_&_)_(_x% _+4x _+_# x_+_ _)_=_(_x_+_#_)_(_x% _+_'x _+_ x_+_ _)_ _______x& _+_4x% _+_# x+_ x_+_&x% _+_ &x _+_& x_+_% _=_ ______________________________x& _+_'x% _+_ x _+_ x_+_x% _+_'x _+_ x_+_ _ ________x& _+# x% _+_%4x _+_*4x_+_% _=_x& _+_# x% _+%4x _+_*&x_+_ _ _________________________________*4x_+_% _=_*&x_+_ _ ________________________________*4x_–_*&x_=_ _–_% _ __________________________________________ x_=_-_*_ ____________________________________________x_=_-_*_ _ _

_Cbser!e_that_(_N# _+_G# _)_with_in_the_cubes_on_

____________________________2. .6._is_x_+_ _+_x_+_%_=_ x_+_*_and_ _ ________________________N _+_G _on_the_ri ht_hand_side__ ______________________________is_x_+_#_+_x_+_&_=_ x_+_*._ _ ______By_!edic_1ormula_we_ha!e_ x_+_*_=_ ________x_=_-_*_ _ ._

_

6ol!e_the_1ollowin _by_usin _!edic_method_$_ _

_______________#._ _______________________________(x_+_%)% ________x+#_ ________________________________________=________ _______________________________(x_+_*)% ________x+ _ _ _______________

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64_ _

_

_ .__ _______________________________(x_-_*)% _________x_-_%_ _________________________________________=________ _______________________________(x_-_ )% _________x_-_' _

_

4._:nurupye_-_6unyamanyat __

_/he_6utra_:nurupye_6unyamanyat_says_$_ 51_one_is_in_ratio,_the_other_one_is_?ero ._

@e_use_this_6utra_in_sol!in _a_special_type_o1_simultaneous_simple_e9uations_

in_which_the_coe11icients_o1_ one _!ariable_are_in_the_same_ratio_to_each_other_as_the_independent_terms_are_to_each_other._5n_such_a_context_the_6utra_says_the_ other _!ariable_is_?ero_1rom_which_we_ et_two_simple_e9uations_in_the_1irst!ariable_(already_considered)_and_o1_course_ i!e_the_same_!alue_1or_the_!ariable._

"xample_#$ __ ___________________________%x_+__ y_=_ __ ___________________________&x_+_ #y_=_4_

Cbser!e_that_the_y-coe11icients_are_in_the_ratio_ _$_ #_i.e.,_#_$_%,_which_is_

same_as_the_ratio_o1_independent_terms_i.e.,_ _$_4_i.e.,_#_$_%._ ence_the_other_!ariable_x_=_ _and_ y_=_ _or_ #y_=_4_ i!es_y_=_ _ _ _

"xample_ $ _ ___________________________% %x_+_#& y_=_#4#*__ ___________________________'4'x_+_% #y_=_& &*_

/he_!ery_appearance_o1_the_problem_is_1ri htenin ._But_8ust_an_obser!ation_and_anurupye_sunyamanyat_ i!e_the_solution_x_=_*,_because_coe11icient_o1_x_ratio_is_ ________% %_$_'4'_=_#_$_%_and_constant_terms_ratio_is_#4#*_$_& &*_=_#_$_%

____________y_=_ _and_% %_x_=_#4#*_or_'4'_x_=_& &*_ i!es_x_=_*._ _

6ol!e_the_1ollowin _by_anurupye_sunyamanyat._ _ ________#.___# x_+_ y_=_# ____________________ .____%x_+_ y_=_ &_

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65_ _

______________#4x_+_'4y_=#4__________________________# x_+_*y_=_'4_ _ ________%.____&x_–_4y_=_ &________________________&.____ax_+_by_=_bm_ ______________ x_–_'y_=_%4________________________________cx_+_dy_=_dm _

_

5n_sol!in _simultaneous_9uadratic_e9uations,_also_we_can_take_the_help_o1_the_ AsutraD_in_the_1ollowin _way$_

"xample_%_$ _ __________________6ol!e_1or_x_and_y_ ________________________________________________________________x_+_&y ____________________________________x _+_*xy_+_&y _+_&x_-_ y_=_ _ _ _____________x _+_*xy_+_&y _+_&x_-_ y_=_ _can_be_written_as__ _______________(_x_+_y_)_(_x_+_&y_)_+_&x_–_ y_=_ _ ____________# _(_x_+_y_)_+_&x_–_ y_=_ _(_6ince_x_+_&y_=_# _)_ ______________# x_+_# y_+_&x_–_ y_=_ _ _____________________________#&x_+_ y_=_ _ _ _________________Now_x_+_&y_=_# _ ____________#&x_+_ y_=_ _and_&_$_ _$$_# _$_ _ _ ______1rom_the_6utra,_x_=_ _and_&y_=_# ,_i.e.,,_ y=_ _y_=_# &_=_ _ _________/hus _x_=_ _and_y_=_ _is_the_solution._

_

._6ankalana_-_Qya!akalanabhyam __

/his_6utra_means_ by_addition_and_by_subtraction ._5t_can_be_applied_in_sol!in _a_special_type_o1_simultaneous_e9uations_where_the_x_-_coe11icients_and_the_y_-_coe11icients_are_1ound_interchan ed._

"xample_#$ _ ___________________________&*x_–_ %y_=_##%_

___________________________ %x_–_&*y_=_'#_5n_the_con!entional_method_we_ha!e_to_make_e9ual_either_the_coe11icient_o1_x_or_coe11icient_o1_y_in_both_the_e9uations._Lor_that_we_ha!e_to_multiply_e9uation_(_#_)_by_&*_and_e9uation_(_ _)_by_ %_and_subtract_to_ et_the_!alue_o1_x_and_then_substitute_the_!alue_o1_x_in_one_o1_the_e9uations_to_ et_the_!alue_o1_y_or_we_ha!e_to_multiply_e9uation_(_#_)_by_ %_and_e9uation_(_ _)_by_&*_and_then_subtract_to_ et_!alue_o1_y_and_then_substitute_the_!alue_o1_y_in_one_o1_the_

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66_ _

e9uations,_to_ et_the_!alue_o1_x._5t_is_di11icult_process_to_think_o1._

Lrom_6ankalana_–_!ya!akalanabhyam_ _ ________add_them, _ _______________i.e.,_(_&*x_–_ %y_)_+_(_ %x_–_&*y_)_=_##%_+_'#_ _______________i.e.,_4 x_–_4 y_=_ &______ _x_–_y_=_%_ _ ______subtract_one_1rom_other, _ _______________i.e.,_(_&*x_–_ %y_)_–_(_ %x_–_&*y_)_=_##%_–_'#_ _______________i.e.,_ x_+_ y_=_ ________ _x_+_y_=_#_ _ ______and_repeat_the_same_sutra,_we_ et_x_=_ _and_y_=_-_#_ ______Qery_simple_addition_and_subtraction_are_enou h,_howe!er_bi _the_coe11icients_may_be._

"xample_ $ _ _______________________#'**x_–___& 4y_=_ & _ _________________________& 4x_–_#'**y_=_-&'#%_

Ch_R_what_a_problem_R_:nd_still_ _ _______________8ust_add,_ &%#(_x_–_y_)_=_-_ &%#____ _x_–_y_=_-#_ _ _______________subtract,_#& '_(_x_+_y_)_=_ %'*____ _x_+_y_=_*_ _

_______________once_a ain_add,_ x_=_&________x_=_ _ _ _______________subtract_-_ y_=_-_4_______y_=_%__

_

6ol!e_the_1ollowin _problems_usin 6ankalana_–_Qya!akalanabhyam._ _ ____________#.____%x_+_ y_=_# _ ___________________ x_+_%y_=_# _ _

____________ .____*x_–_ #y_=_ 4_ ___________________ #x_–_*y_=_ 4_ _ ____________%.____4*'x_+_'*4y_=_&# 4_ ___________________'*4x_+_4*'y_=_% '_

_

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._7uranapuranabhyam __

/he_6utra_can_be_taken_as_7urana_-_:puranabhyam_which_means_by_the_completion_or_non_-_completion._7urana_is_well_known_in_the_present_

system._@e_can_see_its_application_in_sol!in _the_roots_1or_ eneral_1orm_o1_9uadratic_e9uation._

@e_ha!e_$_ax2 _+_bx_+_c_=_ _ _ __________________x_+_(b a)x_+_c a__=__ ____(_di!idin _by_a_)_ _ ___________________x_+_(b a)x_=_-_c a_ _____ ____completin _the_s9uare_(_i.e.,,_purana_)_on_the_2. .6._ _

_______________x_

+_(b a)x_+_(b &a)_=__-c a_+_(b &a)_ _ _________________[x_+_(b a)] _=__(b _-_&ac)_ _&a _ ______________________________________________________________________ ________________________________________________________________-_b_S_ _–_&ac_ ________7roceedin _in_this_way_we_1inally_ et_x_=__________________ ______________________________________________________________________

____Now_we_apply_purana_to_sol!e_problems._

"xample_#. ___x% _+_4x _+_##_x_+_4_=_ ._

_ ___________________6ince_(x_+_ _)% _=_x% _+_4x _+_# x_+_ _ _______________________:dd_(_x_+_ _)_to_both_sides_ ___________________@e_ et_x% _+_4x _+_##x_+_4_+_x_+_ _=_x_+_ _ _______________________i.e.,,_x% _+_4x _+_# x_+_ _=_x_+_ _ _______________________i.e.,,_(_x_+_ _)% _=_(_x_+_ _)_ ___________________this_is_o1_the_1orm_y% _=_y_1or_y_=_x_+_ _ ___________________solution_y_=_ ,_y_=_#,_y_=_-_#_ _______________________________i.e.,,_x_+_ _=_ ,#,-#_ ___________________which_ i!es_x_=_- ,-#,-%_

"xample_ $ _____x% _+_ x _+_# x_+_# _=_ __ _

____________@e_know_(_x_+_%)% _=_x% _+_'x _+_ x_+_ _ _______6o_addin _on_the_both_sides,_the_term_(x _+_# x_+_# _),_we_ et__ _________________x% _+_ x _+_# x_+_x _+_# x_+_# _=_x _+_# x_+_# __ _________________i.e.,,_x% _+_'x _+_ x_+_ _=_x _+_4x_+_'_+_&x_+_ _ _________________i.e.,,_(_x_+_%_)% _=_(_x_+_%_) _+_&_(_x_+_%_)_–_&_

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_____________________y% _=_y _+_&y_–_&_1or_y_=_x_+_%__ _______________________y_=_#,_ ,_- ._

____ ence_x_=_- ,_-#,_-*_ _________/hus_purana_is_help1ul_in_1actori?ation._ _________Lurther_purana_can_be_applied_in_sol!in _Bi9uadratic_e9uations_also._

_

6ol!e_the_1ollowin _usin _purana_–_apuranabhyam._ _ ____________#.____x% _–_4x _+_##x_–_4_=_ _ ____________ .____x% _+_'x _+_ %x_+_#*_=_ __ ____________%.____x _+_ x_–_%_=_ _ ____________&.____x& _+_&x% _+_4x _+_&x_–_#*_=_ _

_

_'._0alana_-_Halanabhyam __

5n_the_book_on_Qedic_3athematics _6ri_Bharati_Hrishna_/irtha8i_mentioned_the_6utra_ 0alana_-_Halanabhyam _at_only_two_places._/he_6utra_means_

6e9uential_motion ._

i) _5n_the_1irst_instance_it_is_used_to_1ind_the_roots_o1_a_9uadratic_e9uation x _–_##x_–_ _=_ ._6wami8i_called_the_sutra_as_calculus_1ormula._5ts_application_at_

that_point_is_as_1ollows.Now_by_calculus_1ormula_we_say$_#&x–##_=_ST%# _:_Note_1ollows_sayin _e!ery_Uuadratic_can_thus_be_broken_down_into_two_binomial_1actors._:n_explanation_in_terms_o1_1irst_di11erential,_discriminant_with_su11icient_number_o1_examples_are_ i!en_under_the_chapter_AUuadratic_"9uationsD._

ii) _:t_the_6econd_instance_under_the_chapter_ALactori?ation_and_Gi11erential_0alculusD_1or_1actori?in _expressions_o1_%rd,_&th _and_*th _de ree,_the_procedure_is_mentioned_as Qedic_6utras_relatin _to_0alana_–_Halana_–_Gi11erential_0alculus ._

Lurther_other_6utras_# _to_#4_mentioned_below_are_also_used_to_ et_the_re9uired_results._ ence_the_sutra_and_its_!arious_applications_will_be_taken_up_at_a_later_sta e_1or_discussion._

But_sutra_–_#&_is_discussed_immediately_a1ter_this_item._

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Now_the_remainin _sutras_$_ _ ________# ._VWQ:GXN:3_(_/he_de1iciency_)_ ________##._QV:YZ56:3:YZ5 _(_@hole_as_one_and_one_as_whole_)_ ________# ._["YWNV:\_H"N:_0:I:3"]:_(_Iemainder_by_the_last_di it_)_ ________#%._6C7WN/V:GQ:V:3:N/V:3_(_Mltimate_and_twice_the_penultimate_)_ ________#*._^M]5/:6:3M00:V: _(_/he_whole_product_is_the_same_)_ ________#4._^M]:H:_6:3M00:V: _(_0ollecti!ity_o1_multipliers_)_

/he_6utras_ha!e_their_applications_in_sol!in _di11erent_problems_in_di11erent_contexts._Lurther_they_are_used_alon _with_other_6utras._6o_it_is_a_bit_o1_incon!enience_to_deal_each_6utra_under_a_separate_headin _exclusi!ely_and_also_independently._C1_course_they_will_be_mentioned_and_also_be_applied_in_sol!in _the_problems_in_the_1orth_comin _chapter_where!er_necessary._/his_decision_has_been_taken_because_up_to_now,_we_ha!e_treated_each_6utra_independently_and_ha!e_not_continued_with_any_other_6utra_e!en_i1_it_is_necessary._@hen_the_need_1or_combinin _6utras_1or_1illin _the_ aps_in_the_process_arises,_we_may_opt_1or_it._Now_we_shall_deal_the_1ourteenth_6utra,_the_6utra_le1t_so_1ar_untouched._

_

# ._"kanyunena_7ur!ena _

/he_6utra_"kanyunena_pur!ena_comes_as_a_6ub-sutra_to_Nikhilam_which_ i!es_the_meanin _ Cne_less_than_the_pre!ious _or_ Cne_less_than_the_one_be1ore .__ _ _____#)_/he_use_o1_this_sutra_in_case_o1_multiplication_by_','','''.._is_as_1ollows_._ _ ____3ethod_$ __

a)_/he_le1t_hand_side_di it_(di its)_is_(_are)_obtained_by_applyin _the_ekanyunena_pur!ena_i.e._by_deduction_#_1rom_the_le1t_side_di it_(di its)_._

_________e. ._(_i_)_ _x_'J_ _–_#_=_4_(_2. .6._di it_)_

b)_/he_ri ht_hand_side_di it_is_the_complement_or_di11erence_between_the_multiplier_and_the_le1t_hand_side_di it_(di its)_._i.e._ _ _'_I. .6_is_'_-_4_=_%._

c)_/he_two_numbers_ i!e_the_answerJ_i.e._ _ _'_=_4%._

"xample_#$ ___ _x_'__6tep_(_a_) _ i!es_ _–_#_=_ _(_2. .6._Gi it_)_ _________________________________6tep_(_b_) _ i!es_'_–_ _=_ _(_I. .6._Gi it_)_ _________________________________6tep_(_c_) _ i!es_the_answer_ _

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"xample_ $ __#*_x_''___6tep_(_a_)_$ _#*_–_#_=_#&_ _____________________________________6tep_(_b_)_$ _''_–_#&_=_ *_(_or_# _–_#*_)_ _____________________________________6tep_(_c_)_$ _#*_x_''_=_#& *_

"xample_%$ ___ &_x_''_

________:nswer_$__

___________________ _________________________

"xample_&$ _____%*4_x_'''_ ________:nswer_$_

________________ _

"xample_*$ _____ _x_''''_ ________:nswer_$_

___________________ ______

Note_the_process_$_/he_multiplicand_has_to_be_reduced_by_#_to_obtain_the_2 6_and_the_ri htside_is_mechanically_obtained_by_the_subtraction_o1_the_2. .6_1rom_the_multiplier_which_is_practically_a_direct_application_o1_Nikhilam_6utra._ _ ____Now_by_Nikhilam_ _ __________________________ &_–_#__=_ %____2. .6._ _______________________x_''_–_ %_=_ 4____I. .6.__(# – &)_ _____________________________________________________ __________________________________ %_ _ 4____________________________=_

Ieconsider_the_"xample_&$ _ _ _________________________%*4_–_#____=_%**____2. .6._ ______________________x_'''_–_%**_=_4&&____I. .6._ ______________________________________________ ________________________________%**_ _4&&___________________=___%**4&

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and_in_"xample_*$ _ _x_''''_we_write_ _ ___________________ _–__#___=_ _____2. .6._ ________________x_''''_–_ _=_'# ____I. .6._ __________________________________________

_____________________________ _ _'# _______________=__ '# _:l ebraic_proo1_$ __

:s_any_two_di it_number_is_o1_the_1orm_(_# x_+_y_),_we_proceed__ __________________(_# x_+_y_)_x_''_ ________________=_(_# x_+_y_)_x_(_# _–_#_)_ ________________=_# x_._# _–_# x_+_# _.y_–_y_ ________________=_x_._#% _+_y_._# _–_(_# x_+_y_)_ ________________=_x_._#% _+_(_y_–_#_)_._# _+_;_# _–_(_# x_+_y_)<_

____/hus_the_answer_is_a_1our_di it_number_whose_#th _place_is_x,#

th _place_is_(_y_-_#_)_and_the_two_di it_number_which_makes_up_the_#th _and_unit_place_is_the_

number_obtained_by_subtractin _the_multiplicand_1rom_# .(or_apply_nikhilam)._ _ ________/hus_in_% _ _''._/he_#th _place_is_x_i.e._%_ _ ________#th _place_is_(_y_-_#_)_i.e._( _-_#_)_=_4_ _ ________Number_in_the_last_two_places_# -% =4%._ _ ________ ence_answer_is_%44%._

_ ____:pply_"kanyunena_pur!ena_to_1ind_out_the_products_ _ ________#._4&_x_''__________ .__ %_x_'''___________%._% *#_x_''''_ _ ________&._&%_x_'''________*.__ *4_x_''''_________4.__# * _x_''''' __

_

@e_ha!e_dealt_the_cases__

___i)_@hen_the_multiplicand_and_multiplier_both_ha!e_the_same_number_o1_di its__ ________ii)_@hen_the_multiplier_has_more_number_o1_di its_than_the_multiplicand.

5n_both_the_cases_the_same_rule_applies._But_what_happens_when_the_multiplier_has_lesser_di its _

i.e._1or_problems_like_& _ _',_# &_ _',_ 4% *_ _''_etc.,_

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72_ _

Lor_this_let_us_ha!e_a_re-look_in_to_the_process_1or_proper_understandin ._

3ultiplication_table_o1_'. _

____________________________________________a____b_

____________________________ _x_'__=____#____ _ ____________________________%_x_'__=____ ____ _ ____________________________&_x_'__=____%____4_ ____________________________-_-_-_-_-_-_-_-_-_-__ ____________________________ _x_'__=____ ____ _ ____________________________'_x_'__=____ ____#_ __________________________# _x_'__=____'____ _

Cbser!e_the_le1t_hand_side_o1_the_answer_is_always_one_less_than_the_multiplicand_(here_multiplier_is_')_as_read_1rom_0olumn_(a)_and_the_ri ht_hand_side_o1_the_answer_is_the_complement_o1_the_le1t_hand_side_di it_1rom_'_as_read_1rom_0olumn_(b)_3ultiplication_table_when_both_multiplicand_and_multiplier_are_o1_ _di its. _

_________________________a_____b_ ______________##_x_''_=_# ____ '_=_(##–#)_ _''_–_(##–#)_=_# '_ ______________# _x_''_=_##____ _=_(# –#)_ _''_–_(# –#)_=_## _ ______________#%_x_''_=_# ____ _=_(#%–#)_ _''_–_(#%–#)_=_# _ _____________-------------------------------------------------_ ______________# _x_''_=_# ____ __----------------------------_ ______________#'_x_''_=_# ____ #_ ______________ _x_''_=_#'____ _=_( –#)_ _''_–_( –#)_=_#' _

/he_rule_mentioned_in_the_case_o1_abo!e_table_also_holds_ ood_here_

Lurther_we_can_state_that_the_rule_applies_to_all_cases,_where_the_multiplicand_and_the_multiplier_ha!e_the_same_number_o1_di its._

0onsider_the_1ollowin _/ables ._

(i) _ ______________________________________a____b_ _______________________##_x_'_=___'____'_ _______________________# _x_'_=_# ____ _ _______________________#%_x_'_=_##____ _ __________________----------------------_ _______________________# _x_'_=_#4____ _ _______________________#'_x_'_=_# ____#_ _______________________ _x_'_=_# ____ _

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73_ _

(ii) _ _______________________ #_x_'_=_# ____'_ _______________________ _x_'_=_#'____ _ _______________________ %_x_'_=_ ____ _ __________________-----------------------__

_______________________ _x_'_=_ *____ _ _______________________ '_x_'_=_ 4____#_ _______________________% _x_'_=_ ____ _

(iii) _ _______________________%*_x_'_=_%#____*_ _______________________&4_x_'_=_&#____&_ _______________________*%_x_'_=_& ____ _ _______________________4 _x_'_=_4 ____%_ _________________-------------------------so_on._

Lrom_the_abo!e_tables_the_1ollowin _points_can_be_obser!ed$_#)_/able_(i) _has_the_multiplicands_with_#_as_1irst_di it_except_the_last_one._ ere_2. .6_o1_products_are_uni1ormly_ _less_than_the_multiplicands._6o_also_with _x_'

)_/able_(ii) _has_the_same_pattern._ ere_2. .6_o1_products_are_uni1ormly_%_less_than_the_multiplicands._

%)_/able_(iii) _is_o1_mixed_example_and_yet_the_same_result_i.e._i1_%_is_1irst_di ito1_the_multiplicand_then_2. .6_o1_product_is_&_less_than_the_multiplicandJ_i1_&_i1irst_di it_o1_the_multiplicand_then,_2. .6_o1_the_product_is_*_less_than_the_multiplicand_and_so_on._&)_/he_ri ht_hand_side_o1_the_product_in_all_the_tables_and_cases_is_obtained_by_subtractin _the_I. .6._part_o1_the_multiplicand_by_Nikhilam._

Heepin _these_points_in_!iew_we_sol!e_the_problems$_

"xample#_$ _& _ _'_

i)_Gi!ide_the_multiplicand_(& )_o1_by_a_Qertical_line_or_by_the6i n_$ _into_a_ri ht_hand_portion_consistin _o1_as_many_di its_as_the_multiplier._ _ _______________________i.e._& _has_to_be_written_as_& _or_&$ _

ii)_6ubtract_1rom_the_multiplicand_one_more_than_the_whole_excess_portion_on_the_le1t._i.e._le1t_portion_o1_multiplicand_is_&._ _ _______________________one_more_than_it_&_+_#_=_*._ _

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_______________@e_ha!e_to_subtract_this_1rom_multiplicand__ _______________________i.e._write_it_as__ _______________________________________&__$__ __ ___________________________________________$-*__ _________________________________---------------__

_______________________________________%__$__ __ _ ____________/his_ i!es_the_2. .6_part_o1_the_product._

/his_step_can_be_interpreted_as_`take_the_ekanyunena_and_sub_tract_1rom_the_pre!ious`_i.e._the_excess_portion_on_the_le1t._

iii)_6ubtract_the_I. .6._part_o1_the_multiplicand_by_nikhilam_process._i.e._I. .6_o1_multiplicand_is_ _ _ ___________________its_nikhilam_is_ _ _ _______________5t_ i!es_the_I. .6_o1_the_product_ _ _______________i.e._answer_is_%_$_ _$_ _=_% ._ _ _______________/hus_& _ _'_can_be_represented_as_ _ ___________________________&_$_ _ ______________________________$-*_$_ __ ____________________------------------__ ___________________________%_$_ _$_ _=_% ._

"xample_ _$ _____# &_ _'_ _ ______________ ere_3ultiplier_has_one_di it_only_._ _ _______________________@e_write_# _$_&__ _ ______________Now_step_(ii),# _+_#_=_#%_ _ _________________________i.e.____# _$_&__ _________________________________-#_$_%___ _____________________________------------__ ______________6tep_(_iii_)_I. .6._o1_multiplicand_is_&._5ts_Nikhilam_is_4_ _ ________________# &_x_'_is____# _$_&__ ___________________________________-#_$_%_$_4__ ___________________________-----------------__ __________________________________##_$_#_$_4____=____###4_ _

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________/he_process_can_also_be_represented_as__ ________# &_x_'_=_;_# &_–_(_# _+_#_)_<_$_(_# _–_&_)_=_(_# &_–_#%_)_$_4_

"xample_%$ _____#*4%'_x_''_ _

_____6ince_the_multiplier_has_ _di its,_the_answer_is__ _____;#*4%'_–_(#*4_+_#)<_$_(# _–_%')_=_(#*4%'_–_#* )_$_4#_=_#*& 4#_

_

Lind_the_products_in_the_1ollowin _cases._ _ ________#.___* _x_'________ .___4 _x_'________________%.___& _x_''_ _ ________&.___ % _x_'______*.___ & #_x_'''_____4._### ##_x_'' _

_"kanyunena_6utra_is_also_use1ul_in_Iecurrin _Gecimals._@e_can_take_up_this_under_a_separate_treatment._

/hus_we_ha!e_a_ limpse_o1_ma8ority_o1_the_6utras._:t_some_places_some_6utras_are_mentioned_as_6ub-6utras._:ny_how_we_now_proceed_into_the_use_o1_6ub-6utras._:s_already_mentioned_the_book_on_Qedic_3athematics_enlisted_#%_Mpa-6utras._

But_some_approaches_in_the_Qedic_3athematics_book_prompted_some_serious_research_workers_in_this_1ield_to_mention_some_other_Mpa-6utras._@e_can_obser!e_those_approaches_and_de!elopments_also._

_

##._:nurupyena_

/he_upa-6utra_ anurupyena _means_ proportionality ._/his_6utra_is_hi hly_use1ul_to_1ind_products_o1_two_numbers_when_both_o1_them_are_near_the_0ommon_bases_i.e_powers_o1_base_# _._5t_is_!ery_clear_that_in_such_cases_the_expected_

6implicity_ _in_doin _problems_is_absent._

"xample_#$_&4_ _&% _

:s_per_the_pre!ious_methods,_i1_we_select_# _as_base_we_ et_ _ ___________________&4___-*&____/his_is_much_more_di11icult_and_o1_no_use._ ___________________&%___-* _ __________________ _

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Now_by_AanurupyenaD_we_consider_a_workin _base_5n_three_ways._@e_can_sol!e_the_problem._

3ethod_#$ _/ake_the_nearest_hi her_multiple_o1_# ._5n_this_case_it_is_* ._ _

________/reat_it_as_# _ _ _=_* ._Now_the_steps_are_as_1ollows$_i)_0hoose_the_workin _base_near_to_the_numbers_under_consideration._i.e.,_workin _base_is_# _ _ _=_* _ _ii)_@rite_the_numbers_one_below_the_other_ _ _______________i.e._____&____4__ ________________________&____%__ ______________________ _ _iii)_@rite_the_di11erences_o1_the_two_numbers_respecti!ely_1rom_* _a ainst_each_number_on_ri ht_side_ _ _______________i.e.____&4____- &__ _______________________&%____- _ ______________________ _ _i!)_@rite_cross-subtraction_or_cross-_addition_as_the_case_may_be_under_the_line_drawn._ _

______________________ _ _!)_3ultiply_the_di11erences_and_write_the_product_in_the_le1t_side_o1_the_answer._ _ __________________________&4____- &_ __________________________&%____- _

___________________________________ _______________________%'__ _-&_x_– _ _ ____________________________=_ _ _!i)_6ince_base_is_# _ _ _=_* _,_%'_in_the_answer_represents_%' * ._ _ ____ ence_di!ide_%'_by_ _because_* _=_# _ _ _

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/hus_%'_E_ _ i!es_#' _where_#'_is_9uotient_and_#_is_remainder_._/his_#_as_Ieminder_ i!es_one_* _makin _the_2. .6_o1_the_answer_ _+_* _=_ (or_Iemainder_ _x_# _+_ _)_

i.e._I. .6_#'_and_2. .6_ _to ether_ i!e_the_answer#' _@e_represent_it_as_

_ ___________________&4____- &_ ___________________&%____- _ __________________ _ ______________ )__%'__ ___ _ __________________ ____________________ __________________#' _ __ _ _ _______________=_#'_ _ _=_#' __

"xample_ $ __& _ _& ._

@ith_# _ _ _=_* _as_workin _base,_the_problem_is_as_1ollows$_ _ ___________________& ____- _ ___________________& ____- _ __________________ _ _______________ )_& ___ __#4_ __________________ __ ___________________ __ ___#4_ _ ________________& _x_& _=_ #4_

3ethod_ $ _Lor_the_example_#$ _&4 &%._@e_take_the_same_workin _base_* ._@e_treat_it_as_* =* # ._i.e._we_operate_with_# _but_not_with_# _as_in_method_ _ ____________now_

_______________________ _ ________________________________(#'*_+_ )_ _ __=__#' _ _ ____;6ince_we_operate_with_# ,_the_I. .6_portion_shall_ha!e_only_unit_place_. ence_out_o1_the_product_ ,_ _is_carried_o!er_to_le1t_side._/he_2. .6_portion_o1_the_answer_shall_be_multiplied_by_*,_since_we_ha!e_taken_* _=_*_ _# .<_

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78_ _

Now_in_the_example_ $ _& _x_& _we_can_carry_as_1ollows_by_treatin _* _=_*_x_ _

_______________________ _

3ethod_%$ _@e_take_the_nearest_lower_multiple_o1_# _since_the_numbers_are_&4_and_&%_as_in_the_1irst_example,_@e_consider_& _as_workin _base_and_treat_it_as

_# ._ _

___ ________________ _ _ _____6ince_# _is_in_operation_#_is_carried_out_di it_in_# ._

6ince_&_ _# _is_workin _base_we_consider_&'_ _&_on_2. .6_o1_answer_i.e._#'4_and_#_carried_o!er_the_le1t_side,_ i!in _2. .6._o1_answer_as_#' ._ ence_the_answer_is_#' .__

@e_proceed_in_the_same_method_1or_& _ _& _ _

___ ________________ _

2et_us_see_the_all_the_three_methods_1or_a_problem_at_a_ lance_ _ ___ __"xample_%$ _ &_ _ %_ _

_

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_ ___ __3ethod_-_#$ _____@orkin _base_=_# _ _*_=_ _ _ ___________________ &____ &_ ___________________ %____ %_

__________________ _ ______________*)__ _ __# _ _________________ _ _________________*_* _ _# ____=__*_ _* __=_** _ _ ________;6ince_ _ _*_o1_# _is_ _ _*_x_# _=_& _and_& _+_# _=_* <_

3ethod_-_ $ __@orkin _base_ _ _# _=_ _ _

___ ________________________ _

Now_as_ _itsel1_is_nearest_lower_multiple_o1_# _1or_the_problem_under_consideration,_the_case_o1_method_–_%_shall_not_arise._

2et_us_take_another_example_and_try_all_the_three_methods._

"xample_&$ _&' _ _& &_

3ethod_-_#_$ __workin _base_=_# _ _ _=_* _ _ _______________________&' ____- _ _______________________& &____- '4_ ______________________ _ __________________ )__%'4__ __ 4 ______since_# _is_in_operation_ ______________________ _ ______________________#' __ ___ 4 _______=__#' 4 _

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3ethod_ $ _workin _base_=_*_x_# _=_* _ _

___ ____________________ _

3ethod_-_%. _ __________6ince_& _can_also_be_taken_as_workin _base,_treat_& _=_&_ _#workin _base._ _ ____________/hus_

___ ________________________ _ ________No_need_to_repeat_that_practice_in_these_methods_1inally_takes_us_to_workout_all_these_mentally_and_ ettin _the_answers_strai ht_away_in_a_sin le_line._

"xample_*$ ___%'' _ _&'' _ _ _______________@orkin _base_=_# _ _ _=_* _ _ ___________________%'' ____-# _ ___________________&'' ____- _ __________________ _ _______________ )_%''4__ __ &_______since_# , _is_in_operation_ _ __________________#'' __ __ &________=__#'' &_

or_takin _workin _base_=_*_x_# _=_*, _and_ _

___ ________________________ _

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____@hat_happens_i1_we_take_& _i.e._&_ _# _as_workin _base _ ___________________________ ___________%'' ____ _ ___________&'' ____ '' ________6ince_# _is_an_operation_ __________ _

____________&''4_ _#''4_ ________________________________________________________________________ _____:s_# _is_in_operation,_#''4_has_to_be_written_as_#''4_and_& _as_base,_the_2. .6_portion_* _has_to_be_multiplied_by_&._i._e._the_answer_is_ _

___ ___________________ __

:_simpler_example_1or_better_understandin ._

"xample_4$ _* _x_& _ _ __________@orkin _base_* _=_*_x_# _ i!es_ _

___ ____________________ _ _ __________6ince_# _is_in_operation._

Mse_anurupyena_by_selectin _appropriate_workin _base_and_method._ _ ______Lind_the_1ollowin _product._ _

________#.___&4_x_&4___________ ._* _x_* ___________________%.__*&_x_& _ ________&.___# _x_# ___________*._4 _x_& ___________________4._ '_x_ % _ ________ .___& _x_'4___________ ._ '4*_x_''''4________'._&'x&''_ _ ________# ._% '_x_*# _

_

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# ._:dyamadyenantya_–_mantyena_

/he_6utra_ _adyamadyenantya-mantyena _means_ the_1irst_by_the_1irst_and_the_last_by_the_last ._

6uppose_we_are_asked_to_1ind_out_the_area_o1_a_rectan ular_card_board_whose_len th_and_breadth_are_respecti!ely_41t_._&_inches_and_*_1t._ _inches._^enerally_we_continue_the_problem_like_this._

________:rea_=_2en th_ _Breath_ _ ___________________=_4D_&`_ _*D_ `_6ince_#D_=_# `,_con!ersion__ _ ___________________=_(_4_ _# _+_&)_(_*_ _# _+_ )_in_to_sin le_unit_ _ ___________________=_ 4`_4 `_=_*#4 _69._inches._

6ince_#_s9._1t._=# _ _# _=_#&&s9.inches_we_ha!e_area_ _ ___________*#4 ____=____#&&)_*#4 _(%*_ ___________ _ ____________#&&__________________&% _ ___________________________________ _ ____________________________________ & _ ____________________________________ ____i.e.,_%*_69._1t_# _69._inches_ ___________________________________ _ ____________________________________# _

By_Qedic_principles_we_proceed_in_the_way_`the_1irst_by_1irst_and_the_last_by_last _ ________i.e._4D_&`_can_be_treated_as_4x_+_&_and_*D_ `_as_*x_+_ ,__ _ ____________@here_x=_#1t._=_# _inJx _is_s9._1t._

Now_(_4x_+_&_)(*x_+_ _)_ _ ____________=_% x _+_4. .x_+_&.*.x_+_% _ ____________=_% x _+_& x_+_ x_+_% _ ____________=_% x _+_4 ._x_+_% _ ____________=_% x _+_(_*x_+_ _)._x_+_% _@ritin _4 _=_*_x_# _+_ _ ____________=_%*x _+_ ._x_+_% _ ____________=_%*_69._1t._+_ _x_# _69._in_+_% _69._in_ ____________=_%*_69._1t._+_'4_69._in_+_% _69._in_ ____________=_%*_69._1t._+_# _69._in_

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5t_is_interestin _to_know_that_a_mathematically_untrained_and_e!en_uneducated_carpenter_simply_works_in_this_way_by_mental_ar umentation._5t_ oes_in_his_mind_like_this_ _ _______________________4D____&`_

_ _______________________*D____ `_ _ ________Lirst_by_1irst_i.e._4D_ _*D_=_% _s9._1t._ _ ________2ast_by_last_i.e._&`_ _ `_=_% _s9._in._ _ ________Now_cross_wise_4_ _ _+_*_x_&_=_& _+ _=_4 ._

:d8ust_as_many_ # _s_as_possible_towards_le1t_as_ units _i.e._4 _=_*_ _# _+ _twel!e s_as_*_s9uare_1eet_make_the_1irst_% +*_=_%*_s9._1t_J_ _le1t_becomes_ _# _s9uare_inches_and_ o_towards_ri ht_i.e._ _x_# _=_'4_s9._in._towards_ri ht_i!es'4+% _=_# s9.in._

/hus_he_ ot_area_in_some_sort_o1_%*_s9uints_and_another_sort_o1_# _s9._units._i.e._%*_s9._1t_# _s9._in_

:nother_"xample$_ _

___ ________________ _ _ ________Now_# _+_ _=_#&,_# _x_# _+_ &_=_# _+_ &_=_#&&_ _ ________/hus_& _4 _x_% _& _=_#&_69._1t._#&&_69._inches._ _ ________6ince_#&&_s9._in_=_# _ _# _=_#_s9._1t_/he_answer_is_#*_s9._1t._ _ ________@e_can_extend_the_same_principle_to_1ind_!olumes_o1_parallelepiped_also.

_

_

_

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_

5._Lind_the_area_o1_the_rectan les_in_each_o1_the_1ollowin _situations.__ _ ________#).___l_=_%D_ `_,_b_=_ D_&_`____________ ).___l_=_# D_*`_,__b_=_*

_ ________%).___l_=_&_yard_%_1t._b_=_ _yards_*_1t.(#yard_=%1t)__ _ ________&).___l_=_4_yard_4_1t._b_=_*_yards_ _1t._ _ _ ____55._Lind_the_area_o1_the_trape?ium_in_each_o1_the_1ollowin _cases._Iecall_area_= _h_(a_+_b)_where_a,_b_are_parallel_sides_and_h_is_the_distance_between_them._ _ ________#).___a_=_%D_ `,_b_=_ D_&`,_h_=_#D_*`_

_ ________ ).___a_=_*D_4`,_b_=_&D_&`,_h_=_%D_ `_ _ ________%).___a_=_ D_&`,_b_=_&D_4`,_h_=_*D_#`. _

_

Lactori?ation_o1_9uadratics$ _ _ ________/he_usual_procedure_o1_1actori?in _a_9uadratic_is_as_1ollows$_ _ ______________________%x _+_ x_+_&_ ___________________=_%x _+_4x_+_ x_+_&_ ___________________=_%x_(_x_+_ _)_+_ _(_x_+_ _)_ ___________________=_(_x_+_ _)_(_%x_+_ _)_

But_by_mental_process,_we_can_ et_the_result_immediately._/he_steps_are_as_1ollows._

i) ._6plit_the_middle_coe11icient_in_to_two_such_parts_that_the_ratio_o1_the_1irst_coe11icient_to_the_1irst_part_is_the_same_as_the_ratio_o1_the_second_part_to_the_lastcoe11icient._/hus_we_split_the_coe11icient_o1_middle_term_o1_%x _+_ x_+_&_i.e._ _in_to_two_such_parts_4_and_ _such_that_the_ratio_o1_the_1irst_coe11icient_to_the_1irst_part_o1_the_middle_coe11icient_i.e._%$4_and_the_ratio_o1_the_second_pat_to_the_lascoe11icient,_i.e._ $_&_are_the_same._5t_is_clear_that_%$4_=_ $&._ ence_such_split!alid._Now_the_ratio_%$_4_=_ $_&_=_#$ _ i!es_one_1actor_x+ ._ _ii) ._6econd_1actor_is_obtained_by_di!idin _the_1irst_coe11icient_o1_the_9uadratic_bythe_1ist_coe11icient_o1_the_1actor_already_1ound_and_the_last_coe11icient_o1_the_9uadratic_by_the_last_coe11icient_o1_the_1actor._

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________i.e._the_second_1actor_is_ _ _____________________%x__________&_ __________________________+_______=__%x_+_ _ ______________________x__________ _

________ ence_%x _+_ x_+_&_=_(_x_+_ _)_(_%x_+_ _)_

" .#$ _&x _+_# x_+_*_

i)_6plit_# _into_ _and_# _so_that_as_per_rule_&_$_ _=_# _$_*_=_ _$_#i.e.,,_ x_1irst_1actor._ _ii)_Now_ ____________&x ______*_ _________________+_____=__ x_+_*___is_second_1actor._ ___________ x_________#_" . $ _#*x _–_#&xy_–_ y _

i)_6plit_–#&_into_– ,_4_so_that_#*_$_-_ _=_%_$_-_&_and_4_$_-_ _=_%_$_-_&same.i.e.,_(_%x_–_&y_)_is_one_1actor._ _ii)_Now_ ___________#*x ________ y _ __________________+________=__*x_+_ y_is_second_1actor._ _____________%x_________-&y_

/hus_#*x _–_#&xy_–_ y _=_(_%x_–_&y_)_(_*x_+_ y_)._

5t_is_e!ident_that_we_ha!e_applied_two_sub-sutras_ AanurupyenaD _i.e.AproportionalityD_and_ AadyamadyenantyamantyenaD _i.e._Athe_1irst_by_the_1irst_and_the_last_by_the_lastD_to_obtain_the_abo!e_results._

Lactorise_the_1ollowin _9uadratics_applyin _appropriate_!edic_maths_sutras$_ _ ____________#).__%x _+_#&x_+_#*_ _ ____________ ).__4x _–_ %x_+_ _ _ ____________%).__ x _–_ x_+_*_ _ ____________&).__# x _–_ %xy_+_# y_

____

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#%._Va!adunam_/a!adunikrtya_Qar anca_Vo8ayet_

/he_meanin _o1_the_6utra_is_ what_e!er_the_de1iciency_subtract_that_de1icit_1rom_the_number_and_write_alon _side_the_s9uare_o1_that_de1icit ._

/his_6utra_can_be_applicable_to_obtain_s9uares_o1_numbers_close_to_bases_o1_powers_o1_# ._

3ethod-#_$ _Numbers_near_and_less_than_the_bases_o1_powers_o1_# ._ _ ____" _#$ __' _ ere_base_is_# ._ _ ________/he_answer_is_separated_in_to_two_parts_by_aD D__ _ ________Note_that_de1icit_is_# _-_'_=_#_ _ ________3ultiply_the_de1icit_by_itsel1_or_s9uare_it_ _ ________# _=_#._:s_the_de1iciency_is_#,_subtract_it_1rom_the_number_i.e.,_'–#_=_ ._ _ ________Now_put_ _on_the_le1t_and_#_on_the_ri ht_side_o1_the_!ertical_line_or_slai.e.,_ #._ _ ________ ence_ #_is_answer._

" ._ $ __'4 _ ere_base_is_# ._ _ ________6ince_de1icit_is_# -'4=&_and_s9uare_o1_it_is_#4_and_the_de1iciency_subtracted_1rom_the_number_'4_ i!es_'4-&_=_' ,_we_ et_the_answer_' _ _#4__/hus_'4 _=_' #4._

" ._%$ __''& _Base_is_# _ _ ________Ge1icit_is_# _-_''&_=_4._69uare_o1_it_is_%4._ _ ________Ge1iciency_subtracted_1rom_''&_ i!es_''&_-_4_=_' _ _ ________:nswer_is_' _ _ %4____;since_base_is_# <_

" ._&$ ___'' ___Base_is_# , ._ _ ________Ge1icit_=_# _-_'' _=_# ._ _ ________69uare_o1_de1icit_=_# _=_#&&._ _ ________Ge1iciency_subtracted_1rom_number_=_'' _-_# _=_'' 4._

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_ ________:nswer_is_'' 4_ _ #&&____;since_base_is_# , <._

" ._*$ ___ ___Base_is_# ._ _

________Ge1icit_=_# _-_ _=_# ._ _ ________69uare_o1_de1icit_=_# _=_#&&.__ _ ________Ge1iciency_subtracted_1rom_number_= _-_# _=_ 4._ _ ________Now_answer_is_ 4_ _#&&_= &&____;since_base_is_# <_

:l ebraic_proo1$ __

/he_numbers_near_and_less_than_the_bases_o1_power_o1_# _can_be_treated_as_(x-y),_where_x_is_the_base_and_y,_the_de1icit._ _ ________/hus__ ___________(#)_'_=_(# _-#)_( )_'4_=_(_# -&)_(%)_''&_=_(# -4)_ _ ____________(&)_'' _=_(# -# _)_(!)_ _=_(# -# )_ _ ____________(_x_–_y_) _=x _–_ xy_+_y _ ________________________=_x_(_x_–_ y_)_+_y _ ________________________=_x_(_x_–_y_–_y_)_+_y _ ________________________=_Base_(_number_–_de1iciency_)_+_(_de1icit_) _ ________/hus_ _ ____________' * _=_(_# _–_#*) _ __________________=_# _(_' *_–_#*_)_+_(#*) _ __________________=_# _(_' _)_+_ *_ __________________=_' _+_ *_ __________________=_' *._

or_we_can_take_the_identity_a _-_b _=_(a_+_b)_(_a_-_b)_and_proceed_as_ _ _________________a _-_b _=_(a_+_b)_(_a_-_b)._ _ ________ i!es_________a_=_(a_+_b)_(_a_-_b)_+b _ _ ____/hus_1or_a_=_' *_and_b_=_#*J_ _ ________a=_(a_+_b)_(_a_-_b)_+_b _ _ ____' * _=_(_' *_+_#*_)_(_' *_-_#*_)_+_(#*) _

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88_ _

_ ___________=_# _(_' _)_+_ *_ _ ___________=_' *._

3ethod._ _$ _Numbers_near_and_ reater_than_the_bases_o1_powers_o1_# ._ _ ___ __" .(#)$ __#% _._ _ ________5nstead_o1_subtractin _the_de1iciency_1rom_the_number_we_add_and_procas_in_3ethod-#._ _ ________1or_#% _,_base_is_# ,_surplus_is_%._ _ ________6urplus_added_to_the_number_=_#%_+_%_=_#4._ _ ________69uare_o1_surplus_=_% _=_'_ _ ________:nswer_is_#4_ _'_=_#4'._ _ ___ __" .( )$ ___## _ _ ___________Base_=_# ,_6urplus_=_# ,__ _ ________69uare_o1_surplus_=_# _=_#&&_ _ ________add_surplus_to_number_=_## _+_# _=_# &._ _ ________:nswer_is_# &_ _#&&_=_# *&&_ _ ____Cr_think_o1_identity_a _=_(a_+_b)_(a_–_b)_+_b __1or_a_=_## ,_b_=_# $_ _ _______________## _=_(## _+_# )_(## _–_# )_+_# _ _______________________=_# &_(# )_+_#&&_ _______________________=_# & _+_#&&_ _______________________=_# *&&._ _ ____________(x_+_y) _=x _+_ xy_+_y _ ________________________=_x_(_x_+_ y_)_+_y _ ________________________=_x_(_x_+_y_+_y_)_+_y _ _ ________=_Base_(_Number_+_surplus_)_+_(_surplus) _ _ _______ i!es_ ____________## _=# _(_## _+_# _)_+# _ ___________________=_# _(_# &_)_+_#&&_

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___________________=_# & _+_#&&_ ___________________=_# *&&._ _ _____" ._%$ ___# * _ _

_______________=_(_# *_+_ *_)_ _ * _ _ _______________=_# * _ _ 4 *_____;_since_base_is_# , _<_ _ _______________=_# * 4 *._

3ethod_-_%$ _/his_is_applicable_to_numbers_which_are_near_to_multiples_o1_# ,_# ,_# _...._etc._Lor_this_we_combine_the_upa-6utra_ anurupyena _and_

ya!adunam_ta!adunikritya_!ar anca_yo8ayet _to ether._

"xample_#$ ___% __Nearest_base_=_& .__ _ ________@e_treat_& _as_&_x_# ._:s_the_number_is_less_than_the_base_we_procas_1ollows_ _ ________Number_% ,_de1icit_=_& _-_% _=_# _ _ ________6ince_it_is_less_than_base,_deduct_the_de1icit__ _ ____________i.e._% _-_# _=_% 4.__ _ ________multiply_this_result_by_&_since_base_is_&_ _# _=_& ._ _ ___________________% 4_x_&_=_#* &_ _ ________69uare_o1_de1icit_=_# _=_#&&._ _ ________ ence_answer_is_#* &_ _#&&_=_#* *&&___;since_we_ha!e_taken_multiples_o1_# <._

"xample_ $ __& * _Nearest_base_=_* ._ _ ________/reat_* _as_*_x_# _and_proceed_ ______________

_____________ _______

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"xample_%$ ___4 _Nearest_base_=_ _ _

_______________ _ ___ _"xample_&$ ___&#4 _Nearest_(_lower_)_base_=_& _ _ ____________ ere_surplus_=_#4_and_& _=_&_x_# _ _

_________________ __________________

"xample_*$ ___* # _Nearest_lower_base_is_* _=_*_x_# _ _ ________6urplus_=_# _ _

____________ _

_

:pply_ya!adunam_to_1ind_the_1ollowin _s9uares._ _ ____________#.__ _____________ .__' ___________%.__' __________&.__#& __________ _ ____________*.__##4 ________4.__# # ________ .__#' ____________ .__& * __ _ ____________'.__ '4 ______# .__# ________##._'' ______# .__4 #&.

_

6o_1ar_we_ha!e_obser!ed_the_application_o1_ya!adunam_in_1indin _the_s9uares_o1_number. Now with a sli ht modi1ication ya!adunam can also be applied 1or

CONTASS3333S

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