in the formulae [ 1 ] and [ 2 ], the alkyl group represented by r 1 , r 2 or r 3 may be a linear , branched or cyclic alkyl group having a carbon number of 1 to 20 , preferably 1 to 15 , more preferably 1 to 10 . specific examples of the alkyl groups include a methyl group , an ethyl group , a n - propyl group , a 2 - propyl group , a n - butyl group , a 2 - butyl group , an isobutyl group , a tert - butyl group , a n - pentyl group , a 2 - pentyl group , a 2 - methylbutyl group , a 3 - methylbutyl group , a 2 , 2 - dimethylpropyl group , a n - hexyl group , a 2 - hexyl group , a 3 - hexyl group , a 2 - methylpentane - 2 - yl group , a 3 - methylpentane - 3 - yl group , a 2 - methylpentyl group , a 3 - methylpentyl group , a 4 - methylpentyl group , a 2 - methylpentane - 3 - yl group , a heptyl group , an octyl group , a 2 - ethylhexyl group , a nonyl group , a decyl group , a cyclopropyl group , a cyclobutyl group , a cyclopentyl group , a cyclohexyl group , etc . the alkenyl group represented by r 1 , r 2 or r 3 may be such that 1 or more double bond is introduced to the above alkyl groups having 2 or more carbon atoms . specific examples of the alkenyl groups include an ethenyl group , a 1 - propenyl group , a 2 - propenyl group , an isopropenyl group , a 1 - butenyl group , a 2 - butenyl group , a 1 , 3 - butadienyl group , a 2 - pentenyl group , a 2 - hexenyl group , a heptenyl group , an octenyl group , a nonenyl group , a decenyl group , a cyclopropenyl group , a cyclopentenyl group , a cyclohexenyl group , etc . the aryl group represented by r 1 , r 2 or r 3 may be an aryl group having 6 to 14 carbon atoms . specific examples of the aryl groups include a phenyl group , a naphthyl group , an anthryl group , a biphenyl group , etc . the substituent bonding to the alkyl , alkenyl or aryl group , i . e . substituent of a substituted alkyl group , a substituted alkenyl group or a substituted aryl group , may be any group that has no adverse affect on the asymmetric hydrogenation of the invention , and examples thereof include alkyl groups , alkoxy groups , aryl groups , halogen atoms , etc . the meanings and specific examples of the alkyl groups and the aryl groups as the substituent may be the same as those described above . the alkoxy group may be a linear , branched or cyclic group having 1 to 20 , preferably 1 to 10 , more preferably 1 to 6 carbon atoms . specific examples of the alkoxy groups include a methoxy group , an ethoxy group , a n - propoxy group , a 2 - propoxy group , a n - butoxy group , a 2 - butoxy group , an isobutoxy group , a tert - butoxy group , a n - pentyloxy group , a 2 - methylbutoxy group , a 3 - methylbutoxy group , a 2 , 2 - dimethylpropyloxy group , a n - hexyloxy group , a 2 - methylpentyloxy group , a 3 - methylpentyloxy group , a 4 - methylpentyloxy group , a 5 - methylpentyloxy group , a cyclohexyloxy group , etc . examples of the halogen atoms include a fluorine atom , a chlorine atom , a bromine atom , an iodine atom , etc . in the formulae [ 1 ] and [ 2 ], r 1 , r 2 and r 3 represent the above atom or group respectively , and it should be noted that r 1 , r 2 and r 3 is not a hydrogen atom simultaneously based on the definition that at least one of the two carbon atoms marked with * in the formula [ 2 ] represents an asymmetric carbon atom . further , r 3 is a group other than a hydrogen atom when one of r 1 and r 2 is a hydrogen atom , r 3 is a group other than a hydrogen atom and a methyl group when both of r 1 and r 2 are hydrogen atoms , and r 1 and r 2 are different groups other than a hydrogen atom when r 3 is a hydrogen atom . this is because , in the formula [ 2 ], the carbon atom bonding to r 1 and r 2 is not an asymmetric carbon atom in the case where r 1 and / or r 2 is a hydrogen atom , and the carbon atom bonding to r 3 is not an asymmetric carbon atom in the case where r 3 is a hydrogen atom or in the case where both of r 1 and r 2 are hydrogen atoms and r 3 is a methyl group . in the formula [ 3 ], arene represents a benzene or an alkyl - substituted benzene . examples of preferred alkyl - substituted benzenes include p - cymene , hexamethylbenzene , 1 , 3 , 5 - trimethylbenzene , etc . in the formulae [ 3 ] and [ 4 ], the alkaline metal atom represented by m may be a sodium atom , a potassium atom , etc . specific examples of the α , β - unsaturated carboxylic acids represented by the formula [ 1 ], used as a starting material in the methods of the invention , include 2 - methylbutenoic acid , 2 - methyl - 2 - pentenoic acid , 2 - methyl - 2 - hexenoic acid , 2 - ethyl - 2 - hexenoic acid , 2 - methyl - 2 - heptenoic acid , 2 - methyl - 2 - octenoic acid , etc . specific examples of the sulfonated binap — ru complexes represented by the formula [ 3 ] used in the methods of the invention include [ rui ( p - cymene ) {( so 3 na ) 2 — binap }] i , [ rubr ( p - cymene ) {( so 3 na ) 2 — binap }] br , [ rucl ( p - cymene ) {( so 3 na ) 2 — binap }] cl , [ rui ( c 6 h 6 ){( so 3 na ) 2 — binap }] i , [ rubr ( c 6 h 6 ) {( so 3 na ) 2 — binap }] br , [ rucl ( c 6 h 6 ){( so 3 na ) 2 — binap }] cl , etc . the sulfonated binap — ru complexes can be easily produced by the methods described in jp - a - 5 - 170780 . specific examples of the optically active carboxylic acids represented by the formula [ 2 ], obtainable by the methods of the invention , include ( 2r )- methylbutanoic acid , ( 2r )- methylpentanoic acid , ( 2r )- methylhexanoic acid , ( 2r )- ethylhexanoic acid , ( 2r )- methylheptanoic acid , ( 2r )- methyloctanoic acid , ( 2s )- methylbutanoic acid , ( 2s )- methylpentanoic acid , ( 2s )- methylhexanoic acid , ( 2s )- ethylhexanoic acid , ( 2s )- methylheptanoic acid , ( 2s )- methyloctanoic acid , etc . in the methods of the invention , the mole ratio of the sulfonated binap — ru complex represented by the formula [ 3 ] to the α , β - unsaturated carboxylic acid is appropriately selected generally from the range of 1 × 10 − 2 to 3 × 10 − 4 mol / mol , preferably from the range of 1 × 10 − 3 to 2 × 10 − 4 mol / mol . in the methods of the invention , the asymmetric hydrogenation is carried out in an aqueous solvent . the aqueous solvent is water or the two - phase mixed solvent of water and the water - insoluble organic solvent . specific examples of the water - insoluble organic solvents used in the methods of the invention include aliphatic hydrocarbons such as pentane , hexane heptane , octane , decane , and cyclohexane ; halogenated hydrocarbons such as methylene chloride , 1 , 2 - dichloroethane , chloroform , carbon tetrachloride , and 1 , 2 - dichlorobenzene ; ethers such as diethyl ether , diisopropyl ether , dimethoxyethane , ethylene glycol diethyl ether , tert - butyl methyl ether , and cyclopentyl methyl ether ; esters such as methyl acetate , ethyl acetate , n - butyl acetate , and methyl propionate ; etc . these solvents may be used alone or in appropriate combination of two or more solvents thereof . the amount of the water - insoluble organic solvent is appropriately selected generally from the range of 1 to 10 parts by weight , preferably from the range of 2 to 5 parts by weight , per 1 part by weight of the α , β - unsaturated carboxylic acid . water used in the methods of the invention may be distilled water , purified water , ion - exchange water , etc . water is preferably distilled and degassed . the amount of water is appropriately selected generally from the range of 1 to 25 parts by weight , preferably from the range of 1 to 15 parts by weight , per 1 part by weight of the α , β - unsaturated carboxylic acid . the amount of water remarkably affects the asymmetric hydrogenation rate depending on the carbon number of the α , β - unsaturated carboxylic acid . the amount of water may be 1 to 2 parts by weight in the case of tiglic acid having 5 carbon atoms , and the amount is 10 parts or more by weight in the case of 2 - ethylhexenoic acid having 8 carbon atoms . in the asymmetric hydrogenation of the invention , the hydrogen pressure is desirably 0 . 1 mpa or more , and appropriately selected generally from the range of 0 . 5 to 10 mpa , preferably from the range of 1 to 5 mpa , from the viewpoint of economical efficiency , etc . the reaction temperature in the methods of the invention is appropriately selected generally from the range of 30 to 100 ° c ., preferably from the range of 40 to 90 ° c . the reaction time depends on the conditions such as the reaction temperature , the amount of the sulfonated binap — ru complex , the amount of water , and the hydrogen pressure . the reaction time is appropriately selected generally from the range of 1 to 20 hours , preferably from the range of 3 to 10 hours . in the methods of the invention , an aqueous solution of the sulfonated binap — ru complex used in the asymmetric hydrogenation can be recovered and reused . thus , the sulfonated binap — ru complex can be recycled ( reused ) in the methods of the invention . the sulfonated binap — ru complex or the aqueous solution thereof may be recovered by a common operation from the reaction solution ( reaction system ). specifically , the aqueous solution of the sulfonated binap — ru complex may be recovered by separating the water phase from the two - phase reaction solution after the asymmetric hydrogenation . further , the sulfonated binap — ru complex can be easily recovered from the separated water phase by concentration , etc . the recovered aqueous solution of the sulfonated binap — ru complex ( the water phase separated after the asymmetric hydrogenation ) may be directly reused ( recycled ) without aftertreatments and purifications for the asymmetric hydrogenation of the α , β - unsaturated carboxylic acid . the isolated or recovered sulfonated binap — ru complex may be reused for the asymmetric hydrogenation of the α , β - unsaturated carboxylic acid or for other asymmetric hydrogenation after aftertreatment , purification , etc . in the case where the recovered sulfonated binap — ru complex , which may be the water phase containing the sulfonated binap — ru complex recovered from the reaction solution ( reaction system ) or the sulfonated binap — ru complex isolated from the water phase , is recycled for the asymmetric hydrogenation of the α , β - unsaturated carboxylic acid to produce the optically active carboxylic acid , the amount of the sulfonated binap — ru complex may be appropriately controlled if necessary by adding further sulfonated binap — ru complex , etc . thus - obtained optically active carboxylic acid is useful as pharmaceutical intermediates , liquid crystal materials , etc . the present invention will be described in more detail below with reference to examples without intention of restricting the scope of the invention . 10 g ( 0 . 1 mol ) of tiglic acid ( available from tokyo kasei kogyo co ., ltd .) and 8 . 7 mg ( 6 . 6 × 10 − 3 mmol ) of [ rui ( p - cymene ){( r )—( so 3 na ) 2 binap }] i were put in a 200 ml autoclave , and the atmosphere in the autoclave was replaced with nitrogen . 20 ml of methylene chloride , which was degassed and distilled while blocking air flow by nitrogen , and 10 ml of degassed distilled water were added to the mixture , and tiglic acid was reacted at 80 ° c . for 4 hours under the hydrogen pressure of 2 . 5 mpa . the temperature of the autoclave was lowered to the room temperature , hydrogen was discharged , and nitrogen was flowed in the autoclave for approximately 30 minutes to remove the remaining hydrogen . the reaction solution was taken out of the autoclave and left for approximately 30 minutes . the reaction solution was separated into two layers , the oil phase of the lower layer and the water phase of the upper layer . the methylene chloride solution in the lower layer was isolated and the water phase was extracted with methylene chloride once . the methylene chloride solutions were mixed , dried over anhydrous magnesium sulfate , and concentrated to recover the solvent , whereby 9 . 8 g of crude ( 2r )- methylbutanoic acid was obtained . the crude ( 2r )- methylbutanoic acid was distilled to obtain 9 . 3 g of purified ( 2r )- methylbutanoic acid : boiling point 85 ° c ./ 11 mmhg ; gc purity 99 . 7 %; optical purity 94 . 8 % ee ; optical rotation [ α ] d 20 - 19 . 5 ( c 1 . 04 , meoh ); mass spectrum ( 20 ev , m / e ) 29 , 41 , 55 , 56 , 57 , 73 , 74 , 87 , and 103 ( m + + 1 ). 10 g ( 0 . 1 mol ) of tiglic acid ( available from tokyo kasei kogyo co ., ltd .) and 11 . 3 mg ( 1 × 10 − 2 mmol ) of [ rucl ( p - cymene ){( r )—( so 3 na ) 2 binap }] cl were put in a 200 ml autoclave , and the atmosphere in the autoclave was replaced with nitrogen . 40 ml of degassed distilled diisopropyl ether and 20 ml of degassed distilled water were added to the mixture , and tiglic acid was reacted at 80 ° c . for 3 hours under the hydrogen pressure of 2 . 5 mpa . the temperature of the autoclave was lowered to the room temperature , hydrogen was discharged , and nitrogen was flowed in the autoclave for approximately 30 minutes to remove the remaining hydrogen . then , the reaction solution was ejected from a sampling hole of the autoclave into a 100 ml glass syringe having a needle with the inside diameter of 1 . 5 mm under nitrogen flow utilizing the nitrogen pressure , and left for approximately 30 minutes . the reaction solution was separated into two layers , the organic phase of the upper layer and the water phase of the lower layer . the water phase was isolated and returned into the autoclave , and sealed under nitrogen to be reused in the next reaction . on the other hand , the oil phase was isolated , dried over anhydrous magnesium sulfate , and concentrated to recover the solvent , whereby 9 . 61 g of a residue was obtained . the residue was distilled to obtain 9 . 3 g of purified ( 2r )- methylbutanoic acid : boiling point 83 ° c ./ 10 mmhg ; gc purity 99 . 6 %; optical purity 92 . 5 % ee ; optical rotation [ α ] d 20 - 19 . 2 ( c 1 . 07 , meoh ). then , a solution of 10 g ( 0 . 1 mol ) of tiglic acid and 40 ml of degassed distilled diisopropyl ether was added into the autoclave containing the water phase used in the above reaction while blocking the air . the tiglic acid was reacted for 3 hours under the same conditions as the above reaction , and subjected to the aftertreatments in the same manner as above to obtain 10 . 2 g of crude ( 2r )- methylbutanoic acid : gc purity 99 . 47 %; enantiomer excess 92 . 5 % ee . the asymmetric hydrogenation of tiglic acid was repeated 4 times such that the water phase was isolated under nitrogen after the reaction and recycled in the same manner as above . it took 4 hours and 5 hours to complete the third and fourth reactions recycling the water phase , respectively . it was considered that the reaction rate was lowered because the water phase containing the catalyst was mixed in the organic phase and removed with the organic phase . the results of the reactions recycling the water phase are shown in table 1 . 11 . 4 g ( 0 . 1 mol ) of trans - 2 - methyl - 2 - pentenoic acid ( available from tokyo kasei kogyo co ., ltd .) and 59 . 3 mg ( 4 . 5 × 10 − 2 mmol ) of [ rui ( p - cymene ){( r )—( so 3 na ) 2 binap }] i were put in a 200 ml autoclave , and the atmosphere in the autoclave was replaced with nitrogen . 20 ml of degassed distilled water and 22 ml of degassed methylene chloride were added to the mixture , and trans - 2 - methyl - 2 - pentenoic acid was reacted at 80 ° c . for 6 hours under the same hydrogen pressure as example 1 , to obtain 11 . 2 g of crude ( 2r )- methylpentanoic acid . the crude ( 2r )- methylpentanoic acid was distilled to obtain 10 . 5 g of purified ( 2r )- methylpentanoic acid : boiling point 105 ° c ./ 11 mmhg ; gc purity 99 . 1 %; optical purity 89 . 6 % ee ; optical rotation [ α ] d 20 - 17 ( c 1 . 0 , meoh ); mass spectrum ( 20 ev , m / e ) 41 , 43 , 45 , 55 , 56 , 71 , 73 , 74 , 87 , 101 , and 117 ( m + + 1 ) 12 . 8 g ( 0 . 1 mol ) of trans - 2 - methyl - 2 - hexenoic acid ( available from tokyo kasei kogyo co ., ltd .) and 66 mg ( 5 × 10 − 2 mmol ) of [ rui ( p - cymene ){( r )—( so 3 na ) 2 binap }] i were put in a 200 ml autoclave , and the atmosphere in the autoclave was replaced with nitrogen . 89 . 6 ml of degassed distilled water and 25 . 6 ml of degassed methylene chloride were added to the mixture , and trans - 2 - methyl - 2 - hexenoic acid was reacted at 80 ° c . for 5 hours under the same hydrogen pressure as example 1 , to obtain 12 . 9 g of crude ( 2r )- methylhexanoic acid . the crude ( 2r )- methylhexanoic acid was distilled to obtain 11 . 8 g of purified ( 2r )- methylhexanoic acid : boiling point 116 ° c ./ 11 mmhg ; gc purity 99 . 4 %; optical purity 89 . 3 % ee ; optical rotation [ α ] d 20 - 18 . 7 ( c 1 . 05 , meoh ); mass spectrum ( 20 ev , m / e ) 41 , 43 , 55 , 56 , 57 , 69 , 73 , 74 , 75 , 85 , 87 , 101 , 113 , and 131 ( m + + 1 ). 14 . 2 g ( 0 . 1 mol ) of 2 - ethyl - 2 - hexenoic acid ( available from aldrich , trans : 94 %, cis : 4 . 83 %) and 53 mg ( 4 . 66 × 10 − 2 mmol ) of [ rui ( p - cymene ) {( r )—( so 3 na ) 2 binap }] i were put in a 500 ml autoclave , and the atmosphere in the autoclave was replaced with nitrogen . 210 ml of degassed distilled water and 28 . 4 ml of degassed methylene chloride were added to the mixture , and 2 - ethyl - 2 - hexenoic acid was reacted at 80 ° c . for 8 hours under the same hydrogen pressure as example 1 , to obtain 13 . 9 g of crude ( 2r )- ethylhexanoic acid . the crude ( 2r )- ethylhexanoic acid was distilled to obtain 13 . 5 g of purified ( 2r )- ethylhexanoic acid : boiling point 125 ° c ./ 11 mmhg ; gc purity 99 . 1 %; optical purity 86 . 4 % ee ; optical rotation [ α ] d 20 - 9 . 1 ( c 1 . 01 , meoh ); mass spectrum ( 20 ev , m / e ) 41 , 43 , 45 , 55 , 57 , 73 , 87 , 88 , 101 , 115 , 116 , and 145 ( m + + 1 ). 20 g ( 0 . 2 mol ) of tiglic acid and 26 . 3 mg ( 1 × 10 − 2 mmol ) of [ rui ( p - cymene ){( r )—( so 3 na ) 2 binap }] i were put in a 200 ml autoclave , and the atmosphere in the autoclave was replaced with nitrogen . 80 ml of degassed distilled water were added to the mixture , and tiglic acid was reacted at 60 ° c . for 3 hours under the hydrogen pressure of 1 . 8 mpa . the temperature of the autoclave was lowered to the room temperature , hydrogen was discharged , and nitrogen was flowed in the autoclave for approximately 30 minutes to remove the remaining hydrogen . then , the reaction solution was ejected from a sampling hole of the autoclave into a 100 ml glass syringe having a needle with the inside diameter of 1 . 5 mm under nitrogen flow utilizing the nitrogen pressure , and left for approximately 30 minutes . the reaction solution was separated into two layers , the organic phase of the upper layer and the water phase of the lower layer . the water phase was isolated and returned into the autoclave , and sealed under nitrogen to be reused in the next reaction . on the other hand , the oil phase was isolated , dried over anhydrous magnesium sulfate , and concentrated to recover the solvent , whereby a residue was obtained . the residue was distilled to obtain purified ( 2r )- methylbutanoic acid . the results are shown in table 2 . then , a solution of 20 g ( 0 . 2 mol ) of tiglic acid was added into the autoclave containing the water phase used in the above reaction and 0 . 8 mg of [ rui ( p - cymene ) {( r )—( so 3 na ) 2 binap }] i while blocking the air . the tiglic acid was reacted for 3 hours under the same conditions as the above reaction , and subjected to the aftertreatments in the same manner as above to obtain ( 2r )- methylbutanoic acid . the asymmetric hydrogenation of tiglic acid was repeated 10 times such that the water phase was isolated under ogen after the reaction and recycled in the same manner bove . the results of the reactions recycling the water phase shown in table 2 . as the number of recycling times increased , the conversion declined , however , the problem was resolved by extending the reaction time . as for the conversion , no advantage of using distilled material was identified , though as for the optical purity , its elevation was seen and maintained 93 % ee , even in cases in which the number of recycling times increased . the asymmetric hydrogenation was carried out in the same manner as described in example 1 , except that the amount of [ rui ( p - cymene ){( r )—( so 3 na ) 2 binap }] i and reaction time were replaced as those in table 3 , and obtained results as described in table 3 . in the methods of the invention , the asymmetric hydrogenation of the α , β - unsaturated carboxylic acid is carried out in water or the two - phase system of water and an organic solvent to obtain a desired optically active carboxylic acid with high optical purity , whereby the methods do not require complicated operations of isolating the produced optically active carboxylic acid and the sulfonated binap — ru complex to be excellent in workability . further , the methods of the invention can remarkably reduce the costs , can utilize the catalyst efficiently , and are excellent in the workability , because the sulfonated binap — ru complex used in the asymmetric hydrogenation can be recovered and reused without complicated recovering processes . furthermore , the recovered water phase can be directly reused , and thus , the methods require less labor and costs , thereby further improving the workability .