ÿØÿà JFIF 

  
 
  ÿÛ „ 	 ( %"1!%)+...383,7(-.+




-+++--++++---+-+-----+---------------+---+-++7-----ÿÀ  ß â
" 

ÿÄ   

           
 ÿÄ H 
     
 !1AQaq"‘¡2B±ÁÑð#R“Ò Tbr‚²á3csƒ’ÂñDS¢³$CÿÄ 
 


            
ÿÄ %

 
       
!1AQa"23‘ÿÚ 
  ? ôÿ ¨pŸªáÿ —åYõõ\?àÒü©ŠÄï¨pŸªáÿ —åYõõ\?àÓü©ŠÄá 0Ÿªáÿ Ÿå[úƒ	ú®ði~TÁb
qÐ8OÕpÿ ƒOò¤Oè`–RÂáœá™êi€ßÉ< FtŸI“öÌ8úDf´°å}“¾œ6
öFá°y¥jñÇh†ˆ¢ã/ÃÐ:ªcÈ
"Y¡ðÑl>ÿ ”ÏËte:qž\oäŠe÷󲍷˜HT4&ÿ ÓÐü6ö®¿øþßèô	Ÿ•7Ñi’•j|“ñì>b…þS?*Óôÿ ÓÐü*h¥£ír¶ü UãSç‚Ÿ[AÐaè[ûª•õ&õj?†Éö+EzP—WeÒírJFt	‘BŒ†Ï‡%#tE Øz ¥OÛ«!1›üä±Í™%º<!vB™'èÔ6´Ña¿«e`Àà0U[-ÂБ4)M÷YK*³hï÷%ÎÎkÇ«m‘-íWŽLsã?nÂ~«†üö®þ¡ÂlÃaÿ ‰ÿ l©°Õ¥ åÜá¿r ºL[Çâ°(Çè'Vãôl8´ÏRÀvLvPõ:…ÙNçF™÷ë>Íãö]°î(–:@<‹ŒÊö×òÆt¦ãº+‡¦%ÌÁ²h´OƒJŒtMÜ>ÀÜÊw3
Y´•牋4ǍýʏTì>œú=Íwhyë,¾Ôò×õ¿ßÊa»«þˆѪQ|%6ž™A	õ%:øj<>É—ÿ Å_ˆCbõ¥š±ý¯Ýƒï…¶|RëócÍf溪“t.СøTÿ *Ä¿-{†çàczůŽ_–^XþŒ±miB[X±d 1,é”zEù»&
î9gœf™9Ð'.;—™i}!ôšåîqêÛ٤ёý£½ÆA–àôe"A$˝Úsäÿ 
÷Û	#°xŸëí(l
	 »ý3—¥5m!
rt`†0~'j2(]S¦¦kv,ÚÇl¦øJA£Šƒ
J3E8ÙiŽ:cÉžúeZ°€¯\®kÖ(79«Ž:¯X”¾³Š&¡* ….‰Ž(ÜíŸ2¥ª‡×Hi²TF¤ò[¨íÈRëÉ䢍mgÑ.Ÿ<öäS0í„ǹÁU´f#Vß;Õ–…P@3ío<ä-±»Ž.L|kªÀê›fÂ6@»eu‚|ÓaÞÆŸ…¨ááå>åŠ?cKü6ùTÍÆ”†sĤÚ;H2RÚ†õ\Ö·Ÿn'¾ ñ#ºI¤Å´%çÁ­‚â7›‹qT3Iï¨Ö
Ú5I7Ë!ÅOóŸ¶øÝñØôת¦$Tcö‘[«Ö³šÒ';Aþ ¸èíg
A2Z
"i¸vdÄ÷.iõ®§)¿]¤À†–‡É&ä{V¶iŽ”.Ó×Õÿ û?h¬Mt–íª[ÿ Ñÿ ÌV(í}=ibÔ¡›¥¢±b
 Lô¥‡piη_Z<‡z§èŒ)iÖwiÇ 2hÙ3·=’d÷8éŽ1¦¸c¤µ€7›7Ø	ð\á)}	¹fËí›pAÃL%âc2 í§æQz¿;T8sæ°qø)QFMð‰XŒÂ±
N¢aF¨…8¯!U		Z©RÊ ÖPVÄÀÍin™Ì-GˆªÅËŠ›•zË}º±ŽÍFò¹}Uw×#ä5B¤{î}Ð<ÙD
é©¤&‡ïDbàÁôMÁ."
¤‡ú*õ'VŽ|¼´Úgllº¼klz[Æüï÷Aób‡Eÿ dÑ»Xx9ÃÜ£ÁT/`¼¸vI±Ýµ·Ë‚“G³þ*Ÿû´r|*}<¨îºœ @¦mÄ’M¹”.œ«Y–|6ÏU¤jç¥ÕÞqO
˜kDÆÁ¨5ÿ š;Њ¦¦€GÙk \
	–Þ=â¼=SͧµªS°ÚÍpÜãQűÀõ¬?ÃÁ1Ñ•õZà?hóœ€
L¦l{Y*K˜Ù›zc˜–ˆâø+¾	­-Ök¥%ùEÜA'}ˆ><ÊIè“bpÍ/qÞâvoX€w,\úªò6Z[XdÒæ­@Ö—€$òJí#é>'°Ú
ôª˜<)4ryÙ£|óAÅn5žêŸyÒäMÝ2{"}‰–¤l÷ûWX\l¾Á¸góÉOÔ /óñB¤f¸çñ[.P˜ZsÊË*ßT܈§QN¢’¡¨§V¼(Üù*eÕ“”5T¨‹Âê¥FŒã½Dü[8'Ò¥a…Ú¶k7a
*•›¼'Ò·\8¨ª\@\õ¢¦íq+DÙrmÎ…_ªæ»ŠÓœ¡¯’Ré9MÅ×D™lælffc+ŒÑ,ý™ÿ ¯þǤ=Å’Á7µ÷ÚÛ/“Ü
€ñýã¼àí¾ÕÑ+ƒ,uµMâÀÄbm:ÒÎPæ{˜Gz[ƒ¯«® KHà`ßØŠéí¯P8Aq.C‰ à€kòpj´kN¶qô€…Õ,ÜNŠª-­{Zö’æû44‰sŽè‰îVíRœÕm"
6?³D9¡ÇTíÅꋇ`4«¸ÝÁô ï’ýorqКÇZ«x4Žâéþuïf¹µö[P ,Q£éaX±`PÉÍZ ¸äYúgüAx
’6Lê‚xÝÓ*äQ
Ï’¨hÍ
=²,6ï#rÃ<¯–£»ƒ‹,–ê•€ aÛsñ'%Æ"®ÛüìBᝠHÚ3ß°©$“XnœÖ’î2ËTeûìxîß
¦å¿çÉ
ðK§þ{‘t‚Ϋ¬jéîZ[”š7L¥4VÚCE×]m¤Øy”ä4-dz£œ§¸x.*ã
ÊÊ b÷•h:©‡¦s`BTÁRû¾g⻩‹jø sF¢àJøFl‘È•Xᓁà~*j¯ +(ÚÕ6-£¯÷GŠØy‚<Ç’.F‹Hœw(+)ÜÜâÈzÄäT§FߘãÏ;DmVœ3Àu@mÚüXÝü•3B¨òÌÁÛ<·ÃÜz,Ì@õÅ·d2]ü8s÷IôÞ¯^Ç9¢u„~ëAŸï4«M?
K]­ÅàPl@s_ p:°¬ZR”´›JC[CS.h‹ƒïËœ«Æ]–÷ó‚wR×k7X‰k›‘´ù¦=¡«‰¨¨Â')—71ó’c‡Ðúµ
`é.{§p¹ój\Ž{1h{o±Ý=áUÊïGÖŒõ–-BÄm+AZX¶¡
ïHðæ¥JmÙ;…ä¡Ÿˆ¦ ° äšiÉg«$üMk5¤L“’çÊvïâï	,=f“"íἊ5ô¬x6{ɏžID0e¸vçmi'︧ºð9$ò¹÷*£’9ÿ ²TÔ…×>JV¥}Œ}$p[bÔ®*[jzS*8
”·T›Í–ñUîƒwo$áè=LT™ç—~ô·¤ÈÚ$榍q‰„+´kFm)ž‹©i–ËqÞŠ‰à¶ü( ‚•§ •°ò·‡#5ª•µÊ﯅¡X¨šÁ*F#TXJÊ
ušJVÍ&=iÄs1‚3•'fý§5Ñ<=[íÞ­
PÚ;ѱÌ_~Ä££8rÞ
²w;’hDT°>ÈG¬8Á²ÚzŽ®ò®qZcqJêäÞ-ö[ܘbň±çb“Š¶31²n×iƒðÕ;1¶þÉ
ªX‰,ßqÏ$>•î íZ¥Z 1{ç൵+ƒÕµ¥°T$§K]á»Ûï*·¤tMI’ÂZbŽÕiÒ˜}bÓ0£ª5›¨[5Ž^ÝœWøÂÝh°
¢OWun£¤5 a2Z.G2³YL]jåtì”
ä	ÁÓ‘%"©<ÔúÊ°sº UZvä‡ÄiÆÒM .÷V·™ø#kèýií<NçŽôª¦„lË»{ƒ®<
[¦Y‹éƒ}j•*‚±€I‡e&gbéšp9å^!®€ö?ù^àTzo@Q¨Aêœ& [”…üÔ}пF’Hq‚©´{“ëH·?.½%úå»É Á´9¿¼Ò‰Á¼Ì¥Ús¼zW¶aÃhwÖ…ÑÏ$mò´ë:×Ñγ|”x§
µ0Âз$·aÆn µ†šÇ³£‰Ì®ž
õ@kŽ«gtí䢩ÒP×5¤?µÿ m´Ì^ ‡8¸Á8‹”žÆWÑ ¡™„,Ì“»à‹ÃéV¸®á›ÐÇŽBÀ®Ã…Z”ØÛë8;é|Ñ5)7aÕ&2Ù¸÷©ö—Ð;‡¬7"¿"7M·d
ñœÒÌf5”ê ;"Ãp5¯¬3‹ùJÚÝG›½&¯<{òñKÎÖ“3:¦ÀòGPÅS¨;
à‘Ün´öpð±W)z'¬öÀÖ#Xf2& æ
Œ‹é¾"‹\!Ñ9…
0±2A›*ØMÄ $X
끫ØXÖ,X¥ ô'ËnFÍãlqJXçuaJ4¦„ëµ7ê;h"Zã¾Áä£,vÛ‹’N©E(Öu÷)…ÁaÜÇÔkˆ$‘1kmG¨øtÛ>Ì–ª)µT[)BˆõÑ
xB¾B€ÖT¨.¥~ð@VĶr#¸ü*åZNDŽH;âi 
],©£öØpù(šºãö¼T.uCê•4@ÿ GÕÛ)Cx›®0ø#:ÏðFÒbR\(€€Ä
®fã4Þ‰Fä¯HXƒÅ,†öEÑÔÜ]Öv²?tLÃvBY£ú6Êu5ÅAQ³1‘’¬x–HŒÐ‡ ^	¸KwJôÖŽ5×CÚ¨vÜ«/B0$×k°=ðbÇ(Ï)w±A†Á†	11Í=èQšµ626ŒÜ/`G«µ<}—-Ö7KEHÈÉðóȤmݱû±·ø«Snmá=“ä«šmݱŸ¡¶~ó·“äUóJæúòB|E LêŽy´jDÔ$G¢þÐñ7óR8ýÒ…Ç›	WVe#·Ÿ	p·Fx~•ݤF÷0Èÿ K¯æS<6’¡WШ
;´ÿ ¥Êø\Òuî†åÝ–VNœkÒ7oòX¨Á­Ø÷FÎÑä±g÷ÿ M~Çî=p,X´
ÝÌÚÅ‹
’ÃjÖ.ØöÏñ
qïQ¤ÓZE†° =6·]܈ s¸>v•Ž^Ý\wq9r‰Î\¸¡kURÒ$­*‹Nq?Þª*!sŠÆ:TU_u±T+øX¡
®¹¡,ÄâÃBTsÜ$Ø›4m椴zÜK]’’›Pƒ@€#â˜`é¹=I‡fiV•Ôî“nRm+µFPOhÍ0B£
€+¬5c
v•:P'ÒyÎ ‰V~‚Ó†ÖuókDoh$å\*ö%Ю=£«…aȼ½÷Û.-½VŒŠ¼'lyî±1¬3ó#ÞE¿ÔS¤gV£m›<HqÚ,sVºóÆß¼Hïô…¹…Méi‡Ò" µùEîØÏj|ÓøVOuœ"f’aÞÞëx…˜ÌO`dHHš×4K*H$‡5×
ÚÄ
Â×CUÅAŠŒsr»2°ÄÁiÚ±Çéõe•¶_SÕÆÃ3ˆ•-3´‘¸¡‡cgDîÚ¦ÃcX}'êûJè½9ñìÓ­ä±	×ÓûçÁmO”üVÞ¸÷¥ I~“ÒôhhûýÑwxlïZ¹G¡4ž=”)º£ò
®vÆŽ%UqÝ4qž©­hûÎí‡š§é1R±cÜèðà y' °ôy¦£q§Ýï¬;šÖõ„Sÿ ›  z('S†$Ï}*evÚ¾Kýº8ý
¨å	jÛ]´-‚6«¶ÖK+8ÊiS45jrœèËQDüDØ]R”˜SPÃ
xJÕl>
=§\û"—WU¤ÚǼÿ ÂnÁ
GŒÃ
‚õN	D³õNÚíŒÕ;HôyÄÈ©P¹Ä{:?R‘Ô¨âF÷ø£bÅó® JS|‚R÷ivýáâ€Æé¡è³´IئÑT!§˜•Øª‚¬â@q€wnïCWÄ@JU€ê¯m6]Ï:£âx
'+ÒðXvÓ¦Úm=–´7œ $ì“B£~p%ÕŸUþ« N@¼üï~w˜ñø5®—'Ôe»¤5ã//€ž~‰Tþ›Å7•#¤×Íö
pÄ$ùeåì*«ÓŠEØWEÈsßg¦ûvžSsLpºÊW–âµEWöˬH;™!CYõZÃÄf æ#1W.
\uWâ\,\Çfj’<qTbên›Î[vxx£ë 'ö¨1›˜ÀM¼Pÿ H)ƒêêŒA7s,|F“꺸k³9Ìö*ç®;Ö!Ö$Eiž•¹ÒÚ†ýóéÝû¾ÕS®ó$’NÝäŸz¤5r¦ãÄÃD÷Üø!°ø‡Ô&@m™Ì^Ãä­d	q5Lnÿ N;.6½·N|#ä"1Nƒx“ã<3('&ñß
t  ~ªu”1Tb㫨9ê–›–bìd$ߣ=#ÕãÒmU¯eí$EFù5ýYô櫨æ왍Ǘ±ssM]·á¿
0ÕåJRÓªîiƒ+O58ÖñªŠÒx"	\µâá¨i’¤i
—Ö
”
M+M¤ë9‚‰A¦°Qõ¾ßøK~¼Ã‘g…Ö´~÷Ï[3GUœÒ½#…kàÔ®Ò”‰³·dWV‰IP‰Ú8u¹”E
ÖqLj¾êÕCBš{A^Âß;–¨`¯¬ìö
˼	×tìø.tƐm*n¨y4o&Àx¥n¦×î‡aupáÛj8¿m›è¶ã!o½;ß0y^ý×^EÑ¿ÒjzŒ­)vÚÑnÄL
…^ªô× ‡—‚3k	Îý­hï]içå–îÏ*÷ñþ»Ô
CÒjøjÍznˆ´
¹#b'Fô‹
‰v¥'’à'T´ƒHýÍ%M‰ ƒ&ÆÇŒï1 ‘	–Þ
‰i¬s žR-Ÿ kŠ¬á¬7:þ 0ŒÅÒÕ/aÙ¬ÃÝ#Úøœ
©aiVc‰. ¹¦ãµ”
›Yg¦›ÆÎýº°f³7ƒhá·¸­}&D9¡ÂsÉÙÞèŠõØàC™¨ñbFC|´Ü(ŸƒÚÒ-%»'a	Ì¿)ËÇn¿úÿ ÞŽX…4ÊÅH^ôΑí@ù¹Eh¶“L8Çjù
¼ÎåVªóR©Ï5uÃ
V4lZß®=€xÖŸ–ÑÈ
÷”¨°¾__yM1tÉ?uÆþIkÄgæ@þ[¢†°XÃJ£j·:nkÅ¢u‘}âGzö­/IµèŠ¬¼48q¦F°ŽR¼=ûì{´¯RýicS
ÕÛ íNtÍÙï£,w4rêì®»~x(©Uñ§#Ñ&œÕ¤>ÎåÍÓ9’Ö{9eV­[Öjâ²ãu]˜å2›qÑšÕJç0€sÄ|Êëè0튔bÁ>“{×_F`Ø©ºê:µä,v¤ðfc1±"«ÔÍän1#=·
Âøv~H<Q-Ìk7Æ;•L´©ÙC°.7
¯hKL@ävÊâž$´4¸ð‚¦­R–ÂZ7ÕÃj±·¿rÓrªJ_PÕ.#g¤gÀ)©`ª<KžèÝ'ä©èÒ3rwû€Ø¦ÒƒLf¤
,¿I½{Ƕ8‘$KÝU/Oi>½ÐßA¾¿Ü€Óš]Õ; I¾÷ç‚Qi†î¹9ywÔKG˜áñ zQY—§ÃÕZ07§X‚Áh;ÁM)iÌCH-¯T‘ë|A0{Ò½LÚ–TâÖkÜ’dÀ“rmm»”جPF³ÖcbE§T€ÒxKºû’Ó®7±²(\4ŽÃ¸Uu@j™yĵ;³µ!Á¢b.W¤=mõ´êµKk
¸K^ÜÛ#p*Ü14qkZç5ïë†°5Ï%ÍÛ<Õ¤×Ô¥ê†C	Õ´¼ú$ƒÖ“”]Ù¬qÞÚ[4©ý!ûÏ—Áb쳐XµA¬â~`›Çr¸8ìùÝ䫦<>ä÷«?xs´ÇÑ	/á;¹øüÊÈÙà{"@Žïzâ¬[âß‚ U_<ÇŸ½4èN˜ú61®qŠu ¦þF£»äJ_ˆÙÎ~ ÞAã–Ý„Ï—rŠD;xTž‘ô`É«…suãO`?³à™ô
Lý#Íc5öoæØ‚y´´÷«ZR§<&JÇ+éâô´€i!Àˆ0æAoàðLèÖ-2ŸõW.’t^–(KÁmHµV@xÜÇy®Ñø­â^:Ú3w· 7½¹°ñ¸â¹®:',«Mœ—n­Á+Ãbš
LÈ‘ÄnRÓÅœ%¦²‰¨ùQ:¤f‚ "PÕtô¸…cæl…&˜Ú˜Ôkv‹ž+vŠ,=¢v­6—X
y*¥t£«<™:“aï<QÝ&Ó]i4音äzçáíHÚÕxcòãæäÝÔm퐡 `Â)C×ehÀdHP6ÅMEöZ®Í¨	é¾Tr뢘6*¶;´sÙì]ÊŽŽÛíÉwRÈ´up* Ã	‚m´:Û{·”MUë©õ̤êÔo7:ŽÕ
qÏ'E ˆk^uãT€$Ú$Æ{•ŸBP°al¾Ž©ŒÃ^}äA
,º­$Ü]ºÚ.í:‹¤ÜöXnno7X€¥BÂà ±1ã>Œºîϲ=¦6rO]XI¿Œ÷¤zÚ­›¶ 6÷”w\dü~v®ˆÌk«^m<ÿ ¢‰Õ\)ùºŽ;… lîÙÅEŠ®cѾ@vnMÏ,¼“
ñ•ŽBxðÃzãÇç%3ˆ"}Ù•Åî>
BÉú;Ò]V+P˜F_´ßé>
Øše|ï‡ÄOmFæÇ ãqÞ$/xÐx­z`ï9"œÜij‚!7.\Td…9M‡•iŽ‹¾‘50ÞŽn¥ß4ÉôO¹*í^QêËÜÇÌ8=Ž
§s‰'ÂëÙ«á%Pú[O
†ÅP¯VsŽ°.‰,kc¶ ¬A9n˜XÎ-ÞšN["¹QÕ‰ƒMýÁߺXJæÍaLj¾×Ãmã¾ãÚ uñÒþåQô¦¥
/ÄUx:‚ÍÜ’ Đ©ØÝ3V¨‰ÕnÐ6ó*óúK­«…c ¯U òhsý­jóÔj#,ímŒRµ«lbïUTŒÑ8†Ä0œÏr`ð¡¬É	Ї
ë"À²

™	6¥ f¶¢ÚoܱԷ-<Àî)†a¶ž'Ú»¨TXqØæ¶÷Y
ÄHy˜9ÈIW­YÀuMFë
ºÏ’AqÌ4·/Ú
†ô'i$øä­=Ä
Ý|öK×40è|È6p‘0§)o¥ctî§H+CA-“ xØ|ÐXŠç
l8íºð3Ø:³¤¬KX¯UÿÙ
<?php

declare(strict_types=1);

namespace Brick\Math;

use Brick\Math\Exception\DivisionByZeroException;
use Brick\Math\Exception\MathException;
use Brick\Math\Exception\NumberFormatException;
use Brick\Math\Exception\RoundingNecessaryException;

/**
 * An arbitrarily large rational number.
 *
 * This class is immutable.
 *
 * @psalm-immutable
 */
final class BigRational extends BigNumber
{
    /**
     * The numerator.
     *
     * @var BigInteger
     */
    private $numerator;

    /**
     * The denominator. Always strictly positive.
     *
     * @var BigInteger
     */
    private $denominator;

    /**
     * Protected constructor. Use a factory method to obtain an instance.
     *
     * @param BigInteger $numerator        The numerator.
     * @param BigInteger $denominator      The denominator.
     * @param bool       $checkDenominator Whether to check the denominator for negative and zero.
     *
     * @throws DivisionByZeroException If the denominator is zero.
     */
    protected function __construct(BigInteger $numerator, BigInteger $denominator, bool $checkDenominator)
    {
        if ($checkDenominator) {
            if ($denominator->isZero()) {
                throw DivisionByZeroException::denominatorMustNotBeZero();
            }

            if ($denominator->isNegative()) {
                $numerator   = $numerator->negated();
                $denominator = $denominator->negated();
            }
        }

        $this->numerator   = $numerator;
        $this->denominator = $denominator;
    }

    /**
     * Creates a BigRational of the given value.
     *
     * @param BigNumber|int|float|string $value
     *
     * @return BigRational
     *
     * @throws MathException If the value cannot be converted to a BigRational.
     *
     * @psalm-pure
     */
    public static function of($value) : BigNumber
    {
        return parent::of($value)->toBigRational();
    }

    /**
     * Creates a BigRational out of a numerator and a denominator.
     *
     * If the denominator is negative, the signs of both the numerator and the denominator
     * will be inverted to ensure that the denominator is always positive.
     *
     * @param BigNumber|int|float|string $numerator   The numerator. Must be convertible to a BigInteger.
     * @param BigNumber|int|float|string $denominator The denominator. Must be convertible to a BigInteger.
     *
     * @return BigRational
     *
     * @throws NumberFormatException      If an argument does not represent a valid number.
     * @throws RoundingNecessaryException If an argument represents a non-integer number.
     * @throws DivisionByZeroException    If the denominator is zero.
     *
     * @psalm-pure
     */
    public static function nd($numerator, $denominator) : BigRational
    {
        $numerator   = BigInteger::of($numerator);
        $denominator = BigInteger::of($denominator);

        return new BigRational($numerator, $denominator, true);
    }

    /**
     * Returns a BigRational representing zero.
     *
     * @return BigRational
     *
     * @psalm-pure
     */
    public static function zero() : BigRational
    {
        /**
         * @psalm-suppress ImpureStaticVariable
         * @var BigRational|null $zero
         */
        static $zero;

        if ($zero === null) {
            $zero = new BigRational(BigInteger::zero(), BigInteger::one(), false);
        }

        return $zero;
    }

    /**
     * Returns a BigRational representing one.
     *
     * @return BigRational
     *
     * @psalm-pure
     */
    public static function one() : BigRational
    {
        /**
         * @psalm-suppress ImpureStaticVariable
         * @var BigRational|null $one
         */
        static $one;

        if ($one === null) {
            $one = new BigRational(BigInteger::one(), BigInteger::one(), false);
        }

        return $one;
    }

    /**
     * Returns a BigRational representing ten.
     *
     * @return BigRational
     *
     * @psalm-pure
     */
    public static function ten() : BigRational
    {
        /**
         * @psalm-suppress ImpureStaticVariable
         * @var BigRational|null $ten
         */
        static $ten;

        if ($ten === null) {
            $ten = new BigRational(BigInteger::ten(), BigInteger::one(), false);
        }

        return $ten;
    }

    /**
     * @return BigInteger
     */
    public function getNumerator() : BigInteger
    {
        return $this->numerator;
    }

    /**
     * @return BigInteger
     */
    public function getDenominator() : BigInteger
    {
        return $this->denominator;
    }

    /**
     * Returns the quotient of the division of the numerator by the denominator.
     *
     * @return BigInteger
     */
    public function quotient() : BigInteger
    {
        return $this->numerator->quotient($this->denominator);
    }

    /**
     * Returns the remainder of the division of the numerator by the denominator.
     *
     * @return BigInteger
     */
    public function remainder() : BigInteger
    {
        return $this->numerator->remainder($this->denominator);
    }

    /**
     * Returns the quotient and remainder of the division of the numerator by the denominator.
     *
     * @return BigInteger[]
     */
    public function quotientAndRemainder() : array
    {
        return $this->numerator->quotientAndRemainder($this->denominator);
    }

    /**
     * Returns the sum of this number and the given one.
     *
     * @param BigNumber|int|float|string $that The number to add.
     *
     * @return BigRational The result.
     *
     * @throws MathException If the number is not valid.
     */
    public function plus($that) : BigRational
    {
        $that = BigRational::of($that);

        $numerator   = $this->numerator->multipliedBy($that->denominator);
        $numerator   = $numerator->plus($that->numerator->multipliedBy($this->denominator));
        $denominator = $this->denominator->multipliedBy($that->denominator);

        return new BigRational($numerator, $denominator, false);
    }

    /**
     * Returns the difference of this number and the given one.
     *
     * @param BigNumber|int|float|string $that The number to subtract.
     *
     * @return BigRational The result.
     *
     * @throws MathException If the number is not valid.
     */
    public function minus($that) : BigRational
    {
        $that = BigRational::of($that);

        $numerator   = $this->numerator->multipliedBy($that->denominator);
        $numerator   = $numerator->minus($that->numerator->multipliedBy($this->denominator));
        $denominator = $this->denominator->multipliedBy($that->denominator);

        return new BigRational($numerator, $denominator, false);
    }

    /**
     * Returns the product of this number and the given one.
     *
     * @param BigNumber|int|float|string $that The multiplier.
     *
     * @return BigRational The result.
     *
     * @throws MathException If the multiplier is not a valid number.
     */
    public function multipliedBy($that) : BigRational
    {
        $that = BigRational::of($that);

        $numerator   = $this->numerator->multipliedBy($that->numerator);
        $denominator = $this->denominator->multipliedBy($that->denominator);

        return new BigRational($numerator, $denominator, false);
    }

    /**
     * Returns the result of the division of this number by the given one.
     *
     * @param BigNumber|int|float|string $that The divisor.
     *
     * @return BigRational The result.
     *
     * @throws MathException If the divisor is not a valid number, or is zero.
     */
    public function dividedBy($that) : BigRational
    {
        $that = BigRational::of($that);

        $numerator   = $this->numerator->multipliedBy($that->denominator);
        $denominator = $this->denominator->multipliedBy($that->numerator);

        return new BigRational($numerator, $denominator, true);
    }

    /**
     * Returns this number exponentiated to the given value.
     *
     * @param int $exponent The exponent.
     *
     * @return BigRational The result.
     *
     * @throws \InvalidArgumentException If the exponent is not in the range 0 to 1,000,000.
     */
    public function power(int $exponent) : BigRational
    {
        if ($exponent === 0) {
            $one = BigInteger::one();

            return new BigRational($one, $one, false);
        }

        if ($exponent === 1) {
            return $this;
        }

        return new BigRational(
            $this->numerator->power($exponent),
            $this->denominator->power($exponent),
            false
        );
    }

    /**
     * Returns the reciprocal of this BigRational.
     *
     * The reciprocal has the numerator and denominator swapped.
     *
     * @return BigRational
     *
     * @throws DivisionByZeroException If the numerator is zero.
     */
    public function reciprocal() : BigRational
    {
        return new BigRational($this->denominator, $this->numerator, true);
    }

    /**
     * Returns the absolute value of this BigRational.
     *
     * @return BigRational
     */
    public function abs() : BigRational
    {
        return new BigRational($this->numerator->abs(), $this->denominator, false);
    }

    /**
     * Returns the negated value of this BigRational.
     *
     * @return BigRational
     */
    public function negated() : BigRational
    {
        return new BigRational($this->numerator->negated(), $this->denominator, false);
    }

    /**
     * Returns the simplified value of this BigRational.
     *
     * @return BigRational
     */
    public function simplified() : BigRational
    {
        $gcd = $this->numerator->gcd($this->denominator);

        $numerator = $this->numerator->quotient($gcd);
        $denominator = $this->denominator->quotient($gcd);

        return new BigRational($numerator, $denominator, false);
    }

    /**
     * {@inheritdoc}
     */
    public function compareTo($that) : int
    {
        return $this->minus($that)->getSign();
    }

    /**
     * {@inheritdoc}
     */
    public function getSign() : int
    {
        return $this->numerator->getSign();
    }

    /**
     * {@inheritdoc}
     */
    public function toBigInteger() : BigInteger
    {
        $simplified = $this->simplified();

        if (! $simplified->denominator->isEqualTo(1)) {
            throw new RoundingNecessaryException('This rational number cannot be represented as an integer value without rounding.');
        }

        return $simplified->numerator;
    }

    /**
     * {@inheritdoc}
     */
    public function toBigDecimal() : BigDecimal
    {
        return $this->numerator->toBigDecimal()->exactlyDividedBy($this->denominator);
    }

    /**
     * {@inheritdoc}
     */
    public function toBigRational() : BigRational
    {
        return $this;
    }

    /**
     * {@inheritdoc}
     */
    public function toScale(int $scale, int $roundingMode = RoundingMode::UNNECESSARY) : BigDecimal
    {
        return $this->numerator->toBigDecimal()->dividedBy($this->denominator, $scale, $roundingMode);
    }

    /**
     * {@inheritdoc}
     */
    public function toInt() : int
    {
        return $this->toBigInteger()->toInt();
    }

    /**
     * {@inheritdoc}
     */
    public function toFloat() : float
    {
        return $this->numerator->toFloat() / $this->denominator->toFloat();
    }

    /**
     * {@inheritdoc}
     */
    public function __toString() : string
    {
        $numerator   = (string) $this->numerator;
        $denominator = (string) $this->denominator;

        if ($denominator === '1') {
            return $numerator;
        }

        return $this->numerator . '/' . $this->denominator;
    }

    /**
     * This method is required for serializing the object and SHOULD NOT be accessed directly.
     *
     * @internal
     *
     * @return array{numerator: BigInteger, denominator: BigInteger}
     */
    public function __serialize(): array
    {
        return ['numerator' => $this->numerator, 'denominator' => $this->denominator];
    }

    /**
     * This method is only here to allow unserializing the object and cannot be accessed directly.
     *
     * @internal
     * @psalm-suppress RedundantPropertyInitializationCheck
     *
     * @param array{numerator: BigInteger, denominator: BigInteger} $data
     *
     * @return void
     *
     * @throws \LogicException
     */
    public function __unserialize(array $data): void
    {
        if (isset($this->numerator)) {
            throw new \LogicException('__unserialize() is an internal function, it must not be called directly.');
        }

        $this->numerator = $data['numerator'];
        $this->denominator = $data['denominator'];
    }

    /**
     * This method is required by interface Serializable and SHOULD NOT be accessed directly.
     *
     * @internal
     *
     * @return string
     */
    public function serialize() : string
    {
        return $this->numerator . '/' . $this->denominator;
    }

    /**
     * This method is only here to implement interface Serializable and cannot be accessed directly.
     *
     * @internal
     * @psalm-suppress RedundantPropertyInitializationCheck
     *
     * @param string $value
     *
     * @return void
     *
     * @throws \LogicException
     */
    public function unserialize($value) : void
    {
        if (isset($this->numerator)) {
            throw new \LogicException('unserialize() is an internal function, it must not be called directly.');
        }

        [$numerator, $denominator] = \explode('/', $value);

        $this->numerator   = BigInteger::of($numerator);
        $this->denominator = BigInteger::of($denominator);
    }
}