Gunaxf sbe guvf! Vg jnf ernyyl sha. V’z rkpvgrq sbe gur arkg bar. V’ir hfrq jbeqf n ybg va zl nafjref gb cerfreir ebg13-vat. Fb vg’f zvtug abg or nf fanccl nf vg pbhyq or
1.2
Ol Fgveyvat’f Nccebkvzngvba
Fhccbfr Uneel vf jrvtuvat fhofrgf bs jnaqf. Ur jrvtuf gb fhofrgf p_v naq pw ntnvafg rnpu bgure. Gur nafjre yrgf uvz qvfgvathvfu orgjrra gur frg bs jbeyqf F< va juvpu p_v < pw naq F> (juvpu vf gur vairefr). Ohg ur znl trg hayhpxl, or yrsg jvgu gur ynetre bar, guhf erqhpvat ahzore bs cbffvoyr jbeyqf ol ng zbfg snpgbe bs gjb. Naq fb jr trg gur fnzr erfhyg.
svir snpgbevny vf bar uhaqerq naq gjragl. Gur prvyvat bs gur ybt bs gung vf frira.
V qba’g unir n snibhevgr, ohg yrg’f fnl dhvpxfbeg. Abeznyyl bar cvpxf n enaqbz cvibg, fb vg frrzf lbh’er rdhnyyl yvxryl gb trg nal bs gur cnegvgvba fvmrf (nffhzvat ab qhcyvpngrf). Gur zber rdhny-fvmrq gur erphefvir fhoyvfgf, gur zber jbeyqf jrer ryvzvangrq. Vg frrzf lbh’er whfg ng gur zrepl bs enaqbzarff sbe dhvpxfbeg.
2.2
Rahzrengr gur cbffvoyr fpurzrf. Nfx “vf vg va gur ybjre unys?” ercrngrqyl. Lbh pna ebhaq qbja sbe unyirf
V pbhyqa’g pbzr hc jvgu orggre guna ybt_2(a-fdhnerq). Bar jnl bs trggvat gung vf whfg nfxvat rirelguvat gjvpr. Nqzvggrqyl, lbh pna fxvc qbhoyvat hc nsgre lbh’ir qrgrpgrq gur yvr
Hfvat zl nobir fpurzr, ybt_2(a-gb-gur-cbjre-bs-x)
3.1
Vs a vf vasvavgl, jr trg rknpgyl 2. Fb jr pna gnxr njnl gur raq bs gur frdhrapr nf 2 * 2 gb gur zvahf a, juvpu vf 2 gb gur zvahf (a zvahf bar). Gnxvat gung njnl sebz 2, jr trg 1 naq (2 gb gur zvahf (a zvahf bar) zvahf bar)/(2 gb gur zvahf (a zvahf bar))
Xrrc nfxvat “vf vg gur uvturfg cebonovyvgl bs gur erznvavat pnaqvqngrf?” hagvy gurer’f bayl bar bcgvba yrsg. Gur fhz bs yvxryvarff_enax(v) * c_v npebff 1 gb a. Abg fher jung zber V pna fnl jvgubhg xabjvat gur cf.
1.2.1 Vs lbh qba’g xabj Fgveyvat’f nccebkvzngvba, guvf pna or qbar whfg ol vagrtengvat ybtk sebz 0 gb a.
1.2.3 Gur ybjre obhaq jr cebirq vf 7. Pna lbh npghnyyl rkuvovg na nytbevguz juvpu gnxrf 7 pbzcnevfbaf gubhtu?
2.2 V npghnyyl qba’g xabj gur rknpg nafjre, ohg lbh pna qb orggre guna xybt_2(a). Urer’f nabgure jnl gb guvax nobhg gur ceboyrz jvgu 1 yvr: vavgvnyyl, lbh unir a puvcf ng cbfvgvba 2 ba gur ahzore yvar. Rnpu dhrfgvba, lbh trg gb cvpx n fhofrg F bs gur puvcf naq Serq zbirf rvgure F be F_pbzcyrzrag yrsg ol 1. Puvcf snyy bss gur obneq jura gurl ernpu 0. Ubj znal zbirf qbrf vg gnxr gb rafher bayl bar puvc erznvaf? Sbe rknzcyr, lbh fubhyq or noyr gb jva gur tnzr ba 32 cbffvovyvgvrf naq 1 yvr va 9 zbirf.
3.2 Gur fgengrtl lbh qrfpevorq vf ernyyl onq jura nyy gur c_v ner rdhny gb 1/a; lbh’yy arrq a/2 zbirf ba nirentr vafgrnq bs gur jbefg pnfr ybt_a jr unq orsber.
1.2.3 Fbeg guerr jnaqf. Vs lbh’er yhpxl guvf pbhyq gnxr 2 pbzcnevfbaf. Vs lbh’er hayhpxl, 3. Jrvtu gur zvqqyr jrvtug ntnvafg gur bgure gjb. Guvf gnxrf lbh gb 4 be 5. Vs lbh unir gjb cvyrf bs gjb, cyhf gur zvqqyr, fbeg rnpu cvyr (6 be 7). Vs lbh unir n cvyr bs 3, fbeg vg (8, 9 be 10), naq gur bgure vf qrtrarengryl fbegrq. Fb rvtug gb gra
2.2 BX, V qvqa’g trg na rknpg erfhyg, ohg zl nytbevguz vf gnxr n fhofrg jvgu unys bs gur puvcf ng 1 naq 2 erphefviryl. Fb nsgre lbhe svefg dhrfgvba, lbh unir n unys gung vg’f va hayrff ur yvrq, naq unys gung vg’f va vs ur yvrq. Gura nfx fbzrguvat gung unf unys bs rnpu bs gubfr.
3.2.2 Ur fubhyq perngr gur fznyyrfg pneqvanyvgl fhofrg bs fpurzrf jvgu cebonovyvgl 1⁄2 naq nfx vs vg’f va gung. Erphefr. Vs ur pna’g trg rknpgyl gb bar unys, cevbevgvfr pybfrarff gb bar-unys (fb gur fhofrgf ner snveyl rdhny).
1.2.3. Guvax V’ir tbg vg! Svefg, pbzcner gjb cnvef, gb trg n < p naq o < q. Gura pbzcner n naq o. Pbf V’ir abg fnvq nalguvat gb qvfgvathvfu gurz, V pna fgvpx jvgu avpr abgngvba naq unir n < o. Abj pbzcner r gb o. Jr’er hc gb 4 pbzcnevfbaf.
Vs vg’f fznyyre jr unir n-r-o, o-q naq n-p. Jr pna pbzcner q gb p. Vs q’f yrff, gura jr’er qbar (va 5!) naq n-r-o-q-p. Vs p jnf yrff gura jr unir n-r-o, o-q naq n-p-q. Pbzcner p naq r. Vs vg’f yrff jr’er qbar (va 6!). Vs p vf terngre pbzcner vg gb o, naq gura jr’er qbar.
Vs r’f ynetre jr unir n-o-r, o-q naq n-p. Pbzcner r naq q. Jr gura unir n-o-r-q be n-o-q-r. Gura pbzcner p gb juvpurire vf abg gur gbc. Gura pbzcner p gb rvgure gur frpbaq be gur gbc, qrcraqvat ba gur erfhyg.
Gunaxf sbe guvf! Vg jnf ernyyl sha. V’z rkpvgrq sbe gur arkg bar. V’ir hfrq jbeqf n ybg va zl nafjref gb cerfreir ebg13-vat. Fb vg’f zvtug abg or nf fanccl nf vg pbhyq or
1.2
Ol Fgveyvat’f Nccebkvzngvba
Fhccbfr Uneel vf jrvtuvat fhofrgf bs jnaqf. Ur jrvtuf gb fhofrgf p_v naq pw ntnvafg rnpu bgure. Gur nafjre yrgf uvz qvfgvathvfu orgjrra gur frg bs jbeyqf F< va juvpu p_v < pw naq F> (juvpu vf gur vairefr). Ohg ur znl trg hayhpxl, or yrsg jvgu gur ynetre bar, guhf erqhpvat ahzore bs cbffvoyr jbeyqf ol ng zbfg snpgbe bs gjb. Naq fb jr trg gur fnzr erfhyg.
svir snpgbevny vf bar uhaqerq naq gjragl. Gur prvyvat bs gur ybt bs gung vf frira.
V qba’g unir n snibhevgr, ohg yrg’f fnl dhvpxfbeg. Abeznyyl bar cvpxf n enaqbz cvibg, fb vg frrzf lbh’er rdhnyyl yvxryl gb trg nal bs gur cnegvgvba fvmrf (nffhzvat ab qhcyvpngrf). Gur zber rdhny-fvmrq gur erphefvir fhoyvfgf, gur zber jbeyqf jrer ryvzvangrq. Vg frrzf lbh’er whfg ng gur zrepl bs enaqbzarff sbe dhvpxfbeg.
2.2
Rahzrengr gur cbffvoyr fpurzrf. Nfx “vf vg va gur ybjre unys?” ercrngrqyl. Lbh pna ebhaq qbja sbe unyirf
V pbhyqa’g pbzr hc jvgu orggre guna ybt_2(a-fdhnerq). Bar jnl bs trggvat gung vf whfg nfxvat rirelguvat gjvpr. Nqzvggrqyl, lbh pna fxvc qbhoyvat hc nsgre lbh’ir qrgrpgrq gur yvr
Hfvat zl nobir fpurzr, ybt_2(a-gb-gur-cbjre-bs-x)
3.1
Vs a vf vasvavgl, jr trg rknpgyl 2. Fb jr pna gnxr njnl gur raq bs gur frdhrapr nf 2 * 2 gb gur zvahf a, juvpu vf 2 gb gur zvahf (a zvahf bar). Gnxvat gung njnl sebz 2, jr trg 1 naq (2 gb gur zvahf (a zvahf bar) zvahf bar)/(2 gb gur zvahf (a zvahf bar))
Xrrc nfxvat “vf vg gur uvturfg cebonovyvgl bs gur erznvavat pnaqvqngrf?” hagvy gurer’f bayl bar bcgvba yrsg. Gur fhz bs yvxryvarff_enax(v) * c_v npebff 1 gb a. Abg fher jung zber V pna fnl jvgubhg xabjvat gur cf.
Avpr gevrf, n srj pbzzragf:
1.2.1 Vs lbh qba’g xabj Fgveyvat’f nccebkvzngvba, guvf pna or qbar whfg ol vagrtengvat ybtk sebz 0 gb a.
1.2.3 Gur ybjre obhaq jr cebirq vf 7. Pna lbh npghnyyl rkuvovg na nytbevguz juvpu gnxrf 7 pbzcnevfbaf gubhtu?
2.2 V npghnyyl qba’g xabj gur rknpg nafjre, ohg lbh pna qb orggre guna xybt_2(a). Urer’f nabgure jnl gb guvax nobhg gur ceboyrz jvgu 1 yvr: vavgvnyyl, lbh unir a puvcf ng cbfvgvba 2 ba gur ahzore yvar. Rnpu dhrfgvba, lbh trg gb cvpx n fhofrg F bs gur puvcf naq Serq zbirf rvgure F be F_pbzcyrzrag yrsg ol 1. Puvcf snyy bss gur obneq jura gurl ernpu 0. Ubj znal zbirf qbrf vg gnxr gb rafher bayl bar puvc erznvaf? Sbe rknzcyr, lbh fubhyq or noyr gb jva gur tnzr ba 32 cbffvovyvgvrf naq 1 yvr va 9 zbirf.
3.2 Gur fgengrtl lbh qrfpevorq vf ernyyl onq jura nyy gur c_v ner rdhny gb 1/a; lbh’yy arrq a/2 zbirf ba nirentr vafgrnq bs gur jbefg pnfr ybt_a jr unq orsber.
Gunaxf sbe gur cbvagref. Jvyy erguvax
1.2.3 Fbeg guerr jnaqf. Vs lbh’er yhpxl guvf pbhyq gnxr 2 pbzcnevfbaf. Vs lbh’er hayhpxl, 3. Jrvtu gur zvqqyr jrvtug ntnvafg gur bgure gjb. Guvf gnxrf lbh gb 4 be 5. Vs lbh unir gjb cvyrf bs gjb, cyhf gur zvqqyr, fbeg rnpu cvyr (6 be 7). Vs lbh unir n cvyr bs 3, fbeg vg (8, 9 be 10), naq gur bgure vf qrtrarengryl fbegrq. Fb rvtug gb gra
2.2 BX, V qvqa’g trg na rknpg erfhyg, ohg zl nytbevguz vf gnxr n fhofrg jvgu unys bs gur puvcf ng 1 naq 2 erphefviryl. Fb nsgre lbhe svefg dhrfgvba, lbh unir n unys gung vg’f va hayrff ur yvrq, naq unys gung vg’f va vs ur yvrq. Gura nfx fbzrguvat gung unf unys bs rnpu bs gubfr.
3.2.2 Ur fubhyq perngr gur fznyyrfg pneqvanyvgl fhofrg bs fpurzrf jvgu cebonovyvgl 1⁄2 naq nfx vs vg’f va gung. Erphefr. Vs ur pna’g trg rknpgyl gb bar unys, cevbevgvfr pybfrarff gb bar-unys (fb gur fhofrgf ner snveyl rdhny).
1.2.3 Gurer rkvfgf n fgenvtug naq aneebj cngu gb fbegvat 5 ryrzragf va rknpgyl 7 zbirf. Xrrc ybbxvat sbe pbzcnevfbaf gung qvivqr gur ahzore bs cbffvoyr jbeyqf nf pybfr gb rirayl nf cbffvoyr.
2.2 Gung’f ebhtuyl evtug, tbbq wbo! Sbe shegure qrgnvyf ba guvf ceboyrz, frr gur “Yvne Tnzr” va Nyba&Fcrapre Pu 14.
3.2.2 Rknpgyl! V’yy fnl zber nobhg guvf arkg gvzr ohg lbh’ir ebhtuyl erperngrq gur vqrn bs Uhsszna pbqrf.
Gunaxf sbe gur ercrngrq srrqonpx!
1.2.3. Guvax V’ir tbg vg! Svefg, pbzcner gjb cnvef, gb trg n < p naq o < q. Gura pbzcner n naq o. Pbf V’ir abg fnvq nalguvat gb qvfgvathvfu gurz, V pna fgvpx jvgu avpr abgngvba naq unir n < o. Abj pbzcner r gb o. Jr’er hc gb 4 pbzcnevfbaf.
Vs vg’f fznyyre jr unir n-r-o, o-q naq n-p. Jr pna pbzcner q gb p. Vs q’f yrff, gura jr’er qbar (va 5!) naq n-r-o-q-p. Vs p jnf yrff gura jr unir n-r-o, o-q naq n-p-q. Pbzcner p naq r. Vs vg’f yrff jr’er qbar (va 6!). Vs p vf terngre pbzcner vg gb o, naq gura jr’er qbar.
Vs r’f ynetre jr unir n-o-r, o-q naq n-p. Pbzcner r naq q. Jr gura unir n-o-r-q be n-o-q-r. Gura pbzcner p gb juvpurire vf abg gur gbc. Gura pbzcner p gb rvgure gur frpbaq be gur gbc, qrcraqvat ba gur erfhyg.
V guvax
Gur svefg unys vf evtug, gur xrl vqrn vf gb pbzcner r gb o, naq purpx gung guvf oernxf n havirefr bs 15 cbffvovyvgvrf vagb 7 naq 8. Nsgre gung lbh unir gb or n ovg zber pnershy nobhg znxvat gur evtug pbzcnevfbaf, sbe rknzcyr vg ybbxf yvxr lbh’er nffhzvat n<r<o whfg orpnhfr r<o, jura vg pbhyq nyfb ybbx yvxr r<n<o.