❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❘❡✈✐s✐t❛♥❞♦ ♦s ❆❧❣♦r✐t♠♦s ♣❛r❛ ❖♣❡r❛çõ❡s
❆r✐t♠ét✐❝❛s ❋✉♥❞❛♠❡♥t❛✐s
❊♠♠❛♥✉❡❧ ❈r✐st✐❛♥♦ ▲♦♣❡s ❞❡ ▼♦r❛❡s
❇r❛sí❧✐❛
❊♠♠❛♥✉❡❧ ❈r✐st✐❛♥♦ ▲♦♣❡s ❞❡ ▼♦r❛❡s
❘❡✈✐s✐t❛♥❞♦ ♦s ❆❧❣♦r✐t♠♦s ♣❛r❛ ❖♣❡r❛çõ❡s
❆r✐t♠ét✐❝❛s ❋✉♥❞❛♠❡♥t❛✐s
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❆r❛✉❥♦
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
M827r
Moraes, Emmanuel Cristiano Lopes de
Revisitando os Algoritmos para Operações
Aritméticas Fundamentais / Emmanuel Cristiano Lopes de Moraes; orientador Kellcio Oliveira Araujo. --Brasília, 2015.
94 p.
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❊♠ t✉❞♦ ✈❡❥♦ ♦ ❚❡✉ ❛♠♦r❀ ❊♠ t✉❞♦ ✈❡❥♦ ❚✉❛s ♠ã♦s❀ ❊ ❚❡ ❛❣r❛❞❡ç♦ ó ❈r✐❛❞♦r❀ P♦r q✉❡ t✉❞♦ q✉❡ ❡①✐st❡✱ só ❡①✐st❡✱ ♣♦r q✉❡ ❚✉ és ❙❡♥❤♦r✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛♦ ❙❡♥❤♦r ♣♦r t♦❞♦s ♦s ❜❡♥❡❢í❝✐♦s q✉❡ ♠❡ t❡♠ ❢❡✐t♦ ❡ ♣♦r s✉❛ ❜♦♥❞❛❞❡ ❡ ❛♠♦r q✉❡ ♠❡ s❡❣✉❡♠ ♣♦r t♦❞♦s ♦s ❞✐❛s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❡s♣♦s❛✱ ❆♥❛ P❛✉❧❛✱ q✉❡ ❝♦♠ ♠✉✐t♦ ❡s❢♦rç♦ ❡ ♣❡rs❡✈❡r❛♥ç❛ ♠❡ ❛♣♦✐♦✉ ❞✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ ❝✉rs♦ ❞❡ ♠❡str❛❞♦ ❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❊❧❛ ❝✉✐❞♦✉ ❞♦ ♥♦ss♦ ♠❛✐♦r ❜❡♠✱ ♥♦ss♦s ✜❧❤♦s ■s❛q✉❡ ❡ ▲❡✈✐ q✉❡ sã♦ ❤❡r❛♥ç❛ ❞♦ ❙❡♥❤♦r ❡♠ ♥♦ss❛s ✈✐❞❛s✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ▼ã❡ ❡ ❛♦ ♠❡✉ P❛✐ q✉❡ ♠❡ ❞❡r❛♠ ✈❛❧✐♦s❛s ❧✐çõ❡s ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ♠♦❝✐❞❛❞❡ ❡ ♣♦r s✉❛ ♣r❡♦❝✉♣❛çã♦ ❡ ❝✉✐❞❛❞♦ ❝♦♠ ♠❡✉ ❜❡♠ ❡st❛r ❡ ❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✳
❆❣r❛❞❡ç♦ às ♠✐♥❤❛s ✐r♠ãs✿ ▲✉❛♥❛✱ ▲♦r❡♥❛ ❡ ❊♠❛♥✉❡❧❛ ♣♦r t♦❞❛s ❛s ♣❛❧❛✈r❛s ❞❡ ❜♦♠ â♥✐♠♦ q✉❡ ♣r♦❢❡r✐r❛♠ ❡ ♣♦r t♦❞♦ ♦ ✐♥❝❡♥t✐✈♦ q✉❡ r❡❝❡❜✐ ❞❡❧❛s✳
❆❣r❛❞❡ç♦ ❛♦ ♦r✐❡♥t❛❞♦r ❞❡ss❡ tr❛❜❛❧❤♦✱ Pr♦❢✳ ❉r✳ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❆r❛✉❥♦✱ ♣♦r s✉❛ ❝♦♥tr✐❜✉✐çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ ❛ ♦r✐❡♥t❛çã♦✱ ❛❧é♠ ❞❡ s✉❛s ❝♦rr❡çõ❡s q✉❡ ❢♦r❛♠ ♣r❡♣♦♥❞❡r❛♥t❡s ♣❛r❛ ♦ ♠❡✉ ê①✐t♦✳
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛✲ sí❧✐❛ q✉❡ ❝♦♠ t❛♥t♦ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ❡ ❞❡❞✐❝❛çã♦ ♠✐♥✐str❛r❛♠ ❞✉r❛♥t❡ ♦ ❝✉rs♦ ❞❡ ♠❡s✲ tr❛❞♦✳
❆❣r❛❞❡ç♦ ❛♦s ❜r✐❧❤❛♥t❡s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ q✉❡ ♠✉✐t♦ ♠❡ ❡♥s✐♥❛r❛♠ ❞✉r❛♥t❡ ❛s ♥♦ss❛s ✈✐✈ê♥❝✐❛s✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❡st✉❞♦ ❊♠❡rs♦♥✱ ●✉st❛✈♦✱ ❘✐❝❛r❞♦ ❡ ❯❧②ss❡s q✉❡ ❝♦♠♣❛rt✐❧❤❛r❛♠ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❛♠✐③❛❞❡ ❡ t❡♠♣♦✳
❘❡s✉♠♦
❆ ♣r❡s❡♥t❡ ♣❡sq✉✐s❛ r❡❢❡r❡✲s❡ ❛♦ ❡st✉❞♦ ❞♦s ❛❧❣♦r✐t♠♦s ❝♦♥✈❡♥❝✐♦♥❛✐s ♣❛r❛ ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s✱ ❛ s❛❜❡r✿ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✳ P❛rt❡ ❞♦ tr❛❜❛❧❤♦ é ♦ r❡s✉❧t❛❞♦ ❞♦ ❡s❢♦rç♦ ❞♦ ❛✉t♦r ❡♠ ❝♦♠♣r❡❡♥❞❡r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ t❛✐s ❛❧❣♦r✐t♠♦s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s ❡♥✈♦❧✈✐❞❛s ❡♠ s✉❛s ❡①❡❝✉çõ❡s✳ ➱ t❛♠❜é♠ ♦❜❥❡t✐✈♦ ❞❡ss❛ ♣❡sq✉✐s❛ ♦❢❡rt❛r ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ❞❡ ❡♥s✐♥♦ ❢✉♥✲ ❞❛♠❡♥t❛❧✱ ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ❡ ✉♠ ♣♦✉❝♦ ❡sq✉❡❝✐❞❛ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❛ ❛r✐t♠ét✐❝❛✱ ❛tr❛✈és ❞❛ ♦♣❡r❛çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ❝♦♥✈❡♥❝✐♦♥❛✐s✱ ♦❜t❡♥❞♦✲s❡ ❛ ♣❛rt✐r ❞✐ss♦ ♦✉tr❛ ❜❡✲ ♥❡ss❡✿ ❛ ❝r✐st❛❧✐③❛çã♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❞ã♦ ❝♦♠♣❧❡t✉❞❡ às ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s✳ P❛❧❛✈r❛s✲❝❤❛✈❡
❖♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s✱ ❆❧❣♦r✐t♠♦s✱ ❆r✐t♠ét✐❝❛✳
❆❜str❛❝t
❚❤✐s r❡s❡❛r❝❤ r❡❢❡rs t♦ t❤❡ st✉❞② ♦❢ ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s ❢♦r ❜❛s✐❝ ♦♣❡r❛t✐♦♥s ✲ ❛❞✲ ❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ✳ P❛rt ♦❢ t❤❡ ✇♦r❦ ✐s t❤❡ r❡s✉❧t ♦❢ t❤❡ ❛✉t❤♦r✬s ❡✛♦rt t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❢✉♥❝t✐♦♥✐♥❣ ♦❢ s✉❝❤ ❛❧❣♦r✐t❤♠s ❛♥❞ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦♣❡r❛t✐♦♥s ✐♥✈♦❧✈❡❞ ✐♥ t❤❡✐r ❡①❡❝✉t✐♦♥s✳ ■t ❛❧s♦ ❛✐♠s t♦ ♦✛❡r ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ t❡❛✲ ❝❤❡rs✱ ❡s♣❡❝✐❛❧❧② ❢♦r t❤♦s❡ ✇♦r❦✐♥❣ ✐♥ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧✱ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♥❞ s♦♠❡✇❤❛t ❢♦r❣♦tt❡♥ ♠❡t❤♦❞ t♦ ❞❡✈❡❧♦♣ ❛r✐t❤♠❡t✐❝s✱ t❤r♦✉❣❤ ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s✱ ♦❜t❛✐♥✐♥❣ ❢r♦♠ t❤❛t ❛♥♦t❤❡r ❛❞✈❛♥t❛❣❡✿ t❤❡ ❝r②st❛❧❧✐③❛t✐♦♥ ♦❢ ♣r♦♣❡rt✐❡s t❤❛t ❣✐✈❡ ❝♦♠♣❧❡t❡♥❡ss t♦ t❤❡ ❜❛s✐❝ ♦♣❡r❛t✐♦♥s✳
❑❡②✇♦r❞s
❋✉♥❞❛♠❡♥t❛❧ ♦♣❡r❛t✐♦♥s✱ ❆❧❣♦r✐t❤♠s✱ ❆r✐t❤♠❡t✐❝✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ❯♠ ♠♦❞♦ ♠❛✐s s✉❣❡st✐✈♦ ❞❡ ✈✐s❧✉♠❜r❛r ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷ ❯♠ ♥ú♠❡r♦ a ❡ s❡✉ s✉❝❡ss♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸ ❯♠ ♥ú♠❡r♦ a+b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✹ ❉❛ ❝♦♥t❛❣❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❧♦❝♦s à ❝♦♥t❛❣❡♠ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❜❧♦❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✺ Pr♦❝❡ss♦ ♣❛r❛ ❝♦♥✈❡rt❡r 8 ❡9 ♣❛r❛ ❜❛s❡ ❜✐♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✻ ❖s ♣r✐♠❡✐r♦s ♥♦✈❡ ❛❧❣❛r✐s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✼ ❈♦♥t❛❣❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❧♦❝♦s à ❝♦♥t❛❣❡♠ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦
❢♦r♠❛❞♦ ♣❡❧❛ r❡♠♦çã♦ ❞❡ ❛❧❣✉♥s ❞❡st❡s ❜❧♦❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✽ ❈♦♥t❛❣❡♠ r❡❣r❡ss✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✾ ❈♦♥t❛❣❡♠ ♣r♦❣r❡ss✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✵ 13×131 r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✶✶ ❘❡tâ♥❣✉❧♦ s✉❜❞✐✈✐❞✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✶✷ ❘❡tâ♥❣✉❧♦ r❡❛rr❛♥❥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✶✸ ❘❡tâ♥❣✉❧♦ ❛♣ós ♥♦✈♦ r❡❛rr❛♥❥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✶✹ ❘❡tâ♥❣✉❧♦ s✉❜❞✐✈✐❞♦ ♥♦✈❛♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✶✺ ❘❡tâ♥❣✉❧♦ ❛♣ós ♥♦✈♦ r❡❛rr❛♥❥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✶✻ ❚❛❜❡❧❛ 2×3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✶✼ ❚❛❜❡❧❛ ❝♦♠ ❛s ❞✐❛❣♦♥❛✐s tr❛ç❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✶✽ ❚❛❜❡❧❛ ❝♦♠ ♦s ♣r♦❞✉t♦s ♣❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✶✾ ❙♦♠❛s ❞♦s ♣r♦❞✉t♦s ♣❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✵ ❘❡❣✐str♦ ❞♦ r❡s✉❧t❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✶ ❘❡♣r❡s❡♥t❛♥❞♦ ♦ ♠✉❧t✐♣❧✐❝❛❞♦r13 ❡ ♦ ♠✉❧t✐♣❧✐❝❛♥❞♦ 131 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✷✷ ❙♦♠❛♥❞♦ ❛ ♣❛rt✐r ❞❛s r❡❣✐õ❡s ❞❡ ✐♥t❡rs❡çã♦ ❞❡ ❆ ❛té ❋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✷✸ 12×13❝♦♠ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✷✹ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❡ ❞✐✈✐sã♦ ✉t✐❧✐③❛❞♦ ♥♦s ❊❯❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✺ 123456789987654321÷987654321 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✸
✷ ❆ ❛❞✐çã♦ ✶✻
✷✳✶ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❛ ❛❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❛❞✐çã♦ ✉s❛♥❞♦ ❛ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ▼ét♦❞♦ ❞❛s s♦♠❛s ♣❛r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ ❆ s✉❜tr❛çã♦ ✸✷
✸✳✶ ❆❧❣♦r✐t♠♦ ❞❛ ❝♦♠♣❡♥s❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❛ s✉❜tr❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❞❡ ❛❞✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✹ ❆❧❣♦r✐t♠♦ ❞❡ s✉❜tr❛çã♦ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❞✐r❡✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ✹✼
✹✳✶ ▼ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❡❣í♣❝✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✷ ▼ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✉t✐❧✐③❛❞♦ ♣♦r ❝❛♠♣♦♥❡s❡s r✉ss♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✸ ▼ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❣r❡❣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✹ ▼ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ár❛❜❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✺ ▼ét♦❞♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❝❤✐♥❡s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✻ ❯♠ ♦❧❤❛r ❣❡♦♠étr✐❝♦ s♦❜r❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✼ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✽ ▼✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✾ ❯♠ ❡rr♦ ❝♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✺ ❆ ❞✐✈✐sã♦ ✼✷
✺✳✶ ▼ét♦❞♦ ❞❡ ❞✐✈✐sã♦ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❡❣í♣❝✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✺✳✷ ✏▼ét♦❞♦ ❞❛ ●❛❧é✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✺✳✸ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❛ ❞✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✺✳✹ ❯♠❛ ❡①♣❧✐❝❛çã♦ ♥♦s ❞✐❛s ❞❡ ❤♦❥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽
✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✾✷
✶ ■♥tr♦❞✉çã♦
❚♦❞♦s q✉❡ ❞❡s❡❥❛♠ ❡♥s✐♥❛r✱ ♥ã♦ só ▼❛t❡♠át✐❝❛✱ ♠❛s q✉❛❧q✉❡r ár❡❛ ❞♦ s❛❜❡r ♥❡✲ ❝❡ss✐t❛♠ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❜❡♠ ❢✉♥❞❛♠❡♥t❛❞♦ ❞❛q✉✐❧♦ q✉❡ ✈ã♦ ❧❡❝✐♦♥❛r✳ ❙♦❜r❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦s ❛❧❣♦r✐t♠♦s ❝♦♥✈❡♥❝✐♦♥❛✐s q✉❡ ✉t✐❧✐③❛♠♦s ♣❛r❛ ❛s q✉❛tr♦ ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s✱ ♥ã♦ s❡ t❡♠ t❛♥t❛s ✐♥❢♦r♠❛çõ❡s✳ ❆♣❡♥❛s r❡♣❡t✐♠♦s ❛q✉✐❧♦ q✉❡ ♦✉✈í❛♠♦s q✉❛♥❞♦ ❛❧✉♥♦s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳ ❈♦♠♦ ❡♥s✐♥❛r t❛✐s tó♣✐❝♦s s❡♠ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣r♦❢✉♥❞♦❄ P♦❞❡♠♦s ❡✈✐❞❡♥❝✐❛r ✐ss♦ t❛♠❜é♠ ❛ ♣❛rt✐r ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ♦s ❛❧✉♥♦s tê♠ ❞♦s ❛❧❣♦r✐t♠♦s ♣❛r❛ ♦♣❡r❛çõ❡s ❜ás✐❝❛s✱ ❞❛ ❢❛❧t❛ ❞❡ ❞♦♠í♥✐♦ ❞❛ t❛❜✉❛❞❛ ❡ ❛té ♠❡s♠♦ ♦ q✉❡ ♦s ❧❡✈❛ ❛ r❡♣r♦❞✉③✐r❡♠ t❛♥t♦s ♣r♦❝❡❞✐♠❡♥t♦s r❡❢❡r❡♥t❡s ❛♦ ❛❧❣♦r✐t♠♦ s❡♠✱ ❞❡ ❢❛t♦✱ s❛❜❡r ♦ ♣♦rq✉ê✳ ◆ós ♣r♦❢❡ss♦r❡s s♦❢r❡♠♦s ❛s ♠❡s♠❛s ❧✐♠✐t❛çõ❡s ❞♦s ❛❧✉♥♦s✳ ❯s❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ♣❛r❛ ❛❧❣♦r✐t♠♦ ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷❪✿
❆▲●❖❘■❚▼❖✱ s✳♠✳ ❈♦♥❥✉♥t♦ ♣r❡❞❡t❡r♠✐♥❛❞♦ ❡ ❞❡✜♥✐❞♦ ❞❡ r❡❣r❛s ❡ ♣r♦❝❡ss♦s ❞❡st✐♥❛❞♦s à s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛✱ ❝♦♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡t❛♣❛s✳
■♥❢❡❧✐③♠❡♥t❡✱ ♥❛ ♣rát✐❝❛ ❞♦❝❡♥t❡ t❡♠♦s ❛♣❧✐❝❛❞♦ ❡ss❡ ❝♦♥❥✉♥t♦ ❞❡ r❡❣r❛s ❛ ✜♠ ❞❡ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ s❡♠ s❛❜❡r✱ ❞❡ ❢❛t♦✱ ♦ ❡♥t❡♥❞✐♠❡♥t♦ s♦❜r❡ ❛s ❡t❛♣❛s ❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦✳ ◗✉❛♥❞♦ ♥♦s r❡❢❡r✐♠♦s às ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❡st❛♠♦s ❢❛③❡♥❞♦ ❛❧✉sã♦ às q✉❛tr♦ ♦♣❡r❛çõ❡s✱ q✉❛✐s s❡❥❛♠✿ ❛❞✐çã♦✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✳ ❖ q✉❡ t❡♠♦s ✈✐st♦ ❛ ♣❛rt✐r ❞❡ ♥♦ss❛ ❡①♣❡r✐ê♥❝✐❛ é q✉❡ ♦s ❛❧✉♥♦s ✈✐♥❞♦s ❞❡ sér✐❡s ❞♦ ♣r✐♠❡✐r♦ ❝✐❝❧♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ✶➸ ❛♦ ✺➸ ❛♥♦✱ ♥ã♦ ❞❡s❡♥✈♦❧✈❡r❛♠ ❛ ❤❛❜✐❧✐❞❛❞❡ ❞❡ ✉t✐❧✐③❛r ❝♦rr❡t❛♠❡♥t❡ t❛✐s ❛❧❣♦r✐t♠♦s✳
❆s ❡✈✐❞ê♥❝✐❛s q✉❡ t❡♠♦s ❛ ♣❛rt✐r ❞❛ ♣❡sq✉✐s❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ❬✶✷❪ é q✉❡ ♥❡♠ s❡♠✲ ♣r❡ ♦s ♣r♦❢❡ss♦r❡s ❞❡ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡s❡♥✈♦❧✈❡♠ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣r♦❢✉♥❞♦ ❞❛ ▼❛t❡♠át✐❝❛ ❜ás✐❝❛✱ ✐st♦ é✱ ♥ã♦ t❡♠ ♦ ❡♠❜❛s❛♠❡♥t♦ ❞❡✈✐❞♦ ♣❛r❛ ❧❡❝✐♦♥❛r ❛q✉✐❧♦ q✉❡ s❡ ❞✐s♣õ❡♠ ❛ ❡♥s✐♥❛r✳ ❆ ♣❡sq✉✐s❛ ❛ q✉❛❧ ♥♦s r❡❢❡r✐♠♦s ❛♥❛❧✐s♦✉ ♣r♦❢❡ss♦r❡s ❛♠❡r✐❝❛♥♦s ❡ ❝❤✐♥❡s❡s✳ ➱ ❝❧❛r♦ q✉❡ ♥ã♦ q✉❡r❡♠♦s ❞✐③❡r q✉❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❬✶✷❪ s❡ ❛♠♦❧❞❛ ♣❡r❢❡✐t❛♠❡♥t❡ ❛♦ s✐st❡♠❛ q✉❡ t❡♠♦s ♥♦ ❇r❛s✐❧✱ ♠❛s ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s t✐✈❡♠♦s ❛ ✐♠♣r❡ssã♦ q✉❡ ♦ q✉❡ ❡stá s❡♥❞♦ tr❛t❛❞♦ é ❜❡♠ s❡♠❡❧❤❛♥t❡ ❛♦ q✉❡ t❡♠♦s ❡♠ ♥♦ss♦ ❝♦♥✲ t❡①t♦ ❡❞✉❝❛❝✐♦♥❛❧✳ ❱❡❥❛♠♦s ♦ q✉❡ ❞✐③ ❛ ♣❡sq✉✐s❛❞♦r❛ ❡♠ ❬✶✷❪✿
❖ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ♣r♦❢❡ss♦r❡s s♦❜r❡ ✉♠❛ ♠❛tér✐❛ ♣♦❞❡ ♥ã♦ ♣r♦❞✉③✐r ❛✉t♦♠❛t✐❝❛♠❡♥t❡ ♠ét♦❞♦s ❞❡ ❡♥s✐♥♦ ♣r♦♠✐ss♦r❡s ♦✉ ♥♦✈❛s ❝♦♥❝❡♣çõ❡s ❞❡ ❡♥s✐♥♦✳ ▼❛s s❡♠ ✉♠ ❛♣♦✐♦ só❧✐❞♦ ❞❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♠ét♦❞♦s ♣r♦♠✐ss♦r❡s ♦✉ ♥♦✈❛s ❝♦♥❝❡♣çõ❡s ❞❡ ❡♥s✐♥♦ ♥ã♦ ♣♦❞❡♠ s❡r r❡❛❧✐③❛❞♦s ❝♦♠ s✉❝❡ss♦✳
❉✉r❛♥t❡ ❛ ❡①♣❧❛♥❛çã♦ ♥♦s ❝❛♣ít✉❧♦s✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❤✐stór✐❝❛ s♦✲ ❜r❡ ❛s ♦♣❡r❛çõ❡s ❡ s♦❜r❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦✱ ❛❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ♦✉tr♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ❡♠ ♦✉tr❛s r❡❣✐õ❡s ❡ ❝✐✈✐❧✐③❛çõ❡s ❞❡ é♣♦❝❛s ❛♥t❡r✐♦r❡s✳ ❊♠ ❛❧❣✉♥s ♠♦✲ ♠❡♥t♦s ❛♣r❡s❡♥t❛r❡♠♦s ❛ ♦♣❡r❛çã♦ ❞♦s ❛❧❣♦r✐t♠♦s ❡♠ ♦✉tr❛s ❜❛s❡s ♥✉♠ér✐❝❛s ❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❝♦♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦✳
◆♦ ❈❛♣ít✉❧♦ ✷ ✐♥✐❝✐❛r❡♠♦s ❛ ❞✐s❝✉ssã♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦✶ ❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❛ ❛❞✐çã♦ ❢❡✐t❛ ❝♦♠ tr❛♥s♣♦rt❡ ❛ ✜♠ ❞❡ ❡❧✉❝✐❞❛r ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❛ ♦r❞❡♠ s✉♣❡r✐♦r ♥♦ ❛❧❣♦r✐t♠♦ ❞❛ s✉❜tr❛çã♦✳
◆♦ ❈❛♣ít✉❧♦ ✸ ❢❛r❡♠♦s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ s✉❜tr❛çã♦✷ ❡ ❞❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s s✉❜❥❛❝❡♥t❡s t❛✐s ❝♦♠♦ ♦ r❡❛❣r✉♣❛♠❡♥t♦✱ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ✉♠ ✉♠❛ ♦r❞❡♠ ✐♥❢❡r✐♦r✱ ❛ ❛❞✐çã♦ ❝♦♠ tr❛♥s♣♦rt❡ q✉❡ ❡stá ❧✐❣❛❞❛ ❝♦♠ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ♦r❞❡♠ s✉♣❡r✐♦r ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧✱ ♦✉ s❡❥❛✱ ✶✵ ✉♥✐❞❛❞❡s ❡♠ ✉♠❛ ♦r❞❡♠ ❝♦♠♣õ❡♠ ✶ ✉♥✐❞❛❞❡ ♥❛ ♦r❞❡♠ ✐♠❡❞✐❛t❛♠❡♥t❡ s✉♣❡r✐♦r✳ ❆❜♦r❞❛r❡♠♦s ♦ ❛ss✉♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❬✶✷❪ ❡ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❤✐stór✐❝♦ ❞♦ ❛❧❣♦r✐t♠♦✳
P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ❡❢❡t✉❛r♠♦s ❛ s✉❜tr❛çã♦ r❡❝♦rr❡♠♦s ❢r❡q✉❡♥t❡♠❡♥t❡ ❛♦ s✉❜t❡r❢ú✲ ❣✐♦ ❞❡ ✏♣❡❞✐r ❡♠♣r❡st❛❞♦✑ ♣❛r❛ ❥✉st✐✜❝❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❡ ❛ ❡✜❝✐ê♥❝✐❛ ❞♦ ❛❧❣♦r✐t♠♦ ❡♠ ❝♦♥t❛s ❝♦♠♦✿
9 8 1
− 1 9 8
7 8 3
◆♦ ❈❛♣ít✉❧♦ ✹ ❢❛r❡♠♦s ✉♠❛ ❞✐s❝✉ssã♦ ❛ ♣❛rt✐r ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❤✐stór✐❝♦ ❞♦s ❛❧❣♦r✐t♠♦s✱ ❛❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ♠✉❧t✐♣❧✐❝❛r ♥❛ ❝✐✈✐❧✐③❛çã♦ ❡❣í♣✲ ❝✐❛✱ ♥❛ ❣r❡❣❛✱ ❞❡♥tr❡ ♦✉tr❛s✳ ❊ ❛♥❛❧✐s❛r❡♠♦s ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♣❛r❛ ♠✉❧t✐♣❧✐❝❛r ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s q✉❡ ❥✉st✐✜❝❛♠ s❡✉ ❢✉♥❝✐♦♥❛♠❡♥t♦✳
◆♦ ❈❛♣ít✉❧♦ ✺✱ ❛❜♦r❞❛r❡♠♦s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❤✐stór✐❝♦ ❞❛ ♦♣❡r❛çã♦ ❡ ❞❛ ♥❡❝❡s✲ s✐❞❛❞❡ ❞❡ ❞✐✈✐❞✐r✳ ❆♣r❡s❡♥t❛r❡♠♦s ♦ ❛❧❣♦r✐t♠♦ ❡❣í♣❝✐♦ ❡ ▼ét♦❞♦ ❞❛ ●❛❧é✱ ♦♥❞❡ ❡st❡ ú❧t✐♠♦ é ❞❡s❡♥✈♦❧✈✐❞♦ ❧❛❞♦ ❛ ❧❛❞♦ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❛ ✜♠ ❞❡ ✈❡r✐✜❝❛r ❛
✶❊♠ ❬✽❪✿ ❆r✐t✳ ❖♣❡r❛çã♦ ❞❡ ❛❞✐❝✐♦♥❛r q✉❛♥t✐❞❛❞❡s❀ ❙❖▼❆✳
✷❊♠ ❬✽❪✿ ❆r✐t✳ ❖♣❡r❛çã♦ ❞❡ s✉❜tr❛✐r q✉❛♥t✐❞❛❞❡s❀ ❉■▼■◆❯■➬➹❖ ❬ ❆♥tô♥✳✿ ❛❞✐çã♦✳ ❪✳
s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ❡❧❡s✳ ❋✐♥❛❧♠❡♥t❡✱ ❢❛r❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❛ ♣❛rt✐r ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❊✉❝❧✐❞❡s✳
✷ ❆ ❛❞✐çã♦
P❛r❛ ✐♥✐❝✐❛r♠♦s ♥♦ss❛ ❞✐s❝✉ssã♦ s♦❜r❡ ♦ ❛❧❣♦r✐t♠♦ ❞❛ s✉❜tr❛çã♦ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦ ✈❛♠♦s ♣❡r❝♦rr❡r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❡ ❞❡ s❡✉ ❛❧❣♦r✐t♠♦ ✉s✉❛❧ q✉❡ t❡✈❡ ❝♦♠♦ ♣r❡❝✉rs♦r ♦ á❜❛❝♦ ❞❡ ❝♦♥t❛❞♦r❡s s♦❧t♦s t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ tá❜✉❛ ❞❡ ❝♦♥t❛✲ ❣❡♠✳ ❈♦♥s✐❞❡r❛♠♦s ❡ss❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣r❡♣♦♥❞❡r❛♥t❡ ♣❡❧❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❛❞✐çã♦ ❝♦♠ tr❛♥s♣♦rt❡ ♣❛r❛ ❝♦♠♣♦r ❛ ♦r❞❡♠ s✉♣❡r✐♦r✸✳ ❊ss❛ ✐❞❡✐❛ s❡rá ❝♦♠♣❧❡♠❡♥✲ t❛r ♥♦ ❛♣r❡♥❞✐③❛❞♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❛ s✉❜tr❛çã♦ ♣❛r❛ ♣♦❞❡r♠♦s r❡❛❣r✉♣❛r ♦ ♠✐♥✉❡♥❞♦✹ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♣♦ss❛♠♦s ❞❡❝♦♠♣♦r ❛ ✉♥✐❞❛❞❡ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ♥❡ss❡ r❡❛❣r✉♣❛✲ ♠❡♥t♦ ❛ ✜♠ ❞❡ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❢✉♥❝✐♦♥❡ ❜❡♠✳ ❚❛♠❜é♠ ♣❛r❛ q✉❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ s❡❥❛ ❜❛st❛♥t❡ ❡❧✉❝✐❞❛t✐✈♦ ❡ ❝♦♠ t✉❞♦ ✐ss♦ s❡ ♣♦ss❛ s✉♣❡r❛r ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ✏♣❡❞✐r ❡♠♣r❡st❛❞♦✑ tã♦ ❞✐❢✉♥❞✐❞♦ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❜r❛s✐❧❡✐r❛✳
❯♠ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ t❛♥t♦ ♥❛ ❝♦♥t❛❣❡♠ q✉❛♥t♦ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❞❡✜♥✐çã♦ ❞❛ ❛❞✐çã♦ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❖ ❝♦♥❥✉♥t♦N={1,2,3,4,5,6,7, . . .}♥ã♦ ❡stá ❛q✉✐ ❞❡s❝r✐t♦ ❝♦♠♣❧❡t❛♠❡♥t❡✱ ♠❛s ❛ ♣❛rt✐r ❞❡ss❡s ❡❧❡♠❡♥t♦s ❡ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❡ ✐♠❛❣✐♥❛çã♦ ♣♦❞❡♠♦s ❝♦♥❝❡❜ê✲❧♦✳
❋✐❣✉r❛ ✶✿ ❯♠ ♠♦❞♦ ♠❛✐s s✉❣❡st✐✈♦ ❞❡ ✈✐s❧✉♠❜r❛r ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s
❈♦♥❢♦r♠❡ s✉❣❡r❡ ❬✼❪ ♥❡ss❡ ♠♦❞♦ ♠❛✐s s✉❣❡st✐✈♦✿
❚✉❞♦ ❝♦♠❡ç❛ ❝♦♠ ♦ ♥ú♠❡r♦ ✉♠✱ s✐♠❜♦❧✐③❛❞♦ ♣♦r ✶✱ q✉❡ r❡♣r❡s❡♥t❛ ❛ ✉♥✐✲ ❞❛❞❡✱ ❡ ❝♦♠ ✉♠❛ ❧❡✐✱ s✐♠❜♦❧✐③❛❞❛ ♣❡❧❛s ✢❡❝❤❛s✱ q✉❡ ❛ ❝❛❞❛ ♥ú♠❡r♦✱ ❝♦✲ ♠❡ç❛♥❞♦ ♣❡❧♦ ✶✱ ❢♦r♥❡❝❡ ♦ s❡✉ s✉❝❡ss♦r✱ ✐st♦ é✱ ♦ ♥ú♠❡r♦ q✉❡ ❧❤❡ s❡❣✉❡✳ ❙❛❜❡♠♦s t❛♠❜é♠ q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ ♥✉♥❝❛ t❡r♠✐♥❛❀ ♦✉ s❡❥❛✱ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s sã♦ ❡♠ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛✳
✸◆♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ ♥ã♦ ♣♦❞❡♠♦s t❡r ♠❛✐s q✉❡9✉♥✐❞❛❞❡s✱ ❡♥tã♦ q✉❛♥❞♦ t❡♠♦s10♦✉ ♠❛✐s ✉♥✐❞❛❞❡s
❝♦♠♣♦♠♦s ❛ ♦r❞❡♠ s✉♣❡r✐♦r ❞❛s ❞❡③❡♥❛s ♦♥❞❡ ❝❛❞❛10✉♥✐❞❛❞❡s ❢♦r♠❛♠ ✉♠❛ ♥♦✈❛ ❞❡③❡♥❛✳ ❖ ♠❡s♠♦
♦❝♦rr❡ ❝♦♠ ❛ ❞❡③❡♥❛✱ ❛ ❝❡♥t❡♥❛✱ ♦ ♠✐❧❤❛r ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳
✹❊♠ ❬✽❪✿ ❉■▼■◆❯❊◆❉❖✿ s♠✳ ❆r✐t✳ ◆✉♠❛ s✉❜tr❛çã♦✱ ♥ú♠❡r♦ ❞❡ q✉❡ s❡ s✉❜tr❛✐ ♦✉tr♦❀ ▼■◆❯✲
❊◆❉❖✳
❆ ❛❞✐çã♦ é ✉♠❛ ♦♣❡r❛çã♦ ❜ás✐❝❛ ♥♦s ♥❛t✉r❛✐s✳ ❙❡ ❡s❝♦❧❤❡r♠♦s✱ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧a✱ ♦ s✉❝❡ss♦r ❞❡a s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦ra+ 1✳
❋✐❣✉r❛ ✷✿ ❯♠ ♥ú♠❡r♦ a ❡ s❡✉ s✉❝❡ss♦r
❉❡✜♥✐çã♦ ✶✳ ✭❆❞✐çã♦✮ ❙❡❥❛♠ ❞❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛ ❡ ❜ ❞✐st✐♥t♦s✱ q✉❛✐sq✉❡r✳ P♦❞❡♠♦s ❞❡s❧♦❝❛r ❛ ❞❡ ❜ ♣♦s✐çõ❡s ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ♦❜t❡♥❞♦ ✉♠ ♥ú♠❡r♦ q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r ❛✰❜✳ ❊ss❛ ♦♣❡r❛çã♦ ❡♥tr❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s é ❝❤❛♠❛❞❛ ❞❡ ❛❞✐çã♦ ❡ ♦ ♥ú♠❡r♦ ❛ ✰ ❜ é ❝❤❛♠❛❞♦ s♦♠❛✺ ❞❡ ❛ ❡ ❜✳
❋✐❣✉r❛ ✸✿ ❯♠ ♥ú♠❡r♦a+b
➱ ❜❡♠ ✈❡r❞❛❞❡ q✉❡ ❛ ♠❛♥❡✐r❛ ❝♦♠♦ ❛ ❛❞✐çã♦ ❢♦✐ ❝♦♥❝❡✐t✉❛❞❛ ❡♠ s✉❛ ♦r✐❣❡♠ ♥ã♦ ❡stá ♣❡r❢❡✐t❛♠❡♥t❡ ❞❡✜♥✐❞❛✱ ❥á q✉❡ ♠❡s♠♦ ❡♠ é♣♦❝❛s ♠❛✐s r❡❝❡♥t❡s ♥ã♦ t❡♠♦s ♠✉✐t♦s r❡❣✐str♦s ❡s❝r✐t♦s ❞❡s✈❡♥❞❛♥❞♦ ♦ ♣r♦❝❡ss♦✳ ❖ q✉❡ s❡ s❛❜❡ ❛ r❡s♣❡✐t♦ ❞❛s ♣r✐♠❡✐r❛s ♣rát✐❝❛s ❞❡ ❛❞✐çã♦ s✉r❣✐✉ ❞❡ ❡s♣❡❝✉❧❛çõ❡s ♣❛r❛ ❛ ✉t✐❧✐③❛çã♦ ❞♦s ♣r✐♠❡✐r♦s ❞✐s♣♦s✐t✐✈♦s ❞❡ ❝á❧❝✉❧♦ ❡ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥t❛s ✜♥❛♥❝❡✐r❛s ❤✐stór✐❝❛s✳ ❆ ❡①❡♠♣❧♦ ❞✐ss♦✱ t❡♠♦s ❛ t❛❜✉❧❡t❛ ❞❡ ❛r❣✐❧❛ ❞❡ ❇❛❜✐❧♦♥✐❛♥ ❨❛❧❡ ❈♦❧❧❡❝t✐♦♥ ❞❛t❛❞❛ ❞❡ ✸✵✵✵ ❛✳❈✳
❆té ♦ sé❝✉❧♦ ❳❱■✱ ❛ tá❜✉❛ ❞❡ ❝♦♥t❛❣❡♠ ❡ ♦ á❜❛❝♦ ♣♦st❡r✐♦r ♣❡r♠❛♥❡❝❡r❛♠ ❝♦♠♦ ♣r✐♥❝✐♣❛✐s ❞✐s♣♦s✐t✐✈♦s ❞❡ ❝á❧❝✉❧♦✱ ♠❛s ♦s ❛♥t❡❝❡ss♦r❡s ❞♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ s✉r✲ ❣✐r❛♠ ♥♦ sé❝✉❧♦ ❳❱ ❝♦♥❢♦r♠❡ r❡❧❛t❛❞♦ ❡♠ ❬✶✺❪✱ q✉❛♥❞♦ ❞❡s❝r❡✈❡ ♦ tr❡❝❤♦ ❞❛ ♦❜r❛ ✧t❤❡ ❡❛r❧✐❡st ❛r✐t❤♠❡t✐❝s ✐♥ ❡♥❣❧✐s❤✑ ❞❡ ❛✉t♦r✐❛ ❛♥ô♥✐♠❛✿
✺❊♠ ❬✽❪✿ s❢✳ ❆r✐t✳ ❖♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ♦✉ ♦ s❡✉ r❡s✉❧t❛❞♦
❆q✉✐ ❝♦♠❡ç❛ ♦ ♦❢í❝✐♦ ❞❛ ❛❞✐çã♦✳ ◆❡st❡ ♦❢í❝✐♦✱ ✈♦❝ê ❞❡✈❡ s❛❜❡r q✉❛tr♦ ❝♦✐s❛s✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡✈❡ s❛❜❡r ♦ q✉❡ é ❛❞✐çã♦✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡✈❡ s❛❜❡r q✉❛♥t❛s ❧✐♥❤❛s ❞❡ ✜❣✉r❛s ✈♦❝ê ❞❡✈❡ t❡r✳ ❉❡♣♦✐s✱ ❞❡✈❡ s❛❜❡r q✉❛♥t♦s ❝❛s♦s ❞✐❢❡r❡♥t❡s ❛❝♦♥t❡❝❡♠ ♥❡st❡ ♦❢í❝✐♦✳ ◗✉❛♥t♦ à ♣r✐♠❡✐r❛✱ ✈♦❝ê ❞❡✈❡ s❛❜❡r q✉❡ ❛ ❛❞✐çã♦ é ✉♠ s♦♠❛tór✐♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❡♠ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦✳ ◗✉❛♥t♦ à s❡❣✉♥❞❛✱ ❞❡✈❡ s❛❜❡r q✉❡ t❡rá ❞✉❛s ❧✐♥❤❛s ❞❡ ✜❣✉r❛s✱ ✉♠❛ s♦❜r❡ ❛ ♦✉tr❛ ✭✳✳✳✮✳ ◗✉❛♥t♦ à t❡r❝❡✐r❛✱ ❞❡✈❡ s❛❜❡r q✉❡ ❡①✐st❡♠ q✉❛tr♦ ❝❛s♦s ❞✐❢❡r❡♥t❡s✳ ◗✉❛♥t♦ à q✉❛rt❛✱ ❞❡✈❡ s❛❜❡r q✉❡ ♦ r❡s✉❧t❛❞♦ ❞❡st❡ ♦❢í❝✐♦ é ❞✐③❡r q✉❛❧ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❡ r❡s✉❧t❛ ❞❛ s♦♠❛ ❞❡ss❡s ♥ú♠❡r♦s ❞✐❢❡r❡♥t❡s✳
✶✹✼ ✷✼✶
◆❛ t❡r❝❡✐r❛ ❝♦✐s❛ ❛ s❛❜❡r✱ q✉❛♥❞♦ s❡ r❡❢❡r❡ ❛♦s q✉❛tr♦ ❝❛s♦s t❡♠♦s q✉❡✿
✶✳ ◆❡♥❤✉♠❛ s♦♠❛ ♣❛r❝✐❛❧ é ♠❛✐♦r ❞♦ q✉❡ ✾❀
✷✳ P❡❧♦ ♠❡♥♦s ✉♠❛ s♦♠❛ ♣❛r❝✐❛❧ é ♠❛✐♦r ❞♦ q✉❡ ✾ ✭♦✉ s❡❥❛✱ ✶✰✷✱ ✹✰✼ ♦✉ ✼✰✶ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✮❀
✸✳ P❡❧♦ ♠❡♥♦s ✉♠❛ s♦♠❛ ♣❛r❝✐❛❧ é ✶✵ ♦✉ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✶✵❀
✹✳ ❊①✐st❡ ✉♠ ③❡r♦ ♥❛ ❧✐♥❤❛ s✉♣❡r✐♦r✳
❖♥❞❡✱ é ❝❧❛r♦✱ ✉♠ ❝❛s♦ ❡①❝❧✉✐ ♦s ♦✉tr♦s✳ ❊♠ t❡♠♣♦s ♠❛✐s r❡❝❡♥t❡s✱ ❡ss❡s q✉❛tr♦ ❝❛s♦s ❢♦r❛♠ ❝r✐st❛❧✐③❛❞♦s ❡ ❛ ❞❡s✐❣♥❛çã♦ ❞❡ tr❛♥s♣♦rt❡s é ❢r✉t♦ ❞❡ é♣♦❝❛s ♠❛✐s ♠♦❞❡r♥❛s r❡s✉❧t❛♥❞♦ ♥♦ ♠❡❝❛♥✐s♠♦ ❡✜❝✐❡♥t❡ q✉❡ ❞❡♥♦♠✐♥❛♠♦s ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❡ ❛❞✐çã♦✳
✷✳✶ ❆❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❛ ❛❞✐çã♦
11 4 7
+ 2 7 1
4 1 8
❆ ♣❛ss❛❣❡♠ ❞❛ ❝♦♥t❛❣❡♠ ♣❛r❛ ❛❞✐çã♦ ❢✉♥❝✐♦♥❛ ❝♦♠♦ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ♣r♦❣r❡ssã♦ ♥❛✲ t✉r❛❧ ❡ ❡s♣♦♥tâ♥❡❛✳ P♦❞❡✲s❡ ♣❛ss❛r ❞❛ ❝♦♥t❛❣❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❧♦❝♦s à ❝♦♥t❛❣❡♠ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❜❧♦❝♦s ❢♦r♠❛♥❞♦ ❛ ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s✳
❋✐❣✉r❛ ✹✿ ❉❛ ❝♦♥t❛❣❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❧♦❝♦s à ❝♦♥t❛❣❡♠ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❜❧♦❝♦s
◗✉❛♥❞♦ ♣❡r❝❡❜❡✲s❡ q✉❡ ♦ ú❧t✐♠♦ ♥ú♠❡r♦ ❝♦♥t❛❞♦ ❞❡♥tr♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ é ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❡❧❡♠❡♥t♦s ♥❡ss❡ ❝♦♥❥✉♥t♦ é ♣♦ssí✈❡❧ ❝♦♠❡ç❛r ❛ ✐♥❝❧✉✐r ✉♠❛ q✉❛♥t✐❞❛❞❡ ♥❛ ♦✉tr❛✱ ♦✉ s❡❥❛✱ ❝♦♥t❛r ✉♠ ♦✉tr♦ ❝♦♥❥✉♥t♦ ❞❡ ♦❜❥❡t♦s ❛ ♣❛rt✐r ❞♦ ú❧t✐♠♦ ♥ú♠❡r♦ ❢❛❧❛❞♦ ♥❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❞♦ ♣r✐♠❡✐r♦ ❝♦♥❥✉♥t♦✳
❙❡❣✉♥❞♦ ❛ ♣❡rs♣❡❝t✐✈❛ ❞❡ ❬✶✺❪✱ ♥❛ ❡❞✉❝❛çã♦ ❛♠❡r✐❝❛♥❛ ❛s ❝r✐❛♥ç❛s ❡stã♦ r❡❢♦rç❛♥❞♦ ❡ss❡s ♣r♦❝❡❞✐♠❡♥t♦s ♥❛ ♠❡♠ór✐❛ ♥♦ ✜♥❛❧ ❞♦ ✶➸ ❡ ♥♦ ✷➸ ❛♥♦✳ ❆ t❛❜✉❛❞❛ ❝♦♠ s♦♠❛ ❛té ✺✱ ♣♦r ❡①❡♠♣❧♦✱2 + 3 = 5 ❡1 + 4 = 5✱ sã♦ ❛♣r❡♥❞✐❞♦s ♣r✐♠❡✐r♦ ❡ ♣♦✉❝♦ ❞❡♣♦✐s ❝♦♠ s♦♠❛
❛té ✶✵ ❡✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ✷✵✳ ❯♠ ❢❛t♦ ❡s♣❡❝✉❧❛❞♦ é q✉❡ ❛s ❝r✐❛♥ç❛s ♥♦rt❡✲❛♠❡r✐❝❛♥❛s ❛♣r❡s❡♥t❛♠ ❞✐✜❝✉❧❞❛❞❡✱ ❞❡✈✐❞♦ ❛ ❢♦r♠❛ ✐rr❡❣✉❧❛r ❞❛s ♣❛❧❛✈r❛s✱ ❡ ♥ã♦ t❡♠ ♦ ♠❡s♠♦ ❞❡s❡♠♣❡♥❤♦ ❞❛s ❝r✐❛♥ç❛s ❛s✐át✐❝❛s✱ q✉❡ ♥♦r♠❛❧♠❡♥t❡ ♥♦ ✶➸ ❛♥♦✱ sã♦ ❡♥s✐♥❛❞❛s ❛ ❢❛③❡r ❞❡③ ❛♦ ❛❞✐❝✐♦♥❛r✱ ♣♦r ❡①❡♠♣❧♦✱6 + 5 = 1 + (5 + 5)✳ ❊ss❛ ❡str❛té❣✐❛ é t❛♠❜é♠ ✉t✐❧✐③❛❞❛
q✉❛♥❞♦ s❡ ✉t✐❧✐③❛ ♦ á❜❛❝♦✳ ❆té ❛❧❣✉♥s ❛❞✉❧t♦s ❥❛♣♦♥❡s❡s ❞✐③❡♠ q✉❡ ♣❛r❛ s♦♠❛r 6 + 3
✈✐s✉❛❧✐③❛♠ ♥❛ ❢♦r♠❛ (5 + 1) + 3✳ ❚❛✐s ❢❛t♦s ❞❛ ❛❞✐çã♦ q✉❛♥❞♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♠ ❛s
♥♦çõ❡s✱ ♥❛ ✈❡r❞❛❞❡✱ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✱ sã♦ ❡ss❡♥❝✐❛✐s ♥♦ ♦❢í❝✐♦ ❞❡ ❢❛③❡r ❛❞✐çõ❡s✳
3 + 5 = 5 + 3 ✭❝♦♠✉t❛t✐✈❛✮✱ (3 + 5) + 2 = 3 + (5 + 2) ✭❛ss♦❝✐❛t✐✈❛✮✳
❖❜s❡r✈❡♠♦s ❛ ♥♦t❛ s♦❜r❡ ♦ s✐st❡♠❛ ❝❤✐♥ês ❞❡ ♥ú♠❡r♦✲♣❛❧❛✈r❛ ❡♠ ❬✶✷❪✿
❖ s✐st❡♠❛ ❝❤✐♥ês ❞❡ ♥ú♠❡r♦✲♣❛❧❛✈r❛ ♣♦❞❡ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ❛t❡♥çã♦ ♣❛rt✐✲ ❝✉❧❛r q✉❡ ♦s ♣r♦❢❡ss♦r❡s ❝❤✐♥❡s❡s ❞ã♦ à ❝♦♠♣♦s✐çã♦ ❡ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ✶✵✳ ❊♠ ❝❤✐♥ês✱ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛ ❝❛s❛ ❞♦s ✶✵ tê♠ ❛ ❢♦r♠❛ ❞❡ ✓❞❡③✱ ♥ú✲ ♠❡r♦ ❞❡ ✉♠ ❛❧❣❛r✐s♠♦✔✳ P♦r ❡①❡♠♣❧♦✱ ♦♥③❡ é ✓❞❡③✲✉♠✱✔ ❞♦③❡ é ✓❞❡③✲❞♦✐s✔ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ✭❱✐♥t❡ é ✓❞♦✐s ❞❡③✔✱ tr✐♥t❛ é ✓três ❞❡③✔ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❱✐♥t❡ ❡ ✉♠ é ❝❤❛♠❛❞♦ ✓❞♦✐s ❞❡③✲✉♠✔✱ ✈✐♥t❡ ❡ ❞♦✐s é ✓❞♦✐s ❞❡③✲❞♦✐s✔ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳✮ P♦r ✐ss♦✱ ✓❞❡❝♦♠♣♦r ✉♠ ✶✵✔ t❡♥❞❡ ❛ s❡r ✉♠❛ s♦❧✉çã♦ ó❜✈✐❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✓❈♦♠♦ s❡ s✉❜tr❛✐ ✺ ❞❡ ❞❡③✲❞♦✐s❄✔
❙❡r✐❛ ♣r♦✈❡✐t♦s♦ ❢❛③❡r ✉♠❛ ❛ss♦❝✐❛çã♦ ❞♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ s✐st❡♠❛ ❝❤✐♥ês ❞❡ ♥ú♠❡r♦✲ ♣❛❧❛✈r❛ ❛♦ ♥♦ss♦ s✐st❡♠❛ ❞❡ ♥ú♠❡r♦✲♣❛❧❛✈r❛ ❡♠ ❧í♥❣✉❛ ♣♦rt✉❣✉❡s❛ ❛ ✜♠ ❞❡ ♣r♦♣✐❝✐❛r ❛♦s ❛❧✉♥♦s ❞♦ ✷➸ ❝✐❝❧♦ ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ✉♠❛ ❝♦♥❝❡♣çã♦ ❞✐❢❡r❡♥❝✐❛❞❛✳ ❚❛❧ ❝♦♥❝❡♣çã♦ ♣♦❞❡ s❡r ❜❛st❛♥t❡ ❢❛✈♦rá✈❡❧ ❛♦ s❡ ✉t✐❧✐③❛r ❛ s♦♠❛ ❝♦♠ tr❛♥s♣♦rt❡✱ ♦✉ s❡❥❛✱ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ♦r❞❡♠ s✉♣❡r✐♦r ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧ ♦✉ ❛✐♥❞❛ ❞❡❝♦♠♣♦r ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❡♠ ✉♥✐❞❛❞❡s ❞❡ ♦r❞❡♠ ✐♥❢❡r✐♦r ♥♦ ❛❧❣♦r✐t♠♦ ✉s✉❛❧ ♣❛r❛ s✉❜tr❛çã♦ ❛ ✜♠ ❞❡ r❡❛❣r✉♣❛r ♦ ♠✐♥✉❡♥❞♦✳ ❆ ❡①♣r❡ssã♦ ✏❞❡❝♦♠♣♦r ✉♠❛ ✉♥✐❞❛❞❡ ❡♠ ♦r❞❡♠ s✉♣❡r✐♦r✑ é ✉♠ t❡r♠♦ ❞❛ ❛r✐t♠ét✐❝❛ ❝❤✐♥❡s❛ tr❛❞✐❝✐♦♥❛❧ ❜❛s❡❛❞❛ ♥♦ á❜❛❝♦✳
❱♦❧t❛♥❞♦ à ❛❞✐çã♦ ♥♦ ❛❧❣♦r✐t♠♦ ✉s✉❛❧✿
11 4 7
+ 2 7 1
4 1 8
✭✶✮
❙❡ ✉t✐❧✐③❛r♠♦s ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♣❛r❛ ❡❢❡t✉❛r ❛ ❛❞✐çã♦ ❛❝✐♠❛ ♣♦❞❡rí❛♠♦s ❞❡s❝r❡✈❡r ♦ ♣r♦❝❡ss♦ ✐♥✐❝✐❛♥❞♦ ♣❡❧❛ ❝♦❧✉♥❛ ❞❛s ✉♥✐❞❛❞❡s s♦♠❛♥❞♦ ♦ ✼ ❡ ♦ ✶✱ ♣❛r❛ ✉♠ r❡s✉❧t❛❞♦ ✽✱ ❡ ❡s❝r❡✈❡♥❞♦ ❡ss❡ ❛❧❣❛r✐s♠♦ ♥❛ ❝♦❧✉♥❛ ❞❛s ✉♥✐❞❛❞❡s✳ ❊♠ s❡❣✉✐❞❛ s♦♠❛rí❛♠♦s ♦ ✹ ❡ ♦ ✼ ✭♥❛ ✈❡r❞❛❞❡✱ ✹✵ ❡ ✼✵✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮ ♣❛r❛ ✉♠ r❡s✉❧t❛❞♦ ✶✶ ✭♥❛ ✈❡r❞❛❞❡✱ ✶✶✵✮✳ ▼❛s ♦ ✈❛❧♦r ♣♦s✐❝✐♦♥❛❧ ♥ã♦ ♣❡r♠✐t❡ q✉❡ ♦ ✶✶ r❡♣r❡s❡♥t❡ ❛ ❝♦❧✉♥❛ ❞❛s ❞❡③❡♥❛s✱ ❧♦❣♦ ❞❡✈❡♠♦s ✏tr❛♥s♣♦rt❛r✑ ♦ ❛❧❣❛r✐s♠♦ ✶ ❞❡ss❡ ✶✶ ✭q✉❡✱ ♥❛ ✈❡r❞❛❞❡✱ ✈❛❧❡ ✶✵✵✮ à ❝♦❧✉♥❛ ❞❛s ❝❡♥t❡♥❛s ❡ ❡s❝r❡✈❡r ♦ ❛❧❣❛r✐s♠♦ ✶ ♥❛ ❝♦❧✉♥❛ ❞❛s ❞❡③❡♥❛s ✭ q✉❡ é✱ ♥❛ r❡❛❧✐❞❛❞❡✱ ✶✵✮✳ ❉❡♣♦✐s ❛❞✐❝✐♦♥❛♠♦s ♦ ✶ ✏tr❛♥s♣♦rt❛❞♦✑✱ ♦ ✶ ❡ ♦ ✷ ✭ q✉❡ sã♦✱ ♥❛ ✈❡r❞❛❞❡✱ ✶✵✵✱ ✶✵✵ ❡ ✷✵✵✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮✱ ♥❛ t❡r❝❡✐r❛ ❝♦❧✉♥❛ r❡s✉❧t❛♥❞♦ ❡♠ ✹ ✭♥❛✱ r❡❛❧✐❞❛❞❡✱ ✹✵✵✮✱ ❥á q✉❡ ❡stá ♥❛ ❝♦❧✉♥❛ ❞❛s ❝❡♥t❡♥❛s✳ ❖ r❡s✉❧t❛❞♦ ✜♥❛❧ é ✹✶✽✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ❡♠ ✭✶✮✳
❖ q✉❡ ❡stá ❞❡s❝r✐t♦ ❛❝✐♠❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♣❛r❛ ❛❞✐✲ çã♦ é ✉♠ t❛♥t♦ ❝❛♥s❛t✐✈♦ ♥❛ ❤♦r❛ ❞❡ ❞❡s❝r❡✈❡r ❡ t♦❞♦s✱ ♥♦r♠❛❧♠❡♥t❡✱ ❡stã♦ ❢❛♠✐❧✐❛✲ r✐③❛❞♦s ❝♦♠ ❡ss❡ ♣r♦❝❡ss♦✳ ❆❣♦r❛✱ ❛ r❛③ã♦ ♣❡❧❛ q✉❛❧ ♦ ♣r♦❝❡ss♦ ❢✉♥❝✐♦♥❛ ♥ã♦ é ♠✉✐t♦ ✐♥t✉✐t✐✈❛✳ ◆♦ s✐st❡♠❛ ♣♦s✐❝✐♦♥❛❧ ❞❡ ❜❛s❡ 10 ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❛❧q✉❡r ♥ú♠❡r♦ N
♥❛t✉r❛❧ ❝♦♠♦ ✉♠❛ s♦♠❛ ❧✐♥❡❛r ❞❡ ♣♦tê♥❝✐❛s ❞❡10✳ ❆ s❛❜❡r✱
N =an10n+a
n−110n−1+· · ·+a2102+a1101+a0✳
❙❛❜❡♠♦s q✉❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❢♦r❛♠ r❡♣r❡s❡♥t❛❞♦s ❞❡ ❞✐✈❡rs♦s ♠♦❞♦s ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ♣♦s✐❝✐♦♥❛❧ é ❜❡♠ ❞✐❢✉♥❞✐❞❛ ❡ q✉❛s❡ ✉♥✐✈❡rs❛❧✲ ♠❡♥t❡ ✉t✐❧✐③❛❞❛✳ ❙❡❣✉♥❞♦ ❬✼❪ tr❛t❛✲s❡ ❞❡ ✉♠❛ ✈❛r✐❛♥t❡ ❞♦ s✐st❡♠❛ s❡①❛❣❡s✐♠❛❧ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❜❛❜✐❧ô♥✐♦s✳ ❖ s✐st❡♠❛ ❞❡❝✐♠❛❧ ♣♦s✐❝✐♦♥❛❧ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♥❛ ❈❤✐♥❛ ❡ ♥❛ ❮♥❞✐❛✳ ❚♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❢♦r♠❛❞❛ ♣❡❧♦s10 ❛❧❣❛r✐s♠♦s 1,2,3,4,5,6,7,8,9,0✳
❊①❛t❛♠❡♥t❡ ♣♦r ❡①✐st✐r❡♠ 10 ❛❧❣❛r✐s♠♦s ♥❡ss❡ s✐st❡♠❛ é q✉❡ ❡❧❡ é ❞❡♥♦♠✐♥❛❞♦
❞❡❝✐♠❛❧✳ ❈❛❞❛ ❛❧❣❛r✐s♠♦✱ ❛❧é♠ ❞❡ s❡✉ ✈❛❧♦r ❛❜s♦❧✉t♦✱ ♣♦ss✉✐ ✉♠❛ ✈❛❧♦r❛çã♦ ❞❡✈✐❞❛ à ♣♦s✐çã♦ q✉❡ ♦❝✉♣❛ ❞❡♥tr♦ ❞❛ s❡q✉ê♥❝✐❛✱ ♦✉ s❡❥❛✱ ♦ ❢❛t♦r é ❞❛❞♦ ♣♦r ✉♠❛ ♣♦tê♥❝✐❛ ❞❡10
q✉❡ ✈❛r✐❛ ❛ss✐♠✿
❖ ❛❧❣❛r✐s♠♦ ♠❛✐s à ❞✐r❡✐t❛ t❡♠ ❢❛t♦r 100 = 1❀ ♦ ♣ró①✐♠♦✱ ❧❡♥❞♦ ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛
❡sq✉❡r❞❛✱ t❡♠ ❢❛t♦r 101 = 10❀ ♦ ♣ró①✐♠♦ t❡♠ ❢❛t♦r 102 = 100❀ ♦ ♣ró①✐♠♦ t❡♠ ❢❛t♦r 103 = 1000 ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳
▲♦❣♦✱ ♦ ♥ú♠❡r♦ ✶✶✸✷✱ ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦
1×103+ 1×102+ 3×10 + 2×100✳
❖s ♥ú♠❡r♦s ③❡r♦s ❡s❝r✐t♦s à ❡sq✉❡r❞❛ ❞❡ ✉♠ ♥ú♠❡r♦ sã♦ ✐rr❡❧❡✈❛♥t❡s✱ ❥á q✉❡
001132 = 0×105+ 0×104+ 1×103+ 1×102+ 3×10 + 2×100
= 1×103+ 1×102+ 3×10 + 2×100
= 1132.
❚♦❞♦s ♦s ❛❧❣❛r✐s♠♦s ❞❡ ✉♠ ♥ú♠❡r♦ ♣♦ss✉❡♠ ✉♠❛ ♦r❞❡♠✱ t♦♠❛❞❛ ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❛ ❡sq✉❡r❞❛✳ ◆♦ ♥ú♠❡r♦ ❛❝✐♠❛✱ ✶✶✸✷✱ ♦ ✷ é ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ♦ ✸ é ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ♦ ✶ é ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ ❡ ♦ ♦✉tr♦ ✶ é ❞❡ q✉❛rt❛ ♦r❞❡♠✳
❈❛❞❛ três ♦r❞❡♥s✱ t❛♠❜é♠ t♦♠❛❞❛s ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❡sq✉❡r❞❛✱ ❢♦r♠❛♠ ✉♠❛ ❝❧❛ss❡✳ ❆s ❝❧❛ss❡s ♣♦❞❡♠ ✈✐r s❡♣❛r❛❞❛s ♣♦r ✉♠ ♣♦♥t♦✳ ❖❜s❡r✈❡ ❝♦♠♦ sã♦ ❞❡♥♦♠✐♥❛❞❛s ❛s ❝❧❛ss❡s✿
❈❧❛ss❡ ❞❛s ❯♥✐❞❛❞❡s
❯♥✐❞❛❞❡s✿ ✶➟ ♦r❞❡♠ ❉❡③❡♥❛s✿ ✷➟ ♦r❞❡♠ ❈❡♥t❡♥❛s✿ ✸➟ ♦r❞❡♠
❈❧❛ss❡ ❞♦ ▼✐❧❤❛r
❯♥✐❞❛❞❡s ❞❡ ♠✐❧❤❛r✿ ✹➟ ♦r❞❡♠ ❉❡③❡♥❛s ❞❡ ♠✐❧❤❛r✿ ✺➟ ♦r❞❡♠ ❈❡♥t❡♥❛s ❞❡ ♠✐❧❤❛r✿ ✻➟ ♦r❞❡♠
❈❧❛ss❡ ❞♦ ▼✐❧❤ã♦
❯♥✐❞❛❞❡s ❞❡ ♠✐❧❤ã♦✿ ✼➟ ♦r❞❡♠ ❉❡③❡♥❛s ❞❡ ♠✐❧❤ã♦✿ ✽➟ ♦r❞❡♠ ❈❡♥t❡♥❛s ❞❡ ♠✐❧❤ã♦✿ ✾➟ ♦r❞❡♠
◗✉❛♥❞♦ ♥ã♦ ❡s♣❡❝✐✜❝❛r♠♦s ❛ ❜❛s❡ ♥✉♠ér✐❝❛ ❞❡ ✉♠ s✐st❡♠❛ é ♣♦rq✉❡ ❡st❛♠♦s ♥♦s r❡❢❡r✐♥❞♦ à ❜❛s❡ ❞❡❝✐♠❛❧✳
❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♠♦str❛ ❝♦♠♦ ❡s❝r❡✈❡r q✉❛❧q✉❡r ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡♠ q✉❛❧q✉❡r ❜❛s❡ ❞✳
❚❡♦r❡♠❛ ✶✳ ❙❡❥❛♠ n≥0 ❡ d >1✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s❡q✉ê♥❝✐❛ ✭♦s ✏❞í❣✐t♦s✑ ❞❡
♥ ♥❛ ❜❛s❡ ❞✮ a0, . . . , ak, . . . ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✶✳ ♣❛r❛ t♦❞♦ k✱ 0≤ak< d❀
✷✳ ❡①✐st❡ m t❛❧ q✉❡ s❡ k ≥m✱ ❡♥tã♦ ak= 0❀
✸✳ n =X k≥0
akdk✳
❆ ❞❡♠♦♥str❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✸❪✳
❉❡♠♦♥str❛çã♦✻ ✿
❊s❝r❡✈❡♠♦s n = n0 = n1d+a0, 0 6 a0 < d, n1 = n2d+a1, 0 6 a1 < d ❡
❡♠ ❣❡r❛❧nk =nk+1d+ak, 06ak < d✳ ◆♦ss❛ ♣r✐♠❡✐r❛ ❛✜r♠❛çã♦ é q✉❡ nk = 0♣❛r❛
❛❧❣✉♠ ✈❛❧♦r ❞❡ k✳ ❉❡ ❢❛t♦✱ s❡ n0 < dm✱ ❡♥tã♦ n1 = ⌊
n0
d ⌋ < d
m−1 ❡ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱
♣♦r ✐♥❞✉çã♦ ✱ nk < dm−k❀ ❢❛③❡♥❞♦ k > m t❡♠♦s nk <1 ❞♦♥❞❡ nk = 0✳ ❙❡❣✉❡ ❞❛í q✉❡
ak= 0 ♣❛r❛ k>m✳ ❆ ✐❞❡♥t✐❞❛❞❡ ❞♦ ✐t❡♠3 é ❢❛❝✐❧♠❡♥t❡ ❞❡♠♦♥str❛❞❛ ♣♦r ✐♥❞✉çã♦✳
P❛r❛ ❛ ✉♥✐❝✐❞❛❞❡✱ s✉♣♦♥❤❛X
k>0
akdk =
X
k>0
bkdk✳ ❙❡ ❛s s❡q✉ê♥❝✐❛sak❡bksã♦ ❞✐st✐♥t❛s
❡①✐st❡ ✉♠ ♠❡♥♦r í♥❞✐❝❡ ✱ ❞✐❣❛♠♦s j✱ ♣❛r❛ ♦ q✉❛❧ aj 6= bj✳ P♦❞❡♠♦s ❡s❝r❡✈❡r aj +
X
k>j
akdk−j = bj +
X
k>j
bkdk−j ❞♦♥❞❡ aj ≡ bj(mod d)✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s
0<|aj −bj|< d❡ ♣♦rt❛♥t♦ aj −bj ♥ã♦ ♣♦❞❡ s❡r ♠ú❧t✐♣❧♦ ❞❡d✳
❱♦❧t❛♥❞♦ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ só q✉❡ ✉t✐❧✐③❛♥❞♦ ♦✉tr❛ ❛❜♦r❞❛❣❡♠✱ ✈❛♠♦s ❞❡s❡♥✈♦❧✲ ✈❡r ❛ ❛❞✐çã♦ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡10✳
✷✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❛❞✐çã♦ ✉s❛♥❞♦ ❛ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡ ✶✵
147 = 1×102+ 4×10 + 7✱ 271 = 2×102+ 7×10 + 1✳
❙♦♠❛♥❞♦ ❡ss❡s ♥ú♠❡r♦s✱ ♦❜t❡♠♦s
147+271 = 1×102+4×10+7+2×102+7×10+1 = 1×100+2×100+4×10+7×10+7+1✳
❖ ✉s♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛ ♥♦s ❢♦r♥❡❝❡✿
147 + 271 = (1 + 2)×100 + (4 + 7)×10 + (7 + 1). ✭✷✮
➱ ❢✉♥❞❛♠❡♥t❛❧ q✉❡ t❛♥t♦ ❡♠ ✭✶✮ q✉❛♥t♦ ❡♠ ✭2✮ ❛s ♣♦tê♥❝✐❛s ❞❡ ✶✵ ❡st❡❥❛♠ ❡♠♣❛✲
r❡❧❤❛❞❛s ❡♠ ❝❛❞❛ ✉♠ ❞♦s ♠❡♠❜r♦s ♦✉ ♣❛r❝❡❧❛s✼✳
✻❆ ♥♦t❛çã♦ ⌊x⌋=max{m∈Z;m≤x} r❡♣r❡s❡♥t❛ ❛ ♣❛rt❡ ✐♥t❡✐r❛ ❡ ❛ ♥♦t❛çã♦a ≡b(mod n) é ❛
r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉❡ê♥❝✐❛ ♠ó❞✉❧♦m✳ P❛r❛ ♦❜t❡r ♠❛✐s ✐♥❢♦r♠❛çõ❡s ✈❡r ❬✶✸❪✳
✼❊♠ ❬✽❪✿ ▼❛t✳ ❈❛❞❛ ✉♠❛ ❞❛s q✉❛♥t✐❞❛❞❡s q✉❡ ♣❡r❢❛③❡♠ ✉♠❛ s♦♠❛✱ ♦✉ ✉♠ ✈❛❧♦r t♦t❛❧✳
❖❜s❡r✈❛çã♦✿ ◆♦ ❛❧❣♦r✐t♠♦ ✉s✉❛❧✱ ✐ss♦ ♥ã♦ ♣❛r❡❝❡ s❡r tã♦ s✐❣♥✐✜❝❛t✐✈♦✱ ♠❛s é ✐♠✲ ♣♦rt❛♥t❡ ♣❛r❛ ♠✐♥✐♠✐③❛r ❡rr♦s ❝♦♠✉♥s✱ q✉❡ ♥♦r♠❛❧♠❡♥t❡ sã♦ ❝♦♠❡t✐❞♦s ♥♦ ✐♥í❝✐♦✱ t❛✐s ❝♦♠♦
147 + 271 = 1×1000 + (4 + 2)×100 + (7 + 7)×10 + 1✳
◆♦ ❛❧❣♦r✐t♠♦ ❞❡ ❛❞✐çã♦✱ ❡ss❡ ❡rr♦ ❛♣❛r❡❝❡r✐❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
1 14 7
+ 2 7 1
1 7 4 1
❆❣♦r❛✱ ✈♦❧t❛♥❞♦ à ❝♦♥t❛ ❛♥t❡r✐♦r✱ s♦♠❛♥❞♦ ♦s ❛❧❣❛r✐s♠♦s ♥❛ ♣♦s✐çã♦ ❞❛s ✉♥✐❞❛❞❡s✱ t❡♠♦s✿
147 + 271 = (1 + 2)×100 + (4 + 7)×10 + (7 + 1)×1 ✭✸✮
= (1 + 2)×100 + (4 + 7)×10 + (8)×1.
❖ ♣ró①✐♠♦ ♣❛ss♦ s❡rá s♦♠❛r ♦s ❛❧❣❛r✐s♠♦s ♥❛ ♣♦s✐çã♦ ❞❛s ❞❡③❡♥❛s✿
147 + 271 = (1 + 2)×100 + (4 + 7)×10 + 8×1 ✭✹✮
= (1 + 2)×100 + (11)×10 + 8×1.
❆❣♦r❛✱ ✈❛♠♦s s♦♠❛r ♦s ❛❧❣❛r✐s♠♦s ♥❛ ♣♦s✐çã♦ ❞❛s ❝❡♥t❡♥❛s✳
147 + 271 = (1 + 2)×100 + 11×10 + 8×1 ✭✺✮
= (3)×100 + 11×10 + 8×1.
▼❛s✱ ♥♦ s✐st❡♠❛ ♣♦s✐❝✐♦♥❛❧ ❞❡❝✐♠❛❧✱ ♥ã♦ ♣♦❞❡♠♦s t❡r ✉♠❛ ♦r❞❡♠ r❡♣r❡s❡♥t❛❞❛ ♣♦r ❞♦✐s ❛❧❣❛r✐s♠♦s ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ✭5✮ q✉❛♥❞♦ ♥❛ ♣♦s✐çã♦ ❞❛s ❞❡③❡♥❛s ✈ê✲s❡11×10✱
♦✉ s❡❥❛✱ ♥ã♦ ♣♦❞❡ ♦❝♦rr❡r ♦ s❡❣✉✐♥t❡✿
147 + 271 = 3118✱
❙❡♥❞♦ ✉♠ ❡rr♦ ❝♦♠✉♠ ❡♥❝♦♥tr❛❞♦ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦ ♣♦r ♣❛rt❡ ❞❡ ❛❧❣✉♥s ❡st✉❞❛♥t❡s ♥ã♦ tã♦ ❢❛♠✐❧✐❛r✐③❛❞♦s ❝♦♠ ❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❛❞✐çã♦✳
❆❣♦r❛✱ tr❛❜❛❧❤❛r❡♠♦s ♥❛ ❝♦♠♣♦s✐çã♦ ❞❛ ♦r❞❡♠ s✉♣❡r✐♦r q✉❡ é ❢r❡q✉❡♥t❡♠❡♥t❡ ❝❤❛✲ ♠❛❞♦ ❞❡ ❛❞✐çã♦ ❝♦♠ tr❛♥s♣♦rt❡✿
11×10 = 10×10 + 1×10 ✭✻✮
= 1×100 + 1×10
P♦rt❛♥t♦✱
147 + 271 = (1 + 2)×100 + 1×100 + 1×10 + 8×1 ✭✼✮
= (1 + 2 +✶)×100 + 1×10 + 8×1 ✭✽✮
❙❡ ❡①❛♠✐♥❛r♠♦s ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❡♠ ✭✶✮ q✉❛♥❞♦ t❡♠♦s ♦ ❛❧❣❛r✐s♠♦ ✏✶✑ ❡♠ ❝✐♠❛ ❞♦ ❛❧❣❛r✐s♠♦ ✏✶✑ ❞♦ ♥ú♠❡r♦ 147✱ ✈❡r❡♠♦s q✉❡ é ❡①❛t❛♠❡♥t❡ ♦ ❛❧❣❛r✐s♠♦ ✏✶✑
q✉❡ ❡stá s✉❜❧✐♥❤❛❞♦ ❡♠ ✭✽✮✳ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♠ ❛ s♦♠❛ ❞♦s ❛❧❣❛r✐s♠♦s ♥❛ ♣♦s✐çã♦ ❞❛s ❝❡♥t❡♥❛s t❡r❡♠♦s
147 + 271 = 4×100 + 1×10 + 8×1 ✭✾✮
= 418
✷✳✸ ▼ét♦❞♦ ❞❛s s♦♠❛s ♣❛r❝✐❛✐s
❖✉tr❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❛❞✐çã♦ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞♦ ❡♠ ♥♦t❛çã♦ ✈❡rt✐❝❛❧ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠ét♦❞♦ ❞❡ s♦♠❛s ♣❛r❝✐❛✐s✿
100 + 40 + 7
200 + 70 + 1
300 + 110 + 8
✭✶✵✮
❋❛r❡♠♦s ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞♦s ♥ú♠❡r♦s ❡♥✈♦❧✈✐❞♦s ❡♠ ❝❡♥t❡♥❛s ✐♥t❡✐r❛s ✭♦✉ ❡①❛t❛s✮✱ ❞❡③❡♥❛s t❛♠❜é♠ ✐♥t❡✐r❛s ❡ ✉♥✐❞❛❞❡s✳ ➱ ❝❧❛r♦ q✉❡ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ ♠❡s♠♦ ♣r♦❝❡ss♦ ❝♦♠ ♦r❞❡♥s s✉♣❡r✐♦r❡s ❞❡ ♠✐❧❤❛r✱ ❞❡③❡♥❛ ❞❡ ♠✐❧❤❛r ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳
❈♦♠ ♦ r❡❛❣r✉♣❛♠❡♥t♦ t❡r❡♠♦s ♦ r❡s✉❧t❛❞♦ ✜♥❛❧✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❛ ❝♦♠♣♦s✐çã♦ ❞❛ ♦r❞❡♠ s✉♣❡r✐♦r ❞❛s ❝❡♥t❡♥❛s ❛ ♣❛rt✐r ❞❛ ♦r❞❡♠ ❞❛s ❞❡③❡♥❛s ❞❛rá ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ❛❞✐çã♦ ♦ ♥ú♠❡r♦ 418✳ ❊ss❡ ♣r♦❝❡ss♦ é ❝❤❛♠❛❞♦ ❞❡ ❛❞✐çã♦ ❝♦♠ tr❛♥s♣♦rt❡✳
❖ ♠ét♦❞♦ ❡s❜♦ç❛❞♦ ❡♠ ✭✶✵✮ é ❜❡♠ ♣❛r❡❝✐❞♦ ❝♦♠ ❡str❛té❣✐❛s q✉❡ ✉t✐❧✐③❛♠♦s ♣❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞❡ ❢♦r♠❛ ♠❡♥t❛❧✳ P♦r ❡①❡♠♣❧♦✱ ✈❡❥❛♠♦s ♦ s❡❣✉✐♥t❡ ❝❛s♦ ♦♥❞❡ ✐♥✐❝✐❛♠♦s ❛ ❛❞✐çã♦ ♣❡❧❛ ❝♦❧✉♥❛ ❞❛s ❝❡♥t❡♥❛s
147 + 271 = (1 + 2)×100 + (4 + 7)×10 + (7 + 1)×1
= 3×100 + (4 + 7)×10 + (7 + 1)×1 ✭✶✶✮
❊♠ s❡❣✉✐❞❛ ♣❛ss❛♠♦s à s♦♠❛ ❞♦s ❛❧❣❛r✐s♠♦s ♥❛ ❝♦❧✉♥❛ ❞❛s ❞❡③❡♥❛s
= 3×100 + 11×10 + (7 + 1)×1 ✭✶✷✮
= 410 + (7 + 1)×1
❡✱ ♣♦r ú❧t✐♠♦✱ s♦♠❛♠♦s ♦s ❛❧❣❛r✐s♠♦s ❞❛ ❝♦❧✉♥❛ ❞❛s ✉♥✐❞❛❞❡s✱
147 + 271 = 410 + 8×1 = 418. ✭✶✸✮
❯♠❛ ✈❛r✐❛çã♦ ❜❛st❛♥t❡ s❡♠❡❧❤❛♥t❡ ❞♦ ♣r♦❝❡ss♦ r❡❛❧✐③❛❞♦ ❡♠ ✭11✮ ❡ ✭12✮ s❡r✐❛ ♦
❛❣r✉♣❛♠❡♥t♦ ✏❢♦r♠❛♥❞♦ ✉♠10✑ ❛ ♣❛rt✐r ❞❡ 4 + 7 = 10 + 1 ❜❡♠ ❝♦♠✉♠ ♥❛ ❝♦♥❝❡♣çã♦
❝❤✐♥❡s❛ ❞❡ ❡♥s✐♥♦✱ ♦ q✉❡ ❢r❡q✉❡♥t❡♠❡♥t❡ ❢❛③❡♠♦s ❞❡ ❢♦r♠❛ ♠❡♥t❛❧✳
147 + 271 = (1 + 2)×100 + (4 + 7)×10 + (7 + 1)×1
= 3×100 + (10 + 1)×10 + (7 + 1)×1 ✭✶✹✮
= 300 + 100 + 1×10 + (7 + 1)×1
= 400 + 1×10 + (7 + 1)×1
= 418
❚❛♥t♦ ♥♦ ♣r♦❝❡ss♦ ❞❡s❡♥✈♦❧✈✐❞♦ ❡♠ ✭✶✹✮✱ q✉❛♥t♦ ♥♦s ♣r♦❝❡ss♦s ❞❡s❡♥✈♦❧✈✐❞♦s ❡♠ ✭11✮
❡ ✭12✮ ❡st❛♠♦s ❝♦♠❡ç❛♥❞♦ ❛ ❛❞✐çã♦ ♣❡❧❛ ❝♦❧✉♥❛ ❞❛s ❝❡♥t❡♥❛s✱ ❥✉st❛♠❡♥t❡ ♦ ❝♦♥trár✐♦
❞♦ q✉❡ s❡ ❢❛③ ♥♦ ❛❧❣♦r✐t♠♦ ❤❛❜✐t✉❛❧ ♣❛r❛ ❛ ❛❞✐çã♦ ♦♥❞❡ s❡♠♣r❡ ❝♦♠❡ç❛♠♦s ❛ ❛❞✐❝✐♦♥❛r ♣❡❧❛ ❝♦❧✉♥❛ ❞♦s ❛❧❣❛r✐s♠♦s ❞❛ ✉♥✐❞❛❞❡s✳
❆ ✈❛♥t❛❣❡♠ q✉❡ t❡♠♦s q✉❛♥❞♦ ❝♦♠❡ç❛♠♦s ❛ s♦♠❛r ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ é q✉❡ t❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ❞♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ❞❛ ❛❞✐çã♦✱ ❡♥q✉❛♥t♦ q✉❡ ❝♦♠❡ç❛♥❞♦ ❛ ❛❞✐❝✐♦♥❛r ❞❛ ❞✐r❡✐t❛ ♣❛r❛ ❡sq✉❡r❞❛ ♥ã♦ ♣♦❞❡r❡♠♦s ❡st✐♠❛r ♦ ✈❛❧♦r ❞❛ s♦♠❛✳
◆ã♦ é ❞❡ s❡ ❡s♣❛♥t❛r q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❡ ❛❞✐çã♦ ❢✉♥❝✐♦♥❡ ❡♠ ♦✉tr❛s ❜❛s❡s✱ ❝♦♠ ♣♦✉❝❛s ❛❞❛♣t❛çõ❡s✱ é ❝❧❛r♦✳ P♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❡♠ q✉❛❧q✉❡r s✐st❡♠❛ ❡ ♥ã♦ ❛♣❡♥❛s ♥♦ ❞❡❝✐♠❛❧✳ ❱❛♠♦s✱ ♣❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ❡❢❡t✉❛r ❛ s♦♠❛ ❞❡ ♥✉♠❡r♦s ♥❛t✉r❛✐s r❡♣r❡s❡♥t❛❞♦s ♥❛ ❜❛s❡ ✷✱ ♠❛s ❛♥t❡s ✈❡❥❛♠♦s ❝♦♠♦ tr❛♥s❢♦r♠❛r ✉♠ ♥ú♠❡r♦ ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧ ♣❛r❛ ❜❛s❡ ✷✳
Pr♦♣♦s✐çã♦ ✶✳ ✭❙❡❣✉♥❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦✮ ❙❡❥❛ p(n) ✉♠❛ ❢✉♥çã♦ ♣r♦♣♦s✐❝✐✲
♦♥❛❧ ❝✉❥♦ ✉♥✐✈❡rs♦ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♠❛✐♦r❡s ❞♦ q✉❡ ♦✉ ✐❣✉❛✐s ❛ ✉♠ ✐♥t❡✐r♦ ❞❛❞♦ ❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ s❡❥❛♠ ✈á❧✐❞♦s✿
✭✐✮ p(a) é ✈❡r❞❛❞❡✐r❛❀
✭✐✐✮ ❙❡r é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♠❛✐♦r q✉❡ ❛ ❡p(s)é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦s s❛t✐s❢❛③❡♥❞♦ a ≤s < r✱ ❡♥tã♦ p(r) é ✈❡r❞❛❞❡✐r❛✳
❊♥tã♦ t❡♠♦s q✉❡ p(n) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n≥a✳
❚❡♦r❡♠❛ ✷✳ ❉❛❞♦s a, b∈N ❝♦♠ b >1✱ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s c0, c1, . . . , cn ♠❡♥♦✲ r❡s ❞♦ q✉❡ b✱ ✉♥✐✈♦❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡
a=c0+c1b+c2b2+. . .+cnbn. ✭✶✺✮
❆ ❞❡♠♦♥str❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✻❪✳ ❉❡♠♦♥str❛çã♦✿
❱❛♠♦s ❞❡♠♦♥str❛r ♦ t❡♦r❡♠❛ ✉s❛♥❞♦ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ s♦❜r❡a✳ ❙❡ a= 0 ♦✉ s❡ a= 1✱ ❜❛st❛ t♦♠❛rn = 0 ❡ c0 =a✳
❙✉♣♦♥❞♦ q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ ♠❡♥♦r ❞♦ q✉❡ a✱ ✈❛♠♦s
♣r♦✈á✲❧♦ ♣❛r❛ a✳ P❡❧❛ ❞✐✈✐sã♦✽ ❡✉❝❧✐❞✐❛♥❛✱ ❡①✐st❡♠q ❡r ú♥✐❝♦s t❛✐s q✉❡
a=bq+r✱ ❝♦♠ r < b✳
❈♦♠♦q < a✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ s❡❣✉❡✲s❡ q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s n′ ❡ d0, d1, . . . , dn′✱ ❝♦♠dj < b✱ ♣❛r❛ t♦❞♦ j✱ t❛✐s q✉❡
q =d0+d1b+. . .+dn′bn ′ ✳
▲❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ ❞❡st❛❝❛❞❛s✱ t❡♠♦s q✉❡
a=bq+r=b(d0 +d1b+. . .+dn′b
n′) +r✱
❞♦♥❞❡ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✲s❡ ♣♦♥❞♦ c0 =r✱ n =n
′
+ 1 ❡cj =dj−1 ♣❛r❛ j = 1, . . . , n✳
❆ ✉♥✐❝✐❞❛❞❡ s❡❣✉❡✲s❡ ❢❛❝✐❧♠❡♥t❡ ❞❛s ✉♥✐❝✐❞❛❞❡s ❛❝✐♠❛ ❡st❛❜❡❧❡❝✐❞❛s✳
✽❊♠ ❬✽❪✿ ❆r✐t✳ ❖♣❡r❛çã♦ ❛r✐t♠ét✐❝❛ q✉❡ ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r q✉❛♥t❛s ♣❛r❝❡❧❛s ✭q✉♦❝✐❡♥t❡✮ ❞❡
✉♠❛ ❝❡rt❛ q✉❛♥t✐❞❛❞❡ ✭❞✐✈✐s♦r✮ s♦♠❛❞❛s✱ ❢♦r♠❛♠ ✉♠ ♥ú♠❡r♦ ✭❞✐✈✐❞❡♥❞♦✮✳
❆ ❡①♣r❡ssã♦ ♠♦str❛❞❛ ❡♠ ✭✶✺✮ é ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❡①♣❛♥sã♦ r❡❧❛t✐✈❛ à ❜❛s❡ b✳ ❙❡
t♦♠❛r♠♦s b= 10 ❡♥tã♦ ❡ss❛ ❡①♣❛♥sã♦ é ❝❤❛♠❛❞❛ ❞❡❝✐♠❛❧ ❡ s❡ b = 2✱ ❜✐♥ár✐❛✳
❆ ♣❛rt✐r ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷ t❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❡①♣❛♥sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ q✉❛❧q✉❡r ♥❛ ❜❛s❡b✳ ❇❛st❛ q✉❡ ❛♣❧✐q✉❡♠♦s✱ r❡♣❡t✐❞❛s ✈❡③❡s✱ ❛
❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ ❝♦♠♦ ❛ s❡❣✉✐r✿
a = bq0+r0, r0 < b,
q0 = bq1+r1, r1 < b,
q1 = bq2+r2, r2 < b,
❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❈♦♠♦a > q0 > q1 >· · ·✱ ❡♠ ❛❧❣✉♠ ♠♦♠❡♥t♦ t❡r❡♠♦sqn−1 < b
❡✱ ♣♦rt❛♥t♦✱ ❞❡ qn−1 = bqn+rn ❞❡❝♦rr❡ q✉❡ qn = 0✱ ✐♠♣❧✐❝❛♥❞♦ ❡♠ 0 = qn = qn+1 =
qn+2 =· · ·✱ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ 0 =rn+1 =rn+2 =· · · ✳
❊♥tã♦✱
a = r0+r1b+· · ·+rnbn. ✭✶✻✮
❆ ❡①♣❛♥sã♦ ❡♠ ✭16✮ ♣❡r♠✐t❡ r❡♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥✲
❥✉♥t♦ ❙ ❞❡ sí♠❜♦❧♦s ❝♦♠ ❝❛r❞✐♥❛❧✐❞❛❞❡✾ b✱ ♦✉ s❡❥❛✱ s❡b= 2✱ ❡♥tã♦ ❙={0,1}✳ ◆❛ ❜❛s❡ ❞❡❝✐♠❛❧✱ ♦♥❞❡ b= 10✱ t❡♠♦s ❙ ={0,1,2,3,4,5,6,7,8,9}✳
❘❡♣r❡s❡♥t❛r❡♠♦s ♦s ♥ú♠❡r♦s 8(10) ❡ 9(10) ♥❛ ❜❛s❡ ❜✐♥ár✐❛✳ P❛r❛ t❛♥t♦✱ ❢❛r❡♠♦s ❛s
❞✐✈✐sõ❡s r❡♣❡t✐❞❛s ❞♦s ♥ú♠❡r♦s8(10) ❡9(10) ♣♦r 2✳
❋✐❣✉r❛ ✺✿ Pr♦❝❡ss♦ ♣❛r❛ ❝♦♥✈❡rt❡r8 ❡ 9♣❛r❛ ❜❛s❡ ❜✐♥ár✐❛
✾❊♠✿ ❬✽❪ ▼❛t✳ ◆ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞✐❢❡r❡♥t❡s ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥❥✉♥t♦✳
❆❣♦r❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿
9(10)= 1001(2) = 1×23+ 0×22 + 0×21+ 1×20✱
8(10)= 1000(2) = 1×23+ 0×22 + 0×21+ 0×20✳
❈♦♠♦ ❛ ❜❛s❡ 2 é ✉♠ s✐st❡♠❛ ❞❡ ✈❛❧♦r❡s ♣♦s✐❝✐♦♥❛✐s✱ ♣r❡❝✐s❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ 0 + 0 = 0✱0 + 1 = 1✱ 1 + 0 = 1✱1 + 1 = 10(2)✱ ♦✉ s❡❥❛✱ ❡ss❡s sã♦ ♦s ❢❛t♦s ❞❛ ❛❞✐çã♦ ♣❛r❛
❜❛s❡ ❜✐♥ár✐❛✳
❊①❡♠♣❧♦ ✶✳ ❆❣♦r❛✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♥❛ ❛❞✐çã♦ ♣❛r❛ r❡❛❧✐③❛r ❛ ♦♣❡r❛çã♦9 + 8✱ ♥❛ ❜❛s❡ ✷
1 0 0 1(2)
+ 1 0 0 0(2)
1 0 0 0 1(2)
✭✶✼✮
❱❡❥❛♠♦s ♦ q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ❝♦❧♦❝❛♠♦s✱ ❡♠ ❡①♣❛♥sã♦ ♥❛ ❜❛s❡ ❜✐♥ár✐❛✱ ♦ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ ❡♠ ✭✶✼✮
10001(2) = 1×24+ 0×23+ 0×22+ 0×21 + 1×20 = 16 + 1✳
❖ q✉❡ t❡♠♦s é ❥✉st❛♠❡♥t❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ❛❞✐çã♦9 + 8 = 17 q✉❡ t❡rí❛♠♦s s❡ t✐✈és✲
s❡♠♦s ✉t✐❧✐③❛❞♦ ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♣❛r❛ ❛❞✐çã♦ ♥❛ ❜❛s❡ ❞❡❝✐♠❛❧✳
❊①❡♠♣❧♦ ✷✳ ✭❖❇▼❊P ✷✵✶✹✮ ◆❛ ❝♦♥t❛ ✐♥❞✐❝❛❞❛ ❛ s❡❣✉✐r✱ ❛s ❧❡tr❛s X✱ Y ❡ Z r❡♣r❡✲
s❡♥t❛♠ ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s✳ ◗✉❛❧ é ♦ ❛❧❣❛r✐s♠♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ Z❄
X X X X
+ Y Y Y Y
Z Z Z Z
Y X X X Z
❙♦❧✉çã♦✿
❊st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦✱ ♣❡❧♦ ❡♥✉♥❝✐❛❞♦✱ ♦s ❛❧❣❛r✐s♠♦s X 6= Y 6=Z✳ ◗✉❛♥❞♦ s♦♠❛✲
♠♦s ♦s três ♥ú♠❡r♦s ❞❡ ✹ ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s✱ ♦❜t❡♠♦s ✉♠ ♥ú♠❡r♦ ❞❡ ✺ ❛❧❣❛r✐s♠♦s✱
Y XXXZ✳ ❖ ❛❧❣❛r✐s♠♦ Y ♣♦❞❡ ❛ss✉♠✐r✱ ❡♠ ❛♥á❧✐s❡ ♣r❡❧✐♠✐♥❛r✱ ❞♦✐s ✈❛❧♦r❡s✱ ✐st♦ é✱ Y = 1♦✉Y = 2✳ ❉❡ ❢❛t♦✱ ♦❜s❡r✈❡ ❛s ❝♦♥t❛s ❛❜❛✐①♦ q✉❡ ❡①❡♠♣❧✐✜❝❛♠ ♦ ♥♦ss♦ r❛❝✐♦❝í♥✐♦✿
