❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❊✈♦❧✉çã♦ ❞❛s ■❞❡✐❛s s♦❜r❡
◆ú♠❡r♦s ■♠❛❣✐♥ár✐♦s
†♣♦r
▲❡❛♥❞r♦ ❙❛❧❡s ❆❧♠❡✐❞❛ ❞❡ ❖❧✐✈❡✐r❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲
❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧
♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❊✈♦❧✉çã♦ ❞❛s ■❞❡✐❛s s♦❜r❡
◆ú♠❡r♦s ■♠❛❣✐♥ár✐♦s
♣♦r
▲❡❛♥❞r♦ ❙❛❧❡s ❆❧♠❡✐❞❛ ❞❡ ❖❧✐✈❡✐r❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ➪❧❣❡❜r❛ ❡ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛
❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❏♦ã♦ ❇♦s❝♦ ❇❛t✐st❛ ▲❛❝❡r❞❛ ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ❏❛♠✐❧s♦♥ ❘❛♠♦s ❈❛♠♣♦s ✲ ❯❋P❇
O48e Oliveira, Leandro Sales Almeida de.
Evolução das ideias sobre números imaginários / Leandro Sales Almeida de Oliveira.- João Pessoa, 2015.
51f. : il.
Orientador: Antônio de Andrade e Silva Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Números imaginários. 3. Complexos. 4. Quatérnios. 5. Octônios. 6. Sedênios.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ♦ ♥♦ss♦ P❛✐ ❈❡❧❡st✐❛❧✱ ❝r✐❛❞♦r ❞❡ t♦❞❛s ❛s ❝♦✐s❛s✱ s❡♠ ♦ q✉❛❧ ♥❡♠ ❛ ♠❛t❡♠át✐❝❛ ❡①✐st✐r✐❛✳
❆♦ P❘❖❋▼❆❚✱ ♣♦r ❞❛r ❡st❛ ♦♣♦rt✉♥✐❞❛❞❡ ❛ ♣r♦❢❡ss♦r❡s q✉❡✱ ❝♦♠♦ ❡✉✱ ♣✉❞❡r❛♠ ❢❛③❡r ✉♠ ♠❡str❛❞♦ ♣❛r❛ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✳
❆ ❈❆P❊❙✱ ♣❡❧♦ ✐♥✈❡st✐♠❡♥t♦ q✉❡ ♠❡ ❛❥✉❞♦✉ ❛ s❡❣✉✐r ❝♦♠ ♦ ♣❡r❝✉rs♦ ❞♦ ♠❡str❛❞♦✳ ❆♦s ❞♦❝❡♥t❡s ✐♥t❡❣r❛❞♦s ❛♦ ❝✉rs♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ àq✉❡❧❡s q✉❡ t✐✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ tr♦❝❛r ✐❞❡✐❛s ❞✉r❛♥t❡ ♦s ❞♦✐s ❞♦ ♠❡str❛❞♦✳
❆♦ ♠❡✉ ❖r✐❡♥t❛❞♦r✱ ♦ Pr♦❢❡ss♦r ❆♥❞r❛❞❡✱ q✉❡ só t✐✈❡ ♦ ♣r✐✈✐❧é❣✐♦ ❞❡ ❝♦♥❤❡❝❡r ❛ ♣♦✉❝♦ t❡♠♣♦✱ ♠❛s ♠❡s♠♦ ❛ss✐♠ ♣✉❞❡ ❛♣r❡♥❞❡r ✐♠❡♥s❛♠❡♥t❡ ♥♦s ♥♦ss♦s ♣♦✉❝♦s ❡♥❝♦♥tr♦s q✉❡ t✐✈❡♠♦s✱ ❝♦♠ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡✱ ❤✉♠✐❧❞❛❞❡ ❡ ♠❛♥s✐❞ã♦✳
❊ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❞♦ ✐♥í❝✐♦ ❛♦ ✜♠✱ s❡♠♣r❡ ♠❡ ❞❛♥❞♦ ♦ ❛♣♦✐♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ♣r♦ss❡❣✉✐r ❝♦♠ ♦s ♠❡✉s s♦♥❤♦s ❡ ❝♦♥❝❧✉✐r ❝♦♠ ê①✐t♦ ❡st❡ ♠❡str❛❞♦✳
❉❡❞✐❝❛tór✐❛
❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ❏❛r❧❡②❞❡✱ ❡ ❛♦ ♠❡✉ ✜❧❤♦✱ ▼✐❣✉❡❧✱ q✉❡ sã♦ ♦ ♠❡✉ t❡s♦✉r♦ ✐♥❡st✐♠á✈❡❧✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ s❡rã♦ ❡st✉❞❛❞♦s ♦s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s ❡ ❝♦♠♦ s❡ ❞❡✉ ❛ s✉❛ ❡✈♦❧✉çã♦ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ❊✈♦❧✉çã♦ ❡st❛ q✉❡ ♦❝♦rr❡✉ ❞❡ ❢♦r♠❛ ❜❡♠ ❧❡♥t❛✱ ❛té s❡ ❝❤❡❣❛r ♥♦ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❤♦❥❡ ❝♦♠♦ ♦ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦i✳ ❊♥tr❡t❛♥t♦✱ ❛ ❝r✐❛çã♦
❞♦s ❝♦♠♣❧❡①♦s ♥ã♦ ❢♦✐ ♦ ♣♦♥t♦ ✜♥❛❧ ❞♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s✳ ❊st✉❞♦s s❡❣✉✐♥t❡s ✐♥tr♦❞✉③✐r❛♠ ❝♦♥❝❡✐t♦s ❛✐♥❞❛ ♠❛✐s ❛❜r❛♥❣❡♥t❡s ❝r✐❛♥❞♦ ❝♦♥❥✉♥t♦s ❝♦♠♦ ♦s q✉❛tér♥✐♦s✱ ❡①t❡♥sã♦ ❞❡ q✉❛tr♦ ❞✐♠❡♥sõ❡s ❞♦s ❝♦♠♣❧❡①♦s✳ ❋✐♥❛❧✐③❛✲s❡ ♦ tr❛❜❛❧❤♦✱ ❝♦♠ ❛s ❡①t❡♥sõ❡s ❞❡ ♦✐t♦ ❡ ❞❡③❡ss❡✐s ❞✐♠❡♥sõ❡s ❞♦s ❝♦♠♣❧❡①♦s✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ♦❝tô♥✐♦s ❡ s❡❞ê♥✐♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆❧é♠ ❞❡ s❡r ❛♣r❡s❡♥t❛❞♦ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❛s ❡①t❡♥sõ❡s✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ♥ú♠❡r♦s ❤✐♣❡r❝♦♠♣❧❡①♦s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ◆ú♠❡r♦s❀ ■♠❛❣✐♥ár✐♦s❀ ❈♦♠♣❧❡①♦s❀ ◗✉❛tér♥✐♦s❀ ❖❝tô♥✐♦s❀ ❙❡✲ ❞ê♥✐♦s✳
❆❜str❛❝t
■♥ t❤✐s ♣❛♣❡r ✐t ✇✐❧❧ ❜❡ st✉❞✐❡❞ t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs ❛♥❞ ❤♦✇ t❤❡✐r ❡✈♦❧✉t✐♦♥ ♦✈❡r t✐♠❡ ♦❝❝✉rr❡❞✳ ❙✉❝❤ ❡✈♦❧✉t✐♦♥ ❤❛s ♦❝❝✉rr❡❞ ❛t ❛ s❧♦✇ ♣❛❝❡ ✉♥t✐❧ ✐t r❡❛❝❤❡❞ ❛t ✇❤❛t ✐s ❦♥♦✇♥ t♦❞❛② ❛s t❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r i✳ ❍♦✇❡✈❡r✱ t❤❡ ❝r❡❛t✐♦♥ ♦❢ t❤❡
❝♦♠♣❧❡① ✇❛s ♥♦t t❤❡ ❡♥❞ ♦❢ t❤❡ st✉❞② ♦❢ ✐♠❛❣✐♥❛r② ♥✉♠❜❡rs✳ ❚❤❡s❡ st✉❞✐❡s ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❡✈❡♥ ♠♦r❡ ❝♦♠♣r❡❤❡♥s✐✈❡ ❝♦♥❝❡♣ts ❝r❡❛t✐♥❣ s❡ts ❛s q✉❛t❡r♥✐♦♥s✱ ❡①t❡♥✲ s✐♦♥ ♦❢ ❢♦✉r ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❝♦♠♣❧❡①✳ ■t ✇✐❧❧ ❜❡ ❝♦♥❝❧✉❞❡❞✱ ✇✐t❤ t❤❡ ❡①t❡♥s✐♦♥s ♦❢ ❡✐❣❤t ❛♥❞ s✐①t❡❡♥ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ❦♥♦✇♥ ❛s ♦❝t♦♥✐♦♥s ❛♥❞ s❡❞❡♥✐♦♥s✱ r❡s♣❡❝t✐✈❡❧②✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ✐t ✇✐❧❧ ❜❡ s✉❜♠✐tt❡❞ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡s❡ ❡①t❡♥s✐♦♥s✱ ❛❧s♦ ❦♥♦✇♥ ❛s ❤②♣❡r❝♦♠♣❧❡① ♥✉♠❜❡rs✳
❑❡②✇♦r❞s✿ ◆✉♠❜❡rs❀ ■♠❛❣✐♥❛r②❀ ❈♦♠♣❧❡①❀ ◗✉❛t❡r♥✐♦♥s❀ ❖❝t♦♥✐♦♥s❀ ❙❡❞❡♥✐✲ ♦♥s✳
❙✉♠ár✐♦
✶ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ ❞♦s ◆ú♠❡r♦s ■♠❛❣✐♥ár✐♦s ✲ ❉❡ ❈❛r❞❛♥♦ ❛
●❛✉ss ✶
✶✳✶ ❈❛r❞❛♥♦✱ ❚❛rt❛❣❧✐❛ ❡ ❛ ❞✐s♣✉t❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ ✸✳♦ ❣r❛✉ ✳ ✶
✶✳✷ ❇♦♠❜❡❧❧✐ ❡ s✉❛ ✐❞❡✐❛ ♣✐♦♥❡✐r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❉❡s❝❛rt❡s✱ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹ ❊✉❧❡r✱ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ❞♦♠✐♥♦✉ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✺ ❖s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ❝♦♠♦ P♦♥t♦s ♥♦ P❧❛♥♦✿ ❲❛❧❧✐s✱ ❲❡ss❡❧ ❡ ❆r❣❛♥❞ ✶✶ ✶✳✻ ●❛✉ss ❡ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷ ❍❛♠✐❧t♦♥ ❡ ♦ ❙❡✉ ❉❡s❡❥♦ ❞❡ ■r ❆❧é♠ ✶✻
✷✳✶ ❊ s❡ ❢♦ss❡ ♣♦ssí✈❡❧ ❡①♣❛♥❞✐r ♦s ❝♦♠♣❧❡①♦s ♣❛r❛ ♦ ❡s♣❛ç♦❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳✶ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❚r✐♣❧❡t♦s ✲ ❚❡st❡ ●❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❚r✐♣❧❡t♦s ✲ ❚❡st❡ ❆❧❣é❜r✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✸ ❈♦♥❥❡❝t✉r❛s ♣❛r❛ ij ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✷ ❙✉r❣❡♠ ♦s ◗✉❛tér♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ ❆té ♦♥❞❡ ♣♦❞❡♠ ✐r ♦s ✐♠❛❣✐♥ár✐♦s❄ ✷✽
✸✳✶ ❖s ❖❝tô♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ❆té ♦♥❞❡ ♣♦❞❡♠ ✐r ♦s ✐♠❛❣✐♥ár✐♦s❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸ ❙❡❞ê♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✹ ❆♣❧✐❝❛çõ❡s ❞♦s ◆ú♠❡r♦s ❍✐♣❡r❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹✳✶ ❆♣❧✐❝❛çã♦ ❞♦s ◗✉❛tér♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹✳✷ ❆♣❧✐❝❛çã♦ ❞♦s ❖❝tô♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹✳✸ ❆♣❧✐❝❛çã♦ ❞♦s ❙❡❞ê♥✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✸✾
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ●✐r♦❧❛♠♦ ❈❛r❞❛♥♦ ✭1501−1576✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ◆✐❝♦❧ò ❋♦♥t❛♥❛ ✭❚❛rt❛❣❧✐❛✮ ✭1499−1557✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❋r♦♥t✐s♣í❝✐♦ ❞❛ ❆rs ▼❛❣♥❛ ✭1545✮✱ ♦❜r❛ ❞❡ ❈❛r❞❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ❘❛❢❛❡❧ ❇♦♠❜❡❧❧✐ ✭1526−1572✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✺ ❘❡♥é ❉❡s❝❛rt❡s ✭1642−1727✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✶✳✻ ■s❛❛❝ ◆❡✇t♦♥ ✭1642−1727✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✼ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③ ✭1646−1716✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✽ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭1707−1783✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✾ ❏♦❤♥ ❲❛❧❧✐s ✭1616−1703✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✶✵ ❈❛s♣❛r ❲❡ss❡❧ ✭1745−1818✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✶✳✶✶ ❏❡❛♥ ❘♦❜❡rt ❆r❣❛♥❞ ✭1768−1822✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✶✳✶✷ ❋♦r♠❛ ●❡♦♠étr✐❝❛ ❞♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✸ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ✭1777−1855✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✹ ❈✐t❛çã♦ ❖r✐❣✐♥❛❧ ❞❡ ●❛✉ss✱ ❡s❝r✐t♦ à s✉❛ ♣ró♣r✐❛ ♠ã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✶ ❙✐r ❲✐❧❧✐❛♠ ❘♦✇❛♥ ❍❛♠✐❧t♦♥ ✭1788−1856✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✷✳✷ Pr♦❞✉t♦ ❞❡ ❞✉❛s ❧✐♥❤❛s r❡t❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸ ■❧✉str❛çã♦ ❞❛ ❖❜s❡r✈❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✹ ❚r✐♣❧❡t♦ (rcosθ, rs❡♥θcosφ, rsenθs❡♥φ)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✺ ❚r✐♣❧❡t♦ x+yi+zj✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✻ ■♥t❡r♣r❡t❛çã♦ ❞❡ j ✈✐❛ r♦t❛çõ❡s ♥♦ ♣❧❛♥♦xz✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✼ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ♦ q✉❛❞r❛❞♦ ❞❡ ✉♠ tr✐♣❧❡t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✽ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ (a+ib+jc)(x+ib+jc)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✷✳✾ P❧❛❝❛ ♥❛ ❇r♦♦♠✬s ❇r✐❞❣❡ s♦❜r❡ ♦ ❘♦②❛❧ ❈❛♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✵ ❇❧♦❝♦ ❞❡ ◆♦t❛s ❞❡ ❍❛♠✐❧t♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✶ ❚á❜✉❛ ❞❡ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❛s ❯♥✐❞❛❞❡s ■♠❛❣✐♥ár✐❛s i, j, k✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶ ❏♦❤♥ ●r❛✈❡s ✭1806−1870✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❆rt❤✉r ❈❛②❧❡② ✭1821−1895✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸ ❚á❜✉❛ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♦❝tô♥✐♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✹ ❆❞♦❧❢ ❍✉r✇✐t③ ✭1859−1919✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✺ ❚á❜✉❛ ❞❡ ▼✉❧t✐♣❧✐❝❛çã♦ ❞♦s ❙❡❞ê♥✐♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✻ ❱❡t♦r ❊s❢ér✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
■♥tr♦❞✉çã♦
❆♦ ♣❡♥s❛r♠♦s ♥♦ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦ i✱ ♣r♦✈❛✈❡❧♠❡♥t❡✱ ❧❡♠❜r❛♠♦s ❞♦ ♥♦ss♦ ❡♥✲
s✐♥♦ ♠é❞✐♦ ❡ ❞♦ ♣r♦❢❡ss♦r ♥♦s ✐♥❢♦r♠❛♥❞♦ q✉❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ♠❛✐♦r ❞♦ q✉❡ ♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ♦♥❞❡ ❡①✐st❡ é ♣♦ssí✈❡❧ ❡①tr❛✐r r❛✐③ q✉❛❞r❛❞❛ ❞❡ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦✱ ❡ ♥♦s ❛♣r❡s❡♥t❛ ❛ ✐❣✉❛❧❞❛❞❡i=√−1✳ ❖ q✉❡ ♣♦❞❡ ♣❛ss❛r ❞❡s♣❡r❝❡❜✐❞♦✱ t❛♥t♦ ♣❡❧♦s ❛❧✉♥♦s q✉❛♥t♦ ♣❡❧♦ ♣r♦❢❡ss♦r✱ é q✉❡ ❡st❛ ✐❣✉❛❧❞❛❞❡ ♥ã♦ s❡ ❞❡✉ ♥✉♠ ♣❛ss❡ ❞❡ ♠á✲ ❣✐❝❛ ❝♦♠♦ ♣❛r❡❝❡ ❛♦ s❡r ❛♣r❡s❡♥t❛❞♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❊①✐st❡ ♠✉✐t❛ ❤✐stór✐❛ ♣♦r trás ❞✐ss♦ ❡ é ♦ ♦❜❥❡t✐✈♦ ❞❡ ♥♦ss❛ ♣❡sq✉✐s❛ tr❛③❡r ♦s ❛❝♦♥t❡❝✐♠❡♥t♦s q✉❡ t♦r♥❛r❛♠ ♣♦ssí✈❡❧ ❡st✉❞❛r♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❤♦❥❡✳
❆ ❤✐stór✐❛ ❝♦♠❡ç❛ ❝♦♠ ❛ ❞✐s♣✉t❛ ❡♥tr❡ ❈❛r❞❛♥♦ ❡ ❚❛rt❛❣❧✐❛ ♣❡❧❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ ✸✳♦ ❣r❛✉✱ ♦♥❞❡ ♥❡ss❛ ❞✐s♣✉t❛ s❡ é ❞❡s❝♦❜❡rt❛ ❛ ❢ór♠✉❧❛ ❞❡ r❡s♦❧✉çã♦
❞❡ t❛❧ ❡q✉❛çã♦ ❬✸❪✱❬✻❪✳ ❈♦♠ ♦ ❞❡s❝♦❜r✐♠❡♥t♦ ❞❡st❛ ❢ór♠✉❧❛✱ ❧❡✈❛♥t❛✲s❡ ♦ q✉❡st✐♦♥❛✲ ♠❡♥t♦ s♦❜r❡ r❛í③❡s q✉❛❞r❛❞❛s ♥❡❣❛t✐✈❛s✳ P❛r❛ ♦s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛✱ ✐st♦ ♥ã♦ ❡①✐st✐❛✳ ❊ ❛ ❢ór♠✉❧❛ ❞❡s❝♦❜❡rt❛ ♣❛r❡❝✐❛ ❢❛③❡r ♣♦ssí✈❡❧ t❛❧ ❢❛t♦✱ ❞❡✐①❛♥❞♦ ♦s ♠❛t❡✲ ♠át✐❝♦s ✐♥tr✐❣❛❞♦s✳ ❆s t❡♥t❛t✐✈❛s ♣❛r❛ ❡①♣❧✐❝❛r t❛❧ ❢❛t♦ ❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❡♥tã♦ ♣r♦♣♦st♦✱ ♣❛ss♦✉ ♣♦r ♠ã♦s ❞❡ ♠❛t❡♠át✐❝♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♠✉✐t♦ ♣❛r❛ ♦ ❛✈❛♥ç♦ ❞❛ ♠❛t❡♠át✐❝❛ ❬✹❪✳
❚❛♠❜é♠ ♥❡st❡ tr❛❜❛❧❤♦✱ tr❛r❡♠♦s✱ ♦ q✉❡ ♣❛r❛ ✉♠ ❛❧✉♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♣♦❞❡ s❡r ✉♠❛ ♠❡r❛ ❝✉r✐♦s✐❞❛❞❡✱ ♠❛s q✉❡ ♣❛r❛ ❛❧❣✉♥s ♠❛t❡♠át✐❝♦s ❢♦✐ ✉♠❛ ♣❡sq✉✐s❛ ❞❡ ✈ár✐♦s ❛♥♦s✳ ❚r❛t❛✲s❡ ❞❡ ❛❧❣✉♥s ❝♦♥❥✉♥t♦s✱ ❝❤❛♠❛❞♦s ❞❡ ❤✐♣❡r❝♦♠♣❧❡①♦s✱ ❛ss✐♠ ❝♦♥❤❡❝✐❞♦s ♣♦r ❝♦♥t❡r❡♠ ♠❛✐s ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ✱ ❛❧é♠ ❞❛ tr❛❞✐❝✐♦♥❛❧i✳ ❈♦♠❡ç❛r❡♠♦s
❝♦♠ ❛ ♣r✐♠❡✐r❛ ❡①♣❛♥sã♦✱ ❝♦♥❤❡❝✐❞❛ ♣♦r ◗✉❛tér♥✐♦s ❞❡ ❍❛♠✐❧t♦♥✳ ◗✉❛tér♥✐♦s ♣❡❧♦ ❢❛t♦ ❞❡ s❡r ✉♠ ❝♦♥❥✉♥t♦ t❡tr❛❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠ três ✉♥✐❞❛❞❡s ✐♠❛❣✐♥ár✐❛s ❞✐st✐♥t❛s✱ ♣♦ré♠ ❛♠❜❛s ✐❣✉❛✐s ❛√−1✳ ❍❛♠✐❧t♦♥ ❡st✉❞♦✉ ♣♦r ♠❛✐s ❞❡ ✉♠❛ ❞é❝❛❞❛ ♣❛r❛ ❝❤❡❣❛r ❛ ❡st❛ ♣r✐♠❡✐r❛ ❡①♣❛♥sã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❡❧❡ t❡♥t♦✉ ❝♦♠ ❛♣❡♥❛s ✉♠❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❛ ♠❛✐s✱ ❞❛♥❞♦ ♦r✐❣❡♠ ❛ ✉♠ tr✐♣❧❡t♦✳ ❘❛③ã♦ ♣❡❧❛ q✉❛❧ ❡❧❡ ❞❡♠♦r♦✉ t❛♥t♦ t❡♠♣♦ ♣❛r❛ ❝❤❡❣❛r ❛ ❝♦♥❝❧✉sã♦ ❞❡✜♥✐t✐✈❛ ❞♦s q✉❛tér♥✐♦s✳ ❊❧❡ q✉❡r✐❛ ✉♠ ❝♦♥❥✉♥t♦
♦♥❞❡ ♣✉❞❡ss❡ s❡r ❞❡s❝r✐t♦ ❡♠ três ❞✐♠❡♥sõ❡s✱ ♣❛r❛ s❡ ❛♣❧✐❝❛r ❛ ❣❡♦♠❡tr✐❛✱ t❛❧ ❝♦♠♦ ♦s ❝♦♠♣❧❡①♦s ❢✉♥❝✐♦♥❛✈❛♠ ♣❡r❢❡✐t❛♠❡♥t❡ ♣❛r❛ ❞✉❛s ❞✐♠❡♥sõ❡s✳ P♦ré♠✱ ❍❛♠✐❧t♦♥ ❡♥❢r❡♥t♦✉ ♣r♦❜❧❡♠❛s ❝♦♠ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ tr✐♣❧❡t♦s✱ ♣♦✐s ♥ã♦ ❝❤❡❣❛✈❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❛♥t♦ ❞❡✈❡r✐❛ ✈❛❧❡r ♦ ♣r♦❞✉t♦ ij✱ ❛s ❞✉❛s ✉♥✐❞❛❞❡s ✐♠❛❣✐♥ár✐❛s ❛té ❡♥tã♦✳ ❖
♣r♦❜❧❡♠❛ só ❢♦✐ r❡s♦❧✈✐❞♦ ❝♦♠ ❛ ✐♥s❡rçã♦ ❞❡ ✉♠❛ t❡r❝❡✐r❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛✳ ❉❛í ♦s tr✐♣❧❡t♦s ♣❛ss❛r❛♠ ❛ s❡r q✉❛tér♥✐♦s ❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞❛ ❛♣❧✐❝❛çã♦ à ❣❡♦♠❡tr✐❛ ❢♦r❛♠ r❡s♦❧✈✐❞♦s ❬✼❪✱❬✶✵❪✳
❆ ♣ró①✐♠❛ ❡①t❡♥sã♦✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♦❝tô♥✐♦s✱ ❢♦✐ ❞❡s❝r✐t❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ●r❛✈❡s✱ ❛♠✐❣♦ ❞❡ ❍❛♠✐❧t♦♥✱ ♠❛s ❡st❡ ♥ã♦ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ♣✉❜❧✐❝á✲❧❛✱ ❢❡✐t♦ ❛tr✐❜✉í❞♦ ❛ ❈❛②❧❡②✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♦❝tô♥✐♦s ♣♦ss✉✐ s❡t❡ ✉♥✐❞❛❞❡s ✐♠❛❣✐♥ár✐❛s ❞✐st✐♥t❛s ❬✶❪✳
❖ ú❧t✐♠♦ ❝♦♥❥✉♥t♦ ❡st✉❞❛❞♦ ♥♦ tr❛❜❛❧❤♦✱ sã♦ ♦s s❡❞ê♥✐♦s✱ ✉♠ ❝♦♥❥✉♥t♦ ❤❡①❛❞✐✲ ♠❡♥s✐♦♥❛❧ ❝♦♠ q✉✐♥③❡ ✉♥✐❞❛❞❡s ✐♠❛❣✐♥ár✐❛s ❞✐st✐♥t❛s ❬✷❪✳
❯♠❛ ❝✉r✐♦s✐❞❛❞❡ s♦❜r❡ ❡st❡s ❝♦♥❥✉♥t♦s é q✉❡ ❛ ❝❛❞❛ ❡①t❡♥sã♦ ❡❧❡s ♣❡r❞❡♠ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠ r❡❧❛çã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ tã♦ ❝♦♠✉♠ ❛♦s ❝♦♠♣❧❡①♦s✳ ❖s q✉❛tér♥✐♦s✱ ♥ã♦ sã♦ ❝♦♠✉t❛t✐✈♦s❀ ♦s ♦❝tô♥✐♦s ♥ã♦ sã♦ ❝♦♠✉t❛t✐✈♦s ♥❡♠ ❛ss♦❝✐❛t✐✈♦s ❡ ♦s s❡❞ê♥✐♦s ♥ã♦ ❢❛③❡♠ ♣❛rt❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❞✐✈✐sã♦✱ ♦✉ s❡❥❛✱ ❡❧❡s ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ♥ã♦ ♥✉❧♦s q✉❡ ♠✉❧t✐♣❧✐❝❛❞♦s r❡s✉❧t❛♠ ♥♦ ❡❧❡♠❡♥t♦ ♥✉❧♦ ❞♦ ❝♦♥❥✉♥t♦ ❬✶❪✱ ❬✷❪✱ ❬✶✷❪✱ ❬✶✸❪✳
❚r❛③❡♠♦s ❛✐♥❞❛✱ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦s ♥ú♠❡r♦s ❤✐♣❡r❝♦♠♣❧❡①♦s✱ ❞❡s❝♦♥❤❡❝✐❞♦s ❞❡ ♠✉✐t♦s✱ ♠❛s ♠✉✐t♦ út❡✐s ❡♠ ♥♦ss❛ ✈✐❞❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞♦s q✉❛tér♥✐♦s ❬✶❪✱ ❬✽❪✱ ❬✾❪✱ ❬✶✹❪✱ ❬✶✺❪✱ ❬✶✻❪✱ ❬✶✼❪✳
❈❛♣ít✉❧♦ ✶
❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ ❞♦s ◆ú♠❡r♦s
■♠❛❣✐♥ár✐♦s ✲ ❉❡ ❈❛r❞❛♥♦ ❛ ●❛✉ss
◆❡st❡ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ❛ ❡✈♦❧✉çã♦ ❞❛s ✐❞❡✐❛s s♦❜r❡ ♦s ♥ú♠❡r♦s ✐♠❛❣✐♥á✲ r✐♦s✳ ■r❡♠♦s ❞❡ s✉❛ ♦r✐❣❡♠ ♥♦s ❝á❧❝✉❧♦s ❢❡✐t♦s ♣♦r ❈❛r❞❛♥♦❀ ♣❛ss❛r❡♠♦s ♣♦r ❞✐✈❡rs❛s ♠❡♥t❡s ❜r✐❧❤❛♥t❡s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡s ♥ú♠❡r♦s ❡ ✐r❡✲ ♠♦s ❛té ♦ ❢❛❜✉❧♦s♦ ●❛✉ss✱ q✉❡ t✐r♦✉ ❞♦ ♠✐st✐❝✐s♠♦ ♦s ♥ú♠❡r♦s q✉❡ ♥♦ ♣❛ss❛❞♦ ❡r❛♠ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ✐♠♣♦ssí✈❡✐s✱ ❡ ♣❡r♣❡t✉♦✉ ❛s ❡①♣r❡ssõ❡s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ♥ú✲ ♠❡r♦ ✐♠❛❣✐♥ár✐♦✳ ❚♦♠❛♥❞♦ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ❡st❛ ♣❡sq✉✐s❛ ♦s tr❛❜❛❧❤♦s ❞❡ ❈❡rr✐ ❬✸❪✱ ❊❜❜✐♥❣❤❛✉s ❬✹❪ ❡ ●❛r❜✐ ❬✻❪✳
✶✳✶ ❈❛r❞❛♥♦✱ ❚❛rt❛❣❧✐❛ ❡ ❛ ❞✐s♣✉t❛ ❞❡ r❡s♦❧✉çã♦ ❞❡
❡q✉❛çõ❡s ❞♦ ✸✳
♦❣r❛✉
◗✉❛♥❞♦ ✉♠ ❡st✉❞❛♥t❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡stá ❛♣r❡♥❞❡♥❞♦ s♦❜r❡ ♦s ♥ú♠❡r♦s ❝♦♠✲ ♣❧❡①♦s✱ ❞❡✈❡ ❛❝❤❛r q✉❡ s✉❛ ❞❡s❝♦❜❡rt❛ s❡ ❞❡✉ ❛tr❛✈és ❞❛ ❡q✉❛çã♦ ❞♦ ✷✳♦ ❣r❛✉✱ q✉❛♥❞♦
♦ s❡✉ ❞✐s❝r✐♠✐♥❛♥t❡ é ♥❡❣❛t✐✈♦✳ P♦ré♠ ✈❡r❡♠♦s q✉❡ ❡st❡ ❡st✉❞♦ ♥ã♦ t❡♠ ❛ ✈❡r ❝♦♠ ❛ ❡q✉❛çã♦ ❞♦ ✷✳♦ ❣r❛✉ ❡ s✐♠ ❝♦♠ ❛ ❞♦ ✸✳♦ ❣r❛✉✱ ❡ q✉❡ ❛ ❛❜s♦rçã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡
♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦ s❡ ❞❡✉ ❞❡ ❢♦r♠❛ ❜❡♠ ❧❡♥t❛✳
❆ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s é ❛❧❣♦ q✉❡ ❢❛s❝✐♥❛ ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞❡ t♦❞❛ ❛ ❤✐stór✐❛✳ ❇❛❜✐❧ô♥✐♦s✱ ❣r❡❣♦s✱ r♦♠❛♥♦s✱ ✐♥❞✐❛♥♦s✱ ❤✐♥❞✉s✱ ❡♥tr❡ ♦✉tr♦s ♣♦✈♦s✱ s❡♠♣r❡ ♣r♦❝✉r❛✈❛♠ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❝r✐❛r ✉♠ ♠ét♦❞♦ ♣❛r❛ s❡ r❡s♦❧✈❡r q✉❛❧q✉❡r t✐♣♦ ❞❡ ❡q✉❛çã♦✳ ◗✉❛♥❞♦ ❢❛❧❛♠♦s ❞❡ ❡q✉❛çã♦ ❞♦ ✷✳♦ ❣r❛✉✱ ♣❡♥s❛♠♦s ❧♦❣♦ ❡♠ ❇❤ás❦❛r❛ ❡
✶✳✶✳ ❈❆❘❉❆◆❖✱ ❚❆❘❚❆●▲■❆ ❊ ❆ ❉■❙P❯❚❆ ❉❊ ❘❊❙❖▲❯➬➹❖ ❉❊
❊◗❯❆➬Õ❊❙ ❉❖ ✸✳♦ ●❘❆❯
s✉❛ té❝♥✐❝❛✳ ❆♣❡s❛r ❞❡❧❡ t❡r ✜❝❛❞♦ ❝♦♠ ❛ ❢❛♠❛✱ q✉❡♠ ❞❡s❝♦❜r✐✉ ❛ ❢ór♠✉❧❛ ♠❡s♠♦ ❢♦✐ ❙r✐❞❤❛r❛✱ ❝♦♥t❡rrâ♥❡♦ ❞❡ ❇❤ás❦❛r❛✳
❖ ❢❛t♦ ❞❡ q✉❡ ♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❡q✉❛çã♦ ❞♦ ✷✳♦ ❣r❛✉ ♣♦❞❡r s❡r ♥❡❣❛t✐✈♦ ♥ã♦
♣❡rt✉r❜❛✈❛ ♦s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛✳ ❊❧❡s✱ s✐♠♣❧❡s♠❡♥t❡ ❞✐③✐❛♠ q✉❡ ❛ ❡q✉❛çã♦ ♥ã♦ t✐♥❤❛ s♦❧✉çã♦✳
➱ ♥♦ ♠❡✐♦ ❞❛ ❞✐s♣✉t❛ ❡♥tr❡ ●✐r♦❧❛♠♦ ❈❛r❞❛♥♦ ✭❋✐❣✉r❛ ✶✳✶✮ ❡ ◆✐❝♦❧ò ❋♦♥t❛♥❛ ✭❚❛rt❛❣❧✐❛✮ ✭❋✐❣✉r❛ ✶✳✷✮ ♣❡❧❛ ❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞♦ ✸✳♦ ❣r❛✉ q✉❡ s❡ ♣❡r❝❡❜❡✉
q✉❡ ♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ❡r❛♠ s✉✜❝✐❡♥t❡s ❡ ❛s ♣r✐♠❡✐r❛s ✐❞❡✐❛s ❞❛ ❝r✐❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s s✉r❣✐r❛♠✳
❋✐❣✉r❛ ✶✳✶✿ ●✐r♦❧❛♠♦ ❈❛r❞❛♥♦ ✭1501−1576✮✳
P♦r ✈♦❧t❛ ❞❡ 1510✱ ❙❝✐♣✐♦♥❡ ❞❡❧ ❋❡rr♦ ❡♥❝♦♥tr♦✉ ✉♠❛ ❢ór♠✉❧❛ ❣❡r❛❧ ❞❡ r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s ❞♦ t✐♣♦
x3 +px+q= 0
❡ r❡✈❡❧♦✉ ❛ s❡✉ ❛❧✉♥♦✱ ❆♥t♦♥✐♦ ▼❛r✐❛ ❋✐♦r✱ q✉❡ ❛♣ós ❛ ♠♦rt❡ ❞❡ s❡✉ ♠❡str❡✱ t❡♥t♦✉ ❛❞q✉✐r✐r ♥♦t♦r✐❡❞❛❞❡ ❝♦♠ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ s❡✉ ♣r♦❢❡ss♦r ❧❛♥ç❛♥❞♦ ✉♠ ❞❡s❛✜♦ ❛ ❚❛r✲ t❛❣❧✐❛ ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞❡ss❡ t✐♣♦✳ ❚❛rt❛❣❧✐❛ ❛❝❡✐t♦✉ ♦ ❞❡s❛✜♦ ❡ ❞✉r❛♥t❡ s✉❛ ♣r❡♣❛r❛çã♦✱ ❚❛rt❛❣❧✐❛ r❡✈❡❧♦✉✿ ✏♠♦❜✐❧✐③❡✐ t♦❞♦ ♦ ❡♥t✉s✐❛s♠♦✱ ❛ ❛♣❧✐❝❛çã♦ ❡ ❛ ❛rt❡ ❞❡ q✉❡ ❢✉✐ ❝❛♣❛③✱ ♦❜❥❡t✐✈❛♥❞♦ ❡♥❝♦♥tr❛r ✉♠❛ r❡❣r❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛q✉❡❧❛s ❡q✉❛çõ❡s✱ ♦ q✉❡ ❝♦♥s❡❣✉✐ ❛ 10 ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ 1535✑❬✻❪✳ P♦ré♠✱ ❡❧❡ ❢♦✐ ❛✐♥❞❛ ♠❛✐s ❧♦♥❣❡ ❡ ❡♥❝♦♥tr♦✉ ❛ ❢ór♠✉❧❛ ❣❡r❛❧ ♣❛r❛ s❡ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ t✐♣♦
x3 +px2+q= 0,
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
❝✉❥❛ r❡s♦❧✉çã♦ ❋✐♦r ❞❡s❝♦♥❤❡❝✐❛ ❡✱ ❛ss✐♠✱ ❚❛rt❛❣❧✐❛ ✈❡♥❝❡✉ ♦ ❞❡s❛✜♦✳
❈❛r❞❛♥♦ ♦✉✈✐✉ ❢❛❧❛r ❞♦ ❢❡✐t♦ ❞❡ ❚❛rt❛❣❧✐❛ ❡ ❧♦❣♦ ❞❡s❡❥♦✉ ❞❡t❡r ❡st❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ❡①♣♦r ♥❛ s✉❛ ❢❛♠♦s❛ ♦❜r❛ q✉❡ ❡st❛✈❛ ♣♦r s❡r ❡s❝r✐t❛✳ ❊❧❡ t❡♥t♦✉ s❡❞✉③✐r ❚❛rt❛✲ ❣❧✐❛ ❛ ❡♥tr❡❣á✲❧♦ ❛ ❢ór♠✉❧❛✱ ♣♦ré♠ ❚❛rt❛❣❧✐❛ ❞❡s❡❥❛✈❛ q✉❡ ❡❧❡ ♣ró♣r✐♦ ✜③❡ss❡ ✐ss♦✱ ♣♦✐s ❡❧❡ ❡r❛ ♦ ❞❡s❝♦❜r✐❞♦r✳ ❈❛r❞❛♥♦ ♥ã♦ ❞❡s✐st✐✉ ❡ ❝♦♥t✐♥✉♦✉ ❛ ✐♥s✐st✐r✱ ❛té q✉❡ ❚❛rt❛❣❧✐❛ ♦ ❞❡✉ ❝♦♠ ❛ ♣r♦♠❡ss❛ ❞❡ q✉❡ ❡❧❡ ♥ã♦ ❛ r❡✈❡❧❛r✐❛ ❛ ♥✐♥❣✉é♠✱ ♠❛s ❡❧❡ ❧♦❣♦ ✈✐✉ q✉❡ ❢♦✐ ✉♠ ❣r❛♥❞❡ ❡rr♦✱ ♣♦✐s ❈❛r❞❛♥♦ ♣✉❜❧✐❝♦✉ ❡ ✜❝♦✉ ❝♦♠ ❛ ❢❛♠❛✳ ❆ ❢ór♠✉❧❛ ❛té ❤♦❥❡ ❧❡✈❛ ♦ ♥♦♠❡ ❞❡ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✳
❋✐❣✉r❛ ✶✳✷✿ ◆✐❝♦❧ò ❋♦♥t❛♥❛ ✭❚❛rt❛❣❧✐❛✮ ✭1499−1557✮✳
P❡r❝❡❜❛♠♦s q✉❡ q✉❛♥❞♦ ❚❛rt❛❣❧✐❛ ❞❡s❝♦❜r✐✉ ❛ ❢ór♠✉❧❛ ♣❛r❛ s❡ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ t✐♣♦
x3+px+q= 0,
❡❧❡ ❞❡s❝♦❜r✐✉ ❝♦♠♦ s❡ r❡s♦❧✈❡ q✉❛❧q✉❡r ❡q✉❛çã♦ ❝♦♠♣❧❡t❛ ❞♦ ✸✳♦ ❣r❛✉
ax3+bx2 +cx+d= 0.
❇❛st❛ ❢❛③❡r ❛ s✉❜st✐t✉✐çã♦ y = x+m ❡ ❝❛❧❝✉❧❛r m✱ ❞❡ ♠♦❞♦ ❛ ❛♥✉❧❛r ♦ t❡r♠♦ ❞❡
❣r❛✉ 2✳
❱❡❥❛♠♦s ❛ ❢ór♠✉❧❛ q✉❡ ❚❛rt❛❣❧✐❛ ❞❡s❝♦❜r✐✉ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞♦ t✐♣♦
x3+px+q= 0
❊✐s ❛ ❢ór♠✉❧❛✿
x= 3
s −q 2+ r q 2 2
+p
3 3 + 3 s −q 2 − r q 2 2
+p
3
3
.
✶✳✶✳ ❈❆❘❉❆◆❖✱ ❚❆❘❚❆●▲■❆ ❊ ❆ ❉■❙P❯❚❆ ❉❊ ❘❊❙❖▲❯➬➹❖ ❉❊
❊◗❯❆➬Õ❊❙ ❉❖ ✸✳♦ ●❘❆❯
P♦r ❡①❡♠♣❧♦✱ ❛ ❡q✉❛çã♦
x3−15x−4 = 0
t❡rá ❝♦♠♦ s♦❧✉çã♦ ❛tr❛✈és ❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦ ♦ s❡❣✉✐♥t❡ ✈❛❧♦r✿
x= 3
q
2 +√−121 + 3
q
2−√−121
❊st❛ ❡q✉❛çã♦ t❡♠ ❝♦♠♦ r❛í③❡s ♦s ✈❛❧♦r❡s
x= 4, x=−2 +√3 ❡ x=−2−√3,
❝♦♠♦ ❡st❛s r❛í③❡s r❡❛✐s ❡st❛r✐❛♠ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♦ r❡s✉❧t❛❞♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛r✲ ❞❛♥♦❄ ❈♦♠♦ s❡ ❡①tr❛✐r✐❛ r❛í③❡s ❝ú❜✐❝❛s ❞❡ ♥ú♠❡r♦s ❞❡ ♥❛t✉r❡③❛ ❞❡s❝♦♥❤❡❝✐❞❛❄ ●❛r❜✐✱ ❡♠ ❖ ❘♦♠❛♥❝❡ ❞❛s ❊q✉❛çõ❡s ❆❧❣é❜r✐❝❛s❬✻❪✱ ❞✐ss❡ q✉❡✿ ✏❚❛rt❛❣❧✐❛ ❤❛✈✐❛ ♠❡✲ ①✐❞♦ ❡♠ ✉♠ ✈❡r❞❛❞❡✐r♦ ✈❡s♣❡✐r♦✱ ❞♦ q✉❛❧ s❛í❛♠ ❡str❛♥❤íss✐♠❛s ❡ ✐♥s♦♥❞á✈❡✐s q✉❡s✲ tõ❡s✑✳ ❖ ✐♥t❡r❡ss❛♥t❡ é q✉❡ ❡♠ s✉❛ ❆rs ▼❛❣♥❛✭❋✐❣✉r❛ ✶✳✸✮✱ ❈❛r❞❛♥♦ ❥á ❤❛✈✐❛ ♦♣❡r❛❞♦ ❝♦♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ s❡♠ s❛❜❡r ❝♦♠ ♦ q✉❡ ❡st❛✈❛ ♠❡①❡♥❞♦✳ ❊❧❡ ❡s❝r❡✈❡✉ q✉❡ é ✐♠♣♦ssí✈❡❧ ❞✐✈✐❞✐r10❡♠ ❞✉❛s ♣❛rt❡s✱ ❞❡ ♠♦❞♦ q✉❡ s❡✉ ♣r♦❞✉t♦ s❡❥❛40✳ ❊♥tr❡t❛♥t♦✱ ❡❧❡ ❞✐ss❡✱ s❡ ♦ ♣r♦❜❧❡♠❛ ♣✉❞❡ss❡ s❡r r❡s♦❧✈✐❞♦✱ s❡r✐❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
(
x+y = 10
xy = 40.
❊❧❡✈❛✲s❡ ❛♦ q✉❛❞r❛❞♦ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛✲s❡ ♣♦r 4❛ s❡❣✉♥❞❛
(
x2+ 2xy+y2 = 100
4xy = 160.
❙✉❜tr❛✐♥❞♦✲s❡ ❛ ♣r✐♠❡✐r❛ ❞❛ s❡❣✉♥❞❛✱
x2−2xy+y2 =−60⇒(x−y)2 =−60⇒x−y=√−60 = 2√−15,
♦✉ s❡❥❛✱
(
x+y = 10
x−y = 2√−15.
P♦rt❛♥t♦✱
2x= 10 + 2√−15⇒x= 5 +√−15 ❡ y= 5−√−15.
❊✐s ❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ❤✐stór✐❛ ❡♠ q✉❡ ♦ ❤♦♠❡♠ ❢❡③ ❛❧❣✉♠❛s ♦♣❡r❛çõ❡s ❝♦♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❖ q✉❡ ♦ ❛✉t♦r✱ ❈❛r❞❛♥♦✱ ❞✐ss❡ ❛ r❡s♣❡✐t♦ ❞❡st❡ ❣r❛♥❞❡ ❛✈❛♥ç♦ ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛✿ ✏❊st❡ r❡s✉❧t❛❞♦ é tã♦ s✉t✐❧ q✉❛♥t♦ ✐♥út✐❧✑❬✻❪✳ ❈❛r❞❛♥♦✱ ❞❡ ❢❛t♦✱ ♥ã♦ s❛❜✐❛ ♦ t❡s♦✉r♦ q✉❡ t✐♥❤❛ ♥❛s ♠ã♦s✳
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
❋✐❣✉r❛ ✶✳✸✿ ❋r♦♥t✐s♣í❝✐♦ ❞❛ ❆rs ▼❛❣♥❛ ✭1545✮✱ ♦❜r❛ ❞❡ ❈❛r❞❛♥♦✳
✶✳✷ ❇♦♠❜❡❧❧✐ ❡ s✉❛ ✐❞❡✐❛ ♣✐♦♥❡✐r❛
❘❛❢❛❡❧ ❇♦♠❜❡❧❧✐✭❋✐❣✉r❛ ✶✳✹✮✱ ♥❛s❝✐❞♦ ❡♠ ❇♦❧♦♥❤❛✱ ■tá❧✐❛✱ ♥♦ ❛♥♦ ❞❡ 1530✱ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❤♦♠❡♠ ❛ t❡♥t❛r ❧✐❞❛r ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡s❝♦♥❤❡❝✐❞♦✱ ❛té ❡♥tã♦✳ ❊❧❡ ❞❡s❡❥❛✈❛ r❡❧❛❝✐♦♥❛r ❛ r❛✐③ x= 4 ❞❛ ❡q✉❛çã♦
x3−15x−4 = 0
❝♦♠ ♦ q✉❡ s❡ ❡r❛ ♦❜t✐❞♦ ❛tr❛✈és ❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦
x= 3
q
2 +√−121 + 3
q
2−√−121.
❊♠ s❡✉ ❧✐✈r♦ ✐♥t✐t✉❧❛❞♦ ▲✬❆❧❣❡❜r❛ ♣❛rt❡ ▼❛❣❣✐♦r❡ ❞❡❧❧✬❆r✐t❤♠❡t✐❝❛❬✹❪✱ ♣♦❞❡♠♦s ❡♥✲ ❝♦♥tr❛r ❝♦♠♦ s❡ ❞❡✉ ♦ s❡✉ ♣❡♥s❛♠❡♥t♦✳ ❊❧❡ r❡s♦❧✈❡✉ ❛ss♦❝✐❛r ♦s ♥ú♠❡r♦s
3
q
2 +√−121 ❡ 3
q
2−√−121
❛ ♥ú♠❡r♦s ♥❛ ❢♦r♠❛
a+√−b ❡ a−√−b,
r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆❡st❡ ❝❛s♦✱
3
q
2 +√−121 =a+√−b ❡ 3
q
2−√−121 =a−√−b.
✶✳✸✳ ❉❊❙❈❆❘❚❊❙✱ ◆❊❲❚❖◆ ❊ ▲❊■❇◆■❩
❆♦ ❝r✐❛r ❛s s❡❣✉✐♥t❡s r❡❣r❛s
(√−1)(√−1) = −1
(√−1)(−√−1) = 1
(−√−1)(−√−1) = −1
(1)(√−1) = √−1
(−1)(√−1) = −√−1
(1)(−√−1) = −√−1
(−1)(−√−1) = √−1
❡
(a+b√−1) + (c+d√−1) = (a+c) + (b+d)√−1,
❇♦♠❜❡❧❧✐ ❝❤❡❣♦✉ ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❡a= 2 ❡b = 1✱ ♣♦✐s
(2 +√−1)3 = 23+ 3.22.√−1 + 3.2.(√−1)2+ (√−1)3 = 2 +√−121
❡
(2−√−1)3 = 23−3.22.√−1 + 3.2.(√−1)2−(√−1)3 = 2−√−121.
❇♦♠❜❡❧❧✐ ❤❛✈✐❛ ❝♦♥str✉í❞♦ ♦s ❛❧✐❝❡r❝❡s ❞❡ t♦❞❛ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❋✐❣✉r❛ ✶✳✹✿ ❘❛❢❛❡❧ ❇♦♠❜❡❧❧✐ ✭1526−1572✮✳
✶✳✸ ❉❡s❝❛rt❡s✱ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③
❖s três ❣ê♥✐♦s ❞♦s q✉❛✐s ❢❛❧❛r❡♠♦s ❛ s❡❣✉✐r✱ ♥ã♦ ♠✉❞❛r❛♠ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♦ q✉❡ s❡ s❛❜✐❛ s♦❜r❡ ♦s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s ♥❛ é♣♦❝❛✱ ♠❛s ✈❛❧❡ ❛ ♣❡♥❛ s❛❜❡r ❛ ♦♣✐♥✐ã♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❛ r❡s♣❡✐t♦ ❞❡st❡s ♥ú♠❡r♦s✳
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
❋✐❣✉r❛ ✶✳✺✿ ❘❡♥é ❉❡s❝❛rt❡s ✭1642−1727✮
❘❡♥é ❉❡s❝❛rt❡s✭❋✐❣✉r❛ ✶✳✺✮✱ ❛♣❡s❛r ❞❡ t❡r ✐♥t❡♥s❛♠❡♥t❡ tr❛❜❛❧❤❛❞♦ ♥❛ ❝♦♥str✉çã♦ ❞❛ t❡♦r✐❛ ❞❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ ❡❧❡ ❡♠ s✉❛ ▲❛ ●é♦♠étr✐❡❬✹❪ ❡♠ 1637 t❛♠❜é♠ ❡st✉❞♦✉ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✳ ❊♠ ✉♠ tr❡❝❤♦ ❞❛ s✉❛ ♦❜r❛✱ ❡❧❡ ❞✐ss❡ q✉❡ ✏◆❡♠ s❡♠♣r❡ ❛s r❛í③❡s ✈❡r❞❛❞❡✐r❛s ✭♣♦s✐t✐✈❛s✮ ♦✉ ❢❛❧s❛s ✭♥❡❣❛t✐✈❛s✮ ❞❡ ✉♠❛ ❡q✉❛çã♦ sã♦ r❡❛✐s✳ ➚s ✈❡③❡s ❡❧❛s sã♦ ✐♠❛❣✐♥ár✐❛s✑❬✹❪✳ ❋♦r❛ ❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ❤✐stór✐❛ q✉❡ t❛❧ ❡①♣r❡ssã♦ ❢♦✐ ✉s❛❞❛ ❡ q✉❡ ♠❛✐s t❛r❞❡ s❡r✐❛ ❞❡s✐❣♥❛❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ♥ú♠❡r♦√−1✳ ❉❡s❝❛rt❡s t❛♠❜é♠ ❝♦♥❢❡ss♦✉ q✉❡ q✉❛♥t✐❞❛❞❡s ✐♠❛❣✐♥ár✐❛s sã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥❝❛♣❛③❡s ❞❡ s❡ ❝♦♥s❡❣✉✐r ✈✐s✉❛❧✐③❛r✳
❋✐❣✉r❛ ✶✳✻✿ ■s❛❛❝ ◆❡✇t♦♥ ✭1642−1727✮
■s❛❛❝ ◆❡✇t♦♥✭❋✐❣✉r❛ ✶✳✻✮ ❝♦♥s✐❞❡r♦✉ ❛s r❛í③❡s ❝♦♠♣❧❡①❛s ❝♦♠♦ s❡♥❞♦ ❛ ✐♥s♦❧✉✲ ❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣r♦❜❧❡♠❛✱ ❡①♣r❡ss❛♥❞♦ q✉❡ ❡q✉❛çõ❡s ❝✉❥❛s r❛í③❡s s❡r✐❛♠ ✐♠❛❣✐♥ár✐❛s ❞❡✈❡r✐❛♠ s❡r tr❛t❛❞❛s ❝♦♠♦ ✐♠♣♦ssí✈❡✐s✱ ♣❛r❛ q✉❡ ♥ã♦ ❛♣r❡s❡♥t❡♠ ❝❛s♦s ❞❡ ♣r♦❜❧❡✲ ♠❛s q✉❡ sã♦ ✐♥s♦❧ú✈❡✐s ♥♦s r❡❛✐s ❝♦♠♦ s❡ ❢♦ss❡♠ ♣♦ssí✈❡✐s✳ ◆♦ t❡♠♣♦ ❞❡ ◆❡✇t♦♥✱
✶✳✹✳ ❊❯▲❊❘✱ ❖ ▼❆❚❊▼➪❚■❈❖ ◗❯❊ ❉❖▼■◆❖❯ ❖❙ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙
♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥ã♦ t✐♥❤❛♠ ✉♠❛ ❛♣❧✐❝❛çã♦ s❡q✉❡r✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ ♥❛ ❢ís✐❝❛✳ ❚❛❧✈❡③✱ s❡ ❤♦✉✈❡ss❡✱ ❡❧❡ ♥ã♦ t❡r✐❛ ❝♦♥s✐❞❡r❛❞♦ ❛s s♦❧✉çõ❡s ✐♠❛❣✐♥ár✐❛s tã♦ ✐♠♣♦ssí✈❡✐s ❛ss✐♠✳
❋✐❣✉r❛ ✶✳✼✿ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③ ✭1646−1716✮
●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③✭❋✐❣✉r❛ ✶✳✼✮✱ ♦❝❛s✐♦♥❛❧♠❡♥t❡✱ ❡s❝r❡✈✐❛ ❛❧❣♦ r❡❧❛❝✐♦♥❛❞♦ ❛♦s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s✳ P♦r ✈♦❧t❛ ❞❡ 1675✱ ♥✉♠❛ ❝❛rt❛ ❞✐r❡❝✐♦♥❛❞❛ ❛ ❍✉②❣❡♥s✱ ❡s❝r❡✈❡✉ ✉♠❛ r❡❧❛çã♦ q✉❡ ❡♥r✐q✉❡❝❡✉ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s✿
q
1 +√−3 +
q
1−√−3 =√6.
❊♠ 1702✱ ♥♦ ♣r✐♠❡✐r♦ ♣❡r✐ó❞✐❝♦ ❝✐❡♥tí✜❝♦ ♣✉❜❧✐❝❛❞♦ ❡♠ ❛❧❡♠ã♦ ♣❡❧♦ ▲❡✐♣③✐❣ ❆❝t❛
❊r✉❞✐t♦r✉♠❬✹❪✱ ▲❡✐❜♥✐③ ❢❛❧❛ q✉❡ ❛s r❛í③❡s ✐♠❛❣✐♥ár✐❛s sã♦ ✉♠ t✐♣♦ ❞❡ r❡s♦rt s✉t✐❧ ❡ ♠❛r❛✈✐❧❤♦s♦✱ ❝❤❡✐♦ ❞❡ ❡s♣ír✐t♦ ❞✐✈✐♥♦✱ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ❤❡r♠❛❢r♦❞✐t❛ ❡♥tr❡ ❡①✐stê♥❝✐❛ ❡ ♥ã♦✲❡①✐stê♥❝✐❛✳ ❊♠ 1712✱ ▲❡✐❜♥✐③ ❥á ❤❛✈✐❛ ✐❞❡♥t✐✜❝❛❞♦ log(−1) ❝♦♠♦ s❡♥❞♦ ✉♠ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦✳
✶✳✹ ❊✉❧❡r✱ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ❞♦♠✐♥♦✉ ♦s ♥ú♠❡r♦s
❝♦♠♣❧❡①♦s
❆♥t❡s ❞♦ s✉íç♦ ▲❡♦♥❤❛r❞ ❊✉❧❡r✭❋✐❣✉r❛ ✶✳✽✮✱ ♥✐♥❣✉é♠ ❤❛✈✐❛ ❞❛❞♦ ✉♠❛ ❝♦♥tr✐❜✉✐✲ çã♦ tã♦ ❣r❛♥❞❡ ❛♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ ❡❧❡✳ ❆s ✐❞❡✐❛s ❞❡ ❊✉❧❡r✱ q✉❛♥t♦ ❛ ❡st❡s ♥ú♠❡r♦s ❡ ❝♦♠♦ ❡❧❡s ♣♦❞❡r✐❛♠ s❡r ✉t✐❧✐③❛❞♦s ❢♦r❛♠ ✈ár✐❛s✳ ❱❡❥❛♠♦s✱ ❡♥tã♦✱ ❛s s✉❛s ❝♦♥tr✐❜✉✐çõ❡s✳
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
❊♠ ♣r✐♥❝í♣✐♦✱ ❢♦✐ ❡❧❡✱ q✉❡♠ ✐♥tr♦❞✉③✐✉ ♦ sí♠❜♦❧♦ q✉❡ ✉s❛♠♦s ❛té ❤♦❥❡ ♣❛r❛ s✉❜s✲ t✐t✉✐r√−1✱ ❛ ❧❡tr❛i✱ ❧❡tr❛ ❡s❝♦❧❤✐❞❛ ❞❡✈✐❞♦ ❛♦ ♥♦♠❡✿ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦✳ ❊❧❡✱ ❡♥tã♦✱
♣❛ss♦✉ ❛ ❡st✉❞❛r ♦s ♥ú♠❡r♦s ❞❛ ❢♦r♠❛ z = a+bi✱ ♦♥❞❡ a ❡ b sã♦ ♥ú♠❡r♦s r❡❛✐s ❡ i2 =−1✳ ❊st❡s ♥ú♠❡r♦s ❢♦r❛♠ ❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❋✐❣✉r❛ ✶✳✽✿ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭1707−1783✮✳
❊✉❧❡r ❞❡✜♥✐✉ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣❛r❛ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ t❡♥❞♦ ❡♠ ✈✐st❛ ♦s ❡st✉❞♦s ❞❡ ❇♦♠❜❡❧❧✐✳ ❱❡r✐✜❝♦✉ q✉❡ sã♦ ✈á❧✐❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ss♦❝✐❛t✐✈❛✱ ❝♦♠✉t❛t✐✈❛ ❡ ❡①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✱ t❛♥t♦ ♣❛r❛ ❛ ❛❞✐çã♦✱ q✉❛♥t♦ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❛❧é♠ ❞❡ ✈❡r✐✜❝❛r t❛♠❜é♠ ❛s ❡①✐stê♥❝✐❛s ❞♦ ❡❧❡♠❡♥t♦ ♦♣♦st♦ ♣❛r❛ ❝❛❞❛ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❡ ❞♦ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ♣❛r❛ ❝❛❞❛ ❝♦♠♣❧❡①♦ ❞✐❢❡r❡♥t❡ ❞♦ ③❡r♦✳ ❆ss✐♠ ❡❧❡ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ✈❡r✐✜❝❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s é ♦ q✉❡ ❝❤❛♠❛♠♦s✱ ♠♦❞❡r♥❛♠❡♥t❡✱ ❞❡ ❝♦r♣♦✳ ❊st❡ ❣ê♥✐♦ ♣❡r❝❡❜❡✉ q✉❡ ❡r❛ ♣♦ssí✈❡❧ r❡❧❛❝✐♦♥❛r ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s à tr✐❣♦♥♦♠❡tr✐❛ ❡ q✉❡ ❡❧❡s ♣♦❞❡r✐❛♠ s❡r ❡s❝r✐t♦s ❞❡ ✉♠❛ ❝❡rt❛ ❢♦r♠❛✱ ❛ q✉❛❧ ❢♦✐ ❝❤❛♠❛❞❛ ❞❡ ❢♦r♠❛ ♣♦❧❛r ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✿
z =ρ(cosθ+isenθ),
♦♥❞❡ ρ=√a2+b2 é ❝❤❛♠❛❞♦ ❞❡ ♠ó❞✉❧♦ ❞❡ z ❡ θ é ♦ ❛r❣✉♠❡♥t♦ ❞❡ z✳ ❊ss❛ r❡❧❛çã♦
❝♦♠ ❛ tr✐❣♦♥♦♠❡tr✐❛ ❡ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❞❡ ❉❡s❝❛rt❡s só tr♦✉①❡ ❜❡♥❡❢í❝✐♦s✱ t❛✐s ❝♦♠♦ ♦ ❞❡s❝♦❜r✐♠❡♥t♦ ❞❡ ❝♦♠♦ s❡ ❡①tr❛✐r r❛í③❡s ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ▲❡♠❜r❡✲ ♠♦s q✉❡ ❡st❛ ❡r❛ ✉♠❛ q✉❡stã♦ q✉❡ ♣❡rt✉r❜❛✈❛ ♦s ♠❛t❡♠át✐❝♦s ❞❡s❞❡ ♦ ❞❡s❝♦❜r✐♠❡♥t♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✳
❈♦♠ r❡s♣❡✐t♦ ❛ r❛í③❡s q✉❛❞r❛❞❛s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✱ ♦ ♣ró♣r✐♦ ❊✉❧❡r✱ ❡♠ s✉❛ ♦❜r❛ ✐♥t✐t✉❧❛❞❛ ✏❱♦❧❧stä♥❞✐❣❡ ❆♥❧❡✐t✉♥❣ ③✉r ❆❧❣❡❜r❛✑✱ ♦✉ tr❛❞✉③✐❞❛ ♣❛r❛ ♦ ✐♥❣❧ês
✶✳✹✳ ❊❯▲❊❘✱ ❖ ▼❆❚❊▼➪❚■❈❖ ◗❯❊ ❉❖▼■◆❖❯ ❖❙ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙
❝♦♠♦ ✏❊❧❡♠❡♥ts ♦❢ ❆❧❣❡❜r❛✑✱ ❞✐③✿ ✏❋✐❝❛ ❝❧❛r♦✱ ♣♦rt❛♥t♦✱ q✉❡ ❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ♥ã♦ ♣♦❞❡♠ s❡r ❝♦♥t❛❞❛s ❡♥tr❡ ♦s ♥ú♠❡r♦s ♣♦ssí✈❡✐s✿ ❝♦♥s❡✲ q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s ❞❡ ❞✐③❡r q✉❡ ❡❧❛s sã♦ ♦s ♥ú♠❡r♦s q✉❡ sã♦ ✐♠♣♦ssí✈❡✐s✳ ❊st❛ ❝✐r❝✉♥stâ♥❝✐❛ ❧❡✈❛✲♥♦s ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦s✱ q✉❡✱ ♣❡❧❛ s✉❛ ♣ró♣r✐❛ ♥❛t✉r❡③❛✱ sã♦ ✐♠♣♦ssí✈❡✐s✱ ❡ q✉❡ sã♦ ❝♦♠✉♠❡♥t❡ ❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s ♦✉ ♥ú♠❡r♦s ❢❛♥t❛s✐♦s♦s✱ ♣♦rq✉❡ ❡❧❡s só ❡①✐st❡♠ ❡♠ ♥♦ss❛ ❢❛♥t❛s✐❛ ♦✉ ♥❛ ♥♦ss❛ ✐♠❛❣✐♥❛çã♦✑❬✹❪✳
❯♠❛ ♦✉tr♦ ❢❡✐t♦ ❞❡ ❊✉❧❡r ❝♦♠ r❡❧❛çã♦ ❛♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❢♦✐ ♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❛ ❢ór♠✉❧❛ ❞❛ r❛✐③n✲és✐♠❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ♥ã♦✲♥✉❧♦✳ ❋ór♠✉❧❛ ❡st❛
q✉❡ ♠♦str❛ t♦❞❛s ❛s n r❛í③❡s ❞✐st✐♥t❛s ❞♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✱ ❞❡s❞❡ q✉❡ ❡st❡❥❛ ♥❛
❢♦r♠❛ ♣♦❧❛r✳ P♦♥❞♦
z =ρ(cosφ+is❡♥φ) ❡ w=r(cosθ+is❡♥θ),
s❡zn
=w✱ ❡♥tã♦
zk=
n √ r cos
θ+ 2kπ n
+is❡♥
θ+ 2kπ n
, ∀ k ∈Z, ❝♦♠ zk=zk+n.
❉❡ ♣♦ss❡ ❞❡st❛ ❢ór♠✉❧❛✱ ♣♦❞❡♠♦s ✈♦❧t❛r ❛ ❡q✉❛çã♦ q✉❡ ✈✐♠♦s ♥❛ s❡çã♦ ✶✳✶
x3−15x−4 = 0
q✉❡✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✱ ♦❜tí♥❤❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
x= 3
q
2 +√−121 + 3
q
2−√−121
♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱
x=√3
2 + 11i+√3
2−11i.
◗✉❡ ♣♦r sé❝✉❧♦s ❧❡✈❛♥t♦✉ q✉❡st✐♦♥❛♠❡♥t♦s ❞❡ q✉❡ ❝♦♠♦ ❛ ❡q✉❛çã♦ ❝♦♠ ✉♠❛ s♦❧✉çã♦ tã♦ ❝❧❛r❛x= 4✱ ♣♦❞❡r✐❛ s❡ ❡q✉✐✈❛❧❡r ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡ s❡ ❡①✐st✐r✐❛♠ ♦✉tr❛s r❛í③❡s✱
❝♦♠♦ ♣❛r❡❝✐❛ ❡①✐st✐r✳ ❋♦✐ só ❝♦♠ ❛ ❢ór♠✉❧❛✱ q✉❡ ❊✉❧❡r ❛❥✉❞♦✉ ❛ ❞❡s❡♥✈♦❧✈❡r✱ q✉❡ ❡st❡s q✉❡st✐♦♥❛♠❡♥t♦s ❢♦r❛♠ r❡s♣♦♥❞✐❞♦s✳ ❆ss✐♠✱ ❡①tr❛✐♥❞♦ ❛s r❛í③❡s ❝ú❜✐❝❛s ❞❡ 2 + 11i
❡2−11i ❡ s♦♠❛♥❞♦✲❛s✱ ❝❤❡❣❛✲s❡ ❛s três r❛í③❡s ❞✐st✐♥t❛s ❞❛ ❡q✉❛çã♦ ❡♠ q✉❡stã♦✿
x= 4, x=−2 +√3 ❡ x=−2−√3,
❊♠❜♦r❛ ♥ã♦ s♦✉❜❡ss❡✱ ❚❛rt❛❣❧✐❛✱ q✉❛♥❞♦ ❞❡s❝♦❜r✐✉ ❛ ❢ór♠✉❧❛ ❞❡ r❡s♦❧✈❡r ❡q✉❛çõs ❞♦ ✸✳♦ ❣r❛✉✱ ❞❡s❝♦❜r✐✉ ❛ ❢ór♠✉❧❛ q✉❡ ❣❡r❛✈❛ t♦❞❛s ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦✳ ❇♦♠❜❡❧❧✐ ❞❡✉
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ♣❛r❛ ❛s q✉❡stõ❡s q✉❡ s❡ s❡❣✉✐r❛♠ ❡ ❊✉❧❡r ❞❡s✈❡♥❞♦✉ t♦t❛❧♠❡♥t❡ ♦ ♠✐stér✐♦ q✉❡ s♦♥❞❛✈❛ s♦❜r❡ ♦s ♥ú♠❡r♦s ✏❞❡s❝♦♥❤❡❝✐❞♦s✑✱ ♣♦r ✐st♦ ❡ ♣♦r ♠✉✐t♦ ♠❛✐s✱ q✉❡ ❡❧❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ❞♦♠✐♥♦✉ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❆✐♥❞❛ ❤á ❛ ❢ór♠✉❧❛ ❞❡ ❊✉❧❡r ❞❡s❡♥✈♦❧✈✐❞❛ ❛♣ós ❡st❡ ❝♦♥❥❡❝t✉r❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s✳ ❯t✐❧✐③❛♥❞♦ ♠ét♦❞♦ ❞❡ sér✐❡s ✐♥✜♥✐t❛s✱ ❡❧❡ ❞❡♠♦♥str♦✉ q✉❡
eiθ = cosθ+is❡♥θ.
◆♦ ❝❛s♦ ❡♠ q✉❡ θ = π✱ t❡♠♦s ❛ ❡①♣r❡ssã♦ q✉❡ ❡♥✈♦❧✈❡ ♦s ❝✐♥❝♦ ♥ú♠❡r♦s ♠❛✐s
✐♠♣♦rt❛♥t❡s ❞❡ t♦❞❛ ❛ ♠❛t❡♠át✐❝❛✿
eiπ+ 1 = 0,
✐❣✉❛❧❞❛❞❡ ❡st❛ q✉❡ é t✐❞❛ ❝♦♠♦ ❛ ♠❛✐s ❜❡❧❛ ❡♥tr❡ t♦❞❛s ❛s ✐❣✉❛❧❞❛❞❡s✱ s❡❣✉♥❞♦ ❋❡②♥♠❛♥ ❬✺❪✳
✶✳✺ ❖s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ❝♦♠♦ P♦♥t♦s ♥♦ P❧❛♥♦✿
❲❛❧❧✐s✱ ❲❡ss❡❧ ❡ ❆r❣❛♥❞
❋✐❣✉r❛ ✶✳✾✿ ❏♦❤♥ ❲❛❧❧✐s ✭1616−1703✮✳
❆s ♣r✐♠❡✐r❛s ♥♦çõ❡s ❞❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ♣♦♥✲ t♦s ♥✉♠ ♣❧❛♥♦ ❢♦r❛♠ tr❛③✐❞❛s ♣❡❧♦ ♠❛t❡♠át✐❝♦ ✐♥❣❧ês ❏♦❤♥ ❲❛❧❧✐s✭❋✐❣✉r❛ ✶✳✾✮✱ ❡♠
✶✳✺✳ ❖❙ ◆Ú▼❊❘❖❙ ❈❖▼P▲❊❳❖❙ ❈❖▼❖ P❖◆❚❖❙ ◆❖ P▲❆◆❖✿ ❲❆▲▲■❙✱ ❲❊❙❙❊▲ ❊ ❆❘●❆◆❉
s✉❛ ♦❜r❛ ✐♥t✐t✉❧❛❞❛ ✏❉❡ ❆❧❣❡❜r❛ ❚r❛❝t❛t✉s✑✱ ♣✉❜❧✐❝❛❞❛ ❡♠1685✳ P♦ré♠✱ s✉❛s ✐❞❡✐❛s ♥ã♦ t✐✈❡r❛♠ ♥❡♥❤✉♠❛ ✐♥✢✉ê♥❝✐❛ ❡♠ s❡✉s ❝♦♥t❡♠♣♦râ♥❡♦s✳
❆ ♣r✐♠❡✐r❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ♣♦♥t♦s ♥✉♠ ♣❧❛♥♦ ❝♦♠♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s q✉❡ ❢♦✐ ❧❡✈❛❞❛ ♠❛✐s ❛ sér✐♦ ❢♦✐ ❛ ❞♦ t♦♣ó❣r❛❢♦ ❈❛s♣❛r ❲❡ss❡❧✭❋✐❣✉r❛ ✶✳✶✵✮✱ ❡♠ s❡✉ ♠❡♠♦r✐❛❧ ✏❖♥ t❤❡ ❛♥❛❧②t✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❞✐r❡❝t✐♦♥ ✲ ❛♥ ❡ss❛②✑ ❞❡ 1798✳ ❖ ♣r✐♠❡✐r♦ ♦❜❥❡t✐✈♦ ❞❡ ❲❡ss❡❧ ❢♦✐ ♣♦❞❡r ♦♣❡r❛r ❝♦♠ s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦s ❡ ❛ss✐♠ ❡❧❡ t❡✈❡ ❛ ✐❞❡✐❛ ❞❡ ♦❧❤❛r ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ ❡ss❡s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s✳ ❲❡ss❡❧ ✐♥tr♦❞✉③✐✉ ✉♠ ❡✐①♦ ❞❡ ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ ✐♥t❡r♣r❡t♦✉ ✈❡t♦r❡s ❝♦♠♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥♦ ♣❧❛♥♦ ✭❋✐❣✉r❛ ✶✳✶✷✮✳ ❆ss✐♠✱ ❡❧❡ ❞❡✜♥✐✉ ❛s ♦♣❡r❛çõ❡s ✉s✉❛✐s ❞❡ ✈❡t♦r❡s ♣❛r❛ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❢❛❧❛♥❞♦✳
❋✐❣✉r❛ ✶✳✶✵✿ ❈❛s♣❛r ❲❡ss❡❧ ✭1745−1818✮✳
❯♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞✐❢❡r❡♥t❡ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❢♦✐ ❞❛❞❛ ♣❡❧♦ s✉íç♦ ❏❡❛♥ ❘♦❜❡rt ❆r❣❛♥❞✭❋✐❣✉r❛ ✶✳✶✶✮ ❡♠ s✉❛ ✏❊ss❛✐ s✉r ✉♥❡ ♠❛♥✐èr❡ ❞❡ r❡♣rés❡♥t❡r ❧❡s q✉❛♥t✐tés ✐♠❛❣✐♥❛✐r❡s ❞❛♥s ❧❡s ❝♦♥str✉❝t✐♦♥s ❣é♦♠❡tr✐q✉❡s✑✳ ❊❧❡ ✐♥t❡r♣r❡t❛ √−1 ❝♦♠♦ ✉♠❛ r♦t❛çã♦ ❛tr❛✈és ❞❡ ✉♠ â♥❣✉❧♦ r❡t♦ ♥♦ ♣❧❛♥♦ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦ ❡ q✉❡ ♦ ♣r♦❞✉t♦
(√−1)(√−1) =−1
é ❡q✉✐✈❛❧❡♥t❡ ❛ ❢❛③❡r ❞✉❛s r♦t❛çõ❡s ❞❡ â♥❣✉❧♦s r❡t♦s✱ ♦✉ s❡❥❛✱ ♦ ♠❡s♠♦ q✉❡ ✉♠❛ r❡✢❡①ã♦✳
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
❋✐❣✉r❛ ✶✳✶✶✿ ❏❡❛♥ ❘♦❜❡rt ❆r❣❛♥❞ ✭1768−1822✮✳
❉❡st❛ ❢♦r♠❛✱ ♦s ♥ú♠❡r♦s ❥á ❡r❛♠ ❝♦♥❤❡❝✐❞♦s t❛♥t♦ ♥❛ s✉❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❝♦♠♦ ♥❛ tr✐❣♦♥♦♠étr✐❝❛✱ ❛❧é♠ ❞❡ s✉❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✳
❋✐❣✉r❛ ✶✳✶✷✿ ❋♦r♠❛ ●❡♦♠étr✐❝❛ ❞♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✳
✶✳✻ ●❛✉ss ❡ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛
❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ✭❋✐❣✉r❛ ✶✳✶✸✮ ❢❡③ ❝♦♠ q✉❡ ❛s ✈✐sõ❡s s♦❜r❡ ♥ú♠❡r♦s ❝♦♠✲ ♣❧❡①♦s ❢♦ss❡♠ ♠✉❞❛❞❛s✳ ❊❧❡ ❥á t✐♥❤❛ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ ♣♦♥t♦s ♥♦ ♣❧❛♥♦✱ ♦ q✉❡ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r ♥✉♠❛ ❝❛rt❛ ❡♠ q✉❡ ●❛✉ss ❡s❝r❡✈❡✉ ♣❛r❛ ❇❡ss❡❧✱ ♦♥❞❡ ❞✐③✿ ✏❆ss✐♠ ❝♦♠♦ s❡ ♣♦❞❡ ♣❡♥s❛r ❡♠ t♦❞♦ ♦ ❞♦♠í♥✐♦ ❞❡ ❣r❛♥❞❡③❛s r❡❛✐s ❝♦♠♦ s❡♥❞♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ✉♠❛ ❧✐♥❤❛ r❡t❛ ✐♥✜♥✐t❛✱ ❞❡ ♠❡s♠♦ ♠♦❞♦ ♦ ❞♦♠í♥✐♦ ❝♦♠♣❧❡t♦ ❞❡ t♦❞❛s ❛s ❣r❛♥❞❡③❛s✱ ♥ú♠❡r♦s r❡❛✐s ❡ ✐♠❛❣✐♥ár✐♦s ♣♦✲ ❞❡♠ s❡r ✈✐s✉❛❧✐③❛❞♦s ❝♦♠♦ ✉♠ ♣❧❛♥♦ ✐♥✜♥✐t♦✱ ♥♦ q✉❛❧ ♦ ♣♦♥t♦ ❞❡✜♥✐❞♦ ♣❡❧❛ ❛❜s❝✐ss❛
✶✳✻✳ ●❆❯❙❙ ❊ ❖ ❚❊❖❘❊▼❆ ❋❯◆❉❆▼❊◆❚❆▲ ❉❆ ➪▲●❊❇❘❆
a ❡ ✉♠❛ ♦r❞❡♥❛❞❛ b✱ r❡♣r❡s❡♥t❛ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ♥ú♠❡r♦ a+bi✑❬✹❪✳ ❊❧❡ ✉s♦✉ ❡st❡
❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦✱ ♦♥❞❡ ♣r♦✈♦✉ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ❡♠ 1799✱ ❛♦s 21 ❛♥♦s ❞❡ ✐❞❛❞❡✳ ❊st❛ t❡s❡ ❛té ❤♦❥❡ é ❝♦♥s✐❞❡r❛❞❛ ❛ ♠❛✐♦r t❡s❡ ❡♠ ♠❛t❡♠át✐❝❛ ❞❡ t♦❞♦s ♦s t❡♠♣♦s✳
❋✐❣✉r❛ ✶✳✶✸✿ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ✭1777−1855✮✳
❖ t❡♦r❡♠❛ ❞✐③ q✉❡ ✏❚♦❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ♦✉ ❝♦♠♣❧❡①♦s t❡♠✱ ♣❡❧♦ ♠❡♥♦s✱ ✉♠❛ r❛✐③ ❝♦♠♣❧❡①❛✑❬✸❪✳ ■ss♦ ✐♠♣❧✐❝❛ ❡♠ q✉❡ t♦❞❛ ❡q✉❛çã♦ ❞❡ ❣r❛✉ n t❡♠ ❡①❛t❛♠❡♥t❡ n r❛í③❡s✱ ❡✈❡♥t✉❛❧♠❡♥t❡ r❡♣❡t✐❞❛s✳ ❈❡rr✐ ❡ ▼♦♥t❡✐r♦ ❡♠
❍✐stór✐❛ ❞♦s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s❬✸❪ ❛✜r♠❛♠✿ ✏❆ss✐♠✱ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ r❡s♦❧✈❡✉ ❛ q✉❡stã♦ ❞❛s s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ❡ ❛✐♥❞❛ ♠♦str♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s é ♦ ♠❡❧❤♦r ❝♦♥❥✉♥t♦ ♣❛r❛ s❡ tr❛t❛r ❞♦ ❛ss✉♥t♦✱ ♣♦✐s ❝♦♥té♠ t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ q✉❛❧q✉❡r ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛✱ ❞❡ q✉❛❧q✉❡r ❣r❛✉✑✳
▼❡s♠♦✱ ❝♦♠ ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛✱ ❛ ✐❞❡✐❛ ❞♦s ♥ú♠❡✲ r♦s ❝♦♠♣❧❡①♦s ♥ã♦ ❤❛✈✐❛ s✐❞♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐ss❡♠✐♥❛❞❛ ❛té 1831✱ q✉❛♥❞♦ ●❛✉ss ♣✉❜❧✐❝♦✉ ✏❚❤❡♦r✐❛ ❘❡s✐❞✉♦r✉♠ ❇✐q✉❛❞r❛t✐❝♦r✉♠✳ ❈♦♠❡♥t❛t✐♦ ❙❡❝✉♥❞❛✑✳ ❆q✉✐ ❡❧❡ ❞❡✜♥❡ ❝❧❛r❛♠❡♥t❡ s❡✉s ♣♦♥t♦s ❞❡ ✈✐st❛ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ q✉❡ s✉♣❡r❛ t♦❞❛s ❛s ♦❜❥❡✲ çõ❡s ❧ó❣✐❝❛s✱ ❝♦♠♣r❛ ❛ ❡①♣r❡ssã♦ ✏♥ú♠❡r♦ ❝♦♠♣❧❡①♦✑ ❡ ❞❡s❝r❡✈❡ ❛ ❛t✐t✉❞❡ ❞❡ s❡✉s ❝♦♥t❡♠♣♦râ♥❡♦s ❛ ❡ss❡s ♥ú♠❡r♦s✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣❛❧❛✈r❛s✿ ✏▼❛s ❡st❡s ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s✱ ❛♦ ❝♦♥trár✐♦ ❞❡ q✉❛♥t✐❞❛❞❡s r❡❛✐s ✲ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡ ♠❡s♠♦ ❛❣♦r❛✱ ♦❝❛✲ s✐♦♥❛❧♠❡♥t❡✱ ❡♠❜♦r❛ ✐♠♣r♦♣r✐❛♠❡♥t❡ ❝❤❛♠❛❞♦ ✐♠♣♦ssí✈❡❧ ✲ ❢♦r❛♠ ❛♣❡♥❛s t♦❧❡r❛❞♦s✱ ❡♠ ✈❡③ ❞❡ ❞❛❞♦ ❝✐❞❛❞❛♥✐❛ ♣❧❡♥❛ ❡✱ ♣♦rt❛♥t♦✱ ❛♣❛r❡❝❡♠ ♠❛✐s ❝♦♠♦✱ ✉♠ ❥♦❣♦ ❥♦❣❛❞♦ ❝♦♠ sí♠❜♦❧♦s ❞❡s♣r♦✈✐❞♦s ❞❡ ❝♦♥t❡ú❞♦ ❡♠ s✐✱ ❛ q✉❡ ✉♠ s❡ ❛❜sté♠ ❛❜s♦❧✉t❛♠❡♥t❡ ❞❡ ❛tr✐❜✉✐r q✉❛❧q✉❡r s✉❜str❛t♦ ✈✐s✉❛❧✐③á✈❡❧✳ ❆♦ ❞✐③❡r ✐st♦ ♥ã♦ s❡ t❡♠ ✈♦♥t❛❞❡ ❞❡ ❞❡♣r❡✲ ❝✐❛r ♦ tr✐❜✉t♦ r✐❝♦ q✉❡ ❡st❡ ❥♦❣♦ ❝♦♠ ♦s sí♠❜♦❧♦s t❡♠ ❝♦♥tr✐❜✉í❞♦ ♣❛r❛ ♦ t❡s♦✉r♦ ❞❛s
❈❆P❮❚❯▲❖ ✶✳ ❆❇❖❘❉❆●❊▼ ❍■❙❚Ó❘■❈❆ ❉❖❙ ◆Ú▼❊❘❖❙ ■▼❆●■◆➪❘■❖❙ ✲ ❉❊ ❈❆❘❉❆◆❖ ❆ ●❆❯❙❙
r❡❧❛çõ❡s ❡♥tr❡ ♦s ♥ú♠❡r♦s r❡❛✐s✳✑❬✹❪
❊ ❡♠ r❡❧❛çã♦ à ❛✉r❛ ❞❡ ♠✐stér✐♦ q✉❡ r♦♥❞❛✈❛ s♦❜r❡ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ●❛✉ss ❞✐③ q✉❡✿ ✏❙❡ ❡st❡ ❛ss✉♥t♦ ❛té ❛❣♦r❛ t❡♠ s✐❞♦ ❝♦♥s✐❞❡r❛❞♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡rr❛❞♦ ❡✱ ❛ss✐♠✱ ❡♥✈♦❧t♦ ❡♠ ♠✐stér✐♦ ❡ ❝❡r❝❛❞♦ ♣❡❧❛ ❡s❝✉r✐❞ã♦✱ é ❡♠ ❣r❛♥❞❡ ♣❛rt❡✱ à ✉♠❛ t❡r♠✐♥♦❧♦❣✐❛ ✐♥❛❞❡q✉❛❞❛✱ q✉❡ ❞❡✈❡ s❡r r❡s♣♦♥s❛❜✐❧✐③❛❞♦✳ ❚❡✈❡ +1,−1 ❡ √−1✱ ❡♠ ✈❡③ ❞❡ s❡r ❝❤❛♠❛❞♦ ❞❡ ♣♦s✐t✐✈♦✱ ♥❡❣❛t✐✈♦ ❡ ✐♠❛❣✐♥ár✐♦ ✭♦✉ ♣✐♦r ❛✐♥❞❛✱ ✐♠♣♦ssí✈❡❧✮ ❛ ✉♥✐❞❛❞❡✱ ❢♦ss❡ ❞❛❞♦ ♦ ♥♦♠❡ ❞❡✱ ❞✐❣❛♠♦s✱ ✉♥✐❞❛❞❡ ❞✐r❡t❛✱ ✐♥✈❡rs❛✱ ❡ ❧❛t❡r❛❧✳ ❊♥tã♦✱ ❞✐✜❝✐❧♠❡♥t❡ t❡r✐❛ t✐❞♦ q✉❛❧q✉❡r ♣♦ss✐❜✐❧✐❞❛❞❡ ♣❛r❛ t❛❧ ♦❜s❝✉r✐❞❛❞❡✳✑❬✹❪
P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ●❛✉ss ✈✐r♦✉ ✉♠ ✈❡r❞❛❞❡✐r♦ ❛❞✈♦❣❛❞♦ ❞♦s ♥ú♠❡r♦s ❝♦♠✲ ♣❧❡①♦s✱ ❧✉t❛♥❞♦ ❝♦♥tr❛ ♦ ♣r❡❝♦♥❝❡✐t♦ q✉❡ ❛✐♥❞❛ ❡①✐st✐❛ ♥❛ é♣♦❝❛✱ ❝♦♥❢♦r♠❡ ❡st❡ r❡❧❛t♦✿ ✏P♦❞❡r✐❛ s❡r tr✐st❡ ❡♠ t✉❞♦ ✐ss♦ q✉❡✱ ❡♥q✉❛♥t♦ q✉❛♥t✐❞❛❞❡s ✐♠❛❣✐♥ár✐❛s ❛✐♥❞❛ ❢♦r❛♠ ❜❛s❡❛❞❛s ❡♠ ✉♠❛ ✜❝çã♦✱ ♥ã♦ ❡r❛♠✱ ♣♦r ❛ss✐♠ ❞✐③❡r✱ t♦t❛❧♠❡♥t❡ ❛❝❡✐t♦ ❡♠ ♠❛t❡✲ ♠át✐❝❛✱ ♠❛s ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞❛s s✐♠ ❝♦♠♦ ❛❧❣♦ ❛ s❡r t♦❧❡r❛❞♦❀ ❡❧❡s ♣❡r♠❛♥❡❝❡r❛♠ ❧♦♥❣❡ ❞❡ s❡r ❞❛❞♦ ♦ ♠❡s♠♦ st❛t✉s q✉❡ ❛s q✉❛♥t✐❞❛❞❡s r❡❛✐s✳ ❏á ♥ã♦ ❡①✐st❡ q✉❛❧q✉❡r ❥✉st✐✜❝❛çã♦ ♣❛r❛ t❛❧ ❞✐s❝r✐♠✐♥❛çã♦✱ ❛❣♦r❛ q✉❡ ❛ ♠❡t❛❢ís✐❝❛ ♦✉ ♥ú♠❡r♦s ✐♠❛❣✐♥ár✐♦s ❢♦✐ ❝♦❧♦❝❛❞♦ ❡♠ ✉♠❛ ✈❡r❞❛❞❡✐r❛ ❧✉③ ❡ q✉❡ t❡♠ s✐❞♦ ❞❡♠♦♥str❛❞♦ q✉❡ tê♠ ❛♣❡♥❛s ❝♦♠♦ ✉♠ ❜♦♠ s✐❣♥✐✜❝❛❞♦ ♦❜❥❡t✐✈♦ r❡❛❧ ❝♦♠♦ ♦s ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✳✑❬✹❪ ❆ ❝✐t❛çã♦ ♦r✐❣✐♥❛❧ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥♦ ❧✐✈r♦ ◆❯▼❇❊❘❙✳ ❬✹❪ ✭❋✐❣✉r❛ ✶✳✶✹✮
❋✐❣✉r❛ ✶✳✶✹✿ ❈✐t❛çã♦ ❖r✐❣✐♥❛❧ ❞❡ ●❛✉ss✱ ❡s❝r✐t♦ à s✉❛ ♣ró♣r✐❛ ♠ã♦✳
❈♦♠ t♦❞♦ ♦ tr❛❜❛❧❤♦ ❡ s✉❛ ❝♦♥st❛♥t❡ ❞❡❢❡s❛ ❛♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ●❛✉ss ❝♦♥s❡❣✉✐✉ r❡♠♦✈❡r t♦❞❛ ❛ ❛✉r❛ ❞❡ ♠✐st✐❝✐s♠♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♣♦✐s s✉❛ ✐♥t❡r♣r❡t❛çã♦ s✐♠♣❧❡s ❝♦♠♦ ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❧✐❜❡rt♦✉ ❡ss❛ ❣r❛♥❞❡③❛ ✜❝tí❝✐❛ ❞❡ t♦❞♦ ♦ ♠✐stér✐♦ ❡ ❡s♣❡❝✉❧❛çõ❡s ❡ ❞❡✉ ❛ ❡❧❡s ✉♠❛ ❝✐❞❛❞❛♥✐❛ ♣❧❡♥❛ ♥❛ ♠❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ♦s ♥ú♠❡r♦s r❡❛✐s ❥á ♣♦ss✉í❛♠✳ ❙✐♠♣❧❡s♠❡♥t❡✱ ♣♦❞❡♠♦s r❡s✉♠✐r q✉❡ ●❛✉ss ❢❡③ ❞♦s ✏✐♠♣♦ssí✈❡✐s✑✱ ♣♦ssí✈❡✐s✳
❈❛♣ít✉❧♦ ✷
❍❛♠✐❧t♦♥ ❡ ♦ ❙❡✉ ❉❡s❡❥♦ ❞❡ ■r ❆❧é♠
◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❛ ❣r❛♥❞✐♦s❛ ❞❡s❝♦❜❡rt❛ ❞❡ ❙✐r ❲✐❧❧✐❛♠ ❘♦✇❛♥ ❍❛✲ ♠✐❧t♦♥❬✼❪✳ ❚r❛t❛✲s❡ ❞❛ ❡①♣❛♥sã♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ✉♠ ❝♦♥❥✉♥t♦ ❞❛ q✉❛rt❛ ❞✐♠❡♥sã♦✱ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ q✉❛tér♥✐♦s ❞❡ ❍❛♠✐❧t♦♥❬✶✼❪✳ ❉✐s❝♦rr❡r❡♠♦s t♦❞♦ ♦ ❝♦♥✲ t❡①t♦ ❤✐stór✐❝♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ t❛❧ ❝♦♥❥✉♥t♦ ❝♦♠♦ s❡ é ❝♦♥❤❡❝✐❞♦ ❡ t❛♠❜é♠ ❞❡ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s❬✶✵❪✳
✷✳✶ ❊ s❡ ❢♦ss❡ ♣♦ssí✈❡❧ ❡①♣❛♥❞✐r ♦s ❝♦♠♣❧❡①♦s ♣❛r❛
♦ ❡s♣❛ç♦❄
❆♣ós ●❛✉ss t❡r ❞✐❢✉♥❞✐❞♦ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❝♦♠❡ç❛✲ r❛♠ ❛ tr❛❜❛❧❤❛r ❝♦♠ ❡❧❡s ❡♠ s✉❛s ♣❡sq✉✐s❛s✳ ❋♦✐ ❡♥tã♦ q✉❡ ❙✐r ❲✐❧❧✐❛♠ ❘♦✇❛♥ ❍❛♠✐❧t♦♥ ✭❋✐❣✉r❛ ✷✳✶✮ s❡ ❡♥❝❛♥t♦✉ ❝♦♠ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❡s♣❡❝✐❛❧♠❡♥t❡ ❝♦♠ ❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❝♦♠♦ ♣♦♥t♦s ♦✉ ✈❡t♦r❡s ♥♦ ♣❧❛♥♦ ❛tr✐❜✉í❞❛ ❛ ❏♦❤♥ ❲❛r✲ r❡♥✳ ■♥s♣✐r❛❞♦ ♣❡❧♦ tr❛❜❛❧❤♦ ❞❡ ❲❛rr❡♥ ❡ ❝♦♠ s✉❛s ✐❞❡✐❛s ❢❡r✈✐❧❤❛♥❞♦ ✱ ❍❛♠✐❧t♦♥ ❡♥tã♦ ❝♦♠❡ç♦✉ ❛ ♣❡♥s❛r s❡ ❡r❛ ♣♦ssí✈❡❧ ❧❡✈❛r ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♣❛r❛ ♦ ❡s♣❛ç♦✳
❋♦✐ q✉❛♥❞♦ ❡❧❡ t❡♥t♦✉ ❛❝r❡s❝❡♥t❛r ✉♠ ♥♦✈♦ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦ ❞✐❢❡r❡♥t❡ ❞♦ ❝♦✲ ♥❤❡❝✐❞♦i✱ ♠❛s ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❡st❡ ♥ú♠❡r♦ ❡❧❡✈❛❞♦ ❛♦ q✉❛❞r❛❞♦ t❛♠❜é♠ ❢♦ss❡ −1✳
❉❛í s✉r❣❡♠ ♦s tr✐♣❧❡t♦s✿
a+bi+cj, ❝♦♠ i2 =−1, j2 =−1 ❡ i6=j.
✭❋♦r♠❛ ❛❧❣é❜r✐❝❛✮✭❋✐❣✉r❛ ✷✳✺✮
