28-N\303\272meros Imagin\303\241rios & Complexos Uma explora\303\247\303\243o da deriva\303\247\303\243o dos n\303\272meros imagin\303\241rios e complexos, opera\303\247\303\265es num\303\251ricas com n\303\272meros imagin\303\241rios e complexos, n\303\272meros conjugados complexos e o gr\303\241fico de n\303\272meros complexos em planos complexos.

restart; with(plots):

Warning, the name changecoords has been redefined

ComplexPlot := proc( z ) local a,b,c,d,Axes,Pt, Run, Rise; if (Re(z) = 0) then b := 1; else b := 1.3*abs(Re(z)); fi; a := - b; if (Re(z) > 0) then a := a/2; else b := b/2; fi; if (Im(z) = 0) then d := 1; else d := 1.3*abs(Im(z)); fi; c := - d; if (Im(z) > 0) then c:= c/2; else d :=d/2; fi; Axes := plot(0, x = a..b, y = c..d, color = black, scaling = constrained): Pt := plottools[disk]([ Re(z), Im(z) ], 1/3, color=red ): Run := plot( [[ 0, Im(z)], [Re(z), Im(z)]], color = blue, linestyle = 2 ); Rise := plot( [[ Re(z), 0],[Re(z), Im(z)] ], color = blue, linestyle = 2 ); plots[display](Axes,Pt, Run, Rise); end proc:

N\303\243o existem n\303\272meros reais que, elevados ao quadrado, resultem em n\303\272meros negativos. x^2 = - 1;

NiMvKiQpJSJ4RyIiIyIiIiEiIg==

x^2 = - 9;

NiMvKiQpJSJ4RyIiIyIiIiEiKg==

x^2 = - 12;

NiMvKiQpJSJ4RyIiIyIiIiEjNw==

Para resolver isso, podemos "inventar" um novo n\303\272mero que pode nos ajudar a dar sentido a essas equa\303\247\303\265es. sqrt(-1);

NiNeIyIiIg==

A simples defini\303\247\303\243o nos fornece capacidade de resolver os tipos de problemas que vimos acima. Quando resolvemos equa\303\247\303\265es com constantes positivas como essa, temos uma resposta positiva e uma negativa. E \303\251 assim que se resolvem equa\303\247\303\265es com constantes negativas. x^2 = + 25; solve(%);

NiMvKiQpJSJ4RyIiIyIiIiIjRA==NiQiIiYhIiY=

x^2 = - 1; solve(%);

NiMvKiQpJSJ4RyIiIyIiIiEiIg==NiReIyIiIiwkRiMhIiI=

x^2 = - 9; solve(%);

NiMvKiQpJSJ4RyIiIyIiIiEiKg==NiQqJiIiJCIiIl4jRiVGJSwkRiMhIiI=

x^2 = - 12; solve(%);

NiMvKiQpJSJ4RyIiIyIiIiEjNw==NiQqJiomIiIjIiIiXiNGJkYmRiYpIiIkI0YmRiVGJiwkKiYsJEYkRiZGJkYoRiYhIiI=

Perceba que i (I no Maple) tem a propriedade de, quando elevado ao quadrado, resultar em -1. I^2;

NiMhIiI=

Claramente esse n\303\243o \303\251 um n\303\272mero real, j\303\241 que nenhum n\303\272mero real tem essa propriedade. Chamamos os n\303\272meros m\303\272ltiplos de i de n\303\272meros imagin\303\241rios. Esse nome n\303\243o \303\251 tido como literal. Gauss, um dos criadores dos n\303\272meros imagin\303\241rios, posteriormente desejou que eles tivessem recebido outro nome para evitar uma compreens\303\243o errada desse nome. No entanto, o nome ficou famoso e agora temos de us\303\241-lo. Apesar de n\303\243o ser evidente, uma coisa \303\251 certa: os n\303\272meros imagin\303\241rios s\303\243o usados em muitas, muitas situa\303\247\303\265es do mundo real. `Exemplos de n\303\272meros imagin\303\241rios :`;I; 3*I; -13*I; sqrt(17)*I;

NiMlRkV4ZW1wbG9zfmRlfm58anl8anZ8aHZtZXJvc35pbWFnaW58anl8anZ8aHZyaW9zfjpHNiNeIyIiIg==NiMqJiIiJCIiIl4jIiIiIiIiNiMsJComIiM4IiIiXiMiIiIiIiIhIiI=NiMqJikiIzwjIiIiIiIjIiIiXiMiIiIiIiI=

Se observarmos as pot\303\252ncias de n\303\272meros imagin\303\241rios, veremos uns padr\303\265es interessantes. Os primeiros quatro t\303\252m quatro valores destintos. No entanto, se observarmos as pot\303\252ncias maiores, veremos que os quatro mesmos valores se repetem. I^k $ k = 1..4 ;

NiZeIyIiIiEiIiwkRiNGJUYk

I^k $ k = 1..20 ;

NjZeIyIiIiEiIiwkRiNGJUYkRiNGJUYmRiRGI0YlRiZGJEYjRiVGJkYkRiNGJUYmRiQ=

Na verdade, i, a qualquer pot\303\252ncia ( de um n\303\272mero natural), \303\251 um desses quatro valores. I^(111); I^(222); I^(333); I^(444);

NiMsJF4jIiIiISIiNiMhIiI=NiNeIyIiIg==NiMiIiI=

E quanto aos expoentes negativos? J\303\241 que 1/i = (i^3)/(i^4) = i^3/1 = i^3 = -i, os expoentes negativos simplesmente s\303\243o os valores negativos do que \303\251 dado pelos expoentes positivos. Isso \303\251 muito incomum, n\303\243o \303\251? 1/I;

NiMsJF4jIiIiISIi

I^(-k) $ k = 1..20 ;

NjYsJF4jIiIiISIiRiZGJEYlRiNGJkYkRiVGI0YmRiRGJUYjRiZGJEYlRiNGJkYkRiU=

Lembre-se de que, quando temos uma equa\303\247\303\243o quadr\303\241tica e usamos a f\303\263rmula quadr\303\241tica para resolv\303\252-la, temos como resultado a raiz quadrada de D = b^2 - 4ac. a*x^2 + b*x + c = 0; solve( %, x);

NiMvLCgqJiUiYUciIiIpJSJ4RyIiIyIiIiIiIiomJSJiRyIiIiUieEciIiIiIiIlImNHIiIiIiIhNiQsJCooIiIjISIiJSJhR0YmLCYlImJHRiYqJCksJiokKUYpRiUiIiJGLyooIiIlRi9GJ0YvJSJjR0YvRiYjRi9GJUYvRi9GL0YvLCQqKEYlRiZGJ0YmLCZGKUYmRipGJkYvRi8=

Isso funciona bem se D for positivo ou zero. 3*x^2 + 7*x - 2 = 0; D = 7^2 - 4*3*(-2); x = solve( %%, x);

NiMvLCgqJiIiJCIiIiklInhHIiIjIiIiIiIiKiYiIigiIiIlInhHIiIiIiIiIiIjISIiIiIhNiMvJSJERyIjdA==NiMvJSJ4RzYkLCYjIiIoIiInISIiKiYiIichIiIpIiN0IyIiIiIiIyIiIiIiIiwmIyIiKCIiJyEiIiomIiInISIiKSIjdCMiIiIiIiMiIiIhIiI=

5*x^2 - 30*x + 45 = 0; D = (-30)^2 - 4*5*45; x = solve( %%, x);

NiMvLCgqJiIiJiIiIiklInhHIiIjIiIiIiIiKiYiI0kiIiIlInhHIiIiISIiIiNYIiIiIiIhNiMvJSJERyIiIQ==NiMvJSJ4RzYkIiIkIiIk

No entanto, se D for negativo, ent\303\243o isso significa que teremos o quadrado de um n\303\272mero negativo com parte da solu\303\247\303\243o. x^2 + 3*x + 5 = 0; D = 3^2 - 4*1*5; x = solve( %%, x);

NiMvLCgqJCklInhHIiIjIiIiIiIiKiYiIiQiIiIlInhHIiIiIiIiIiImIiIiIiIhNiMvJSJERyEjNg==NiMvJSJ4RzYkLCYjIiIkIiIjISIiKiYqJiMiIiIiIiMiIiJeIyIiIiIiIiIiIikiIzYjIiIiIiIjIiIiIiIiLCYjIiIkIiIjISIiKiYsJComIyIiIiIiIyIiIl4jIiIiIiIiIiIiIiIiKSIjNiMiIiIiIiMiIiIhIiI=

N\303\272meros que s\303\243o parte reais e parte imagin\303\241rios s\303\243o chamados de n\303\272meros complexos. N\303\243o s\303\243o nem totalmente reais, nem totalmente imagin\303\241rios - eles s\303\243o uma combina\303\247\303\243o dos dois. Vejamos alguns exemplos. 3 + 4*I;

NiMsJiIiJCIiIiomIiIlIiIiXiMiIiIiIiIiIiI=

-13 + I;

NiMsJiIjOCEiIl4jIiIiIiIi

41 - 10*sqrt(2)*I;

NiMsJiIjVCIiIiomLCQqJiIjNSIiIl4jIiIiIiIiIiIiIiIiKSIiIyMiIiIiIiMiIiIhIiI=

1 + Pi*I;

NiMsJiIiIiIiIiomJSNQaUciIiJeIyIiIiIiIiIiIg==

Podemos tratar n\303\272meros complexos como tratamos n\303\272meros reais e somar, subtrair, multiplicar ou dividi-los - com algumas precau\303\247\303\265es. Subtra\303\247\303\243o e Adi\303\247\303\243o de N\303\272meros Complexos Perceba que as partes reais se combinam e as partes imagin\303\241rias se combinam, mas as duas n\303\243o se misturam. \303\211 bem parecido com a combina\303\247\303\243o de termos semelhantes. Perceba que \303\251 possivel somar dois n\303\272meros complexos e ter um resultado totalmente real ou totalmente imagin\303\241rio. (10 + 20*I) + (3 + 7*I);

NiMsJiIjOCIiIiomIiNGIiIiXiMiIiIiIiIiIiI=

(4 - 4*I) + (-11 + 6*I);

NiMsJiIiKCEiIiomIiIjIiIiXiMiIiIiIiIiIiI=

(10 + 20*I) + (-10 - 8*I);

NiMqJiIjNyIiIl4jIiIiIiIi

(-9 + 17*I) + (9+ 17*I);

NiMqJiIjTSIiIl4jIiIiIiIi

(30 + 40*I) - (35 + 35*I);

NiMsJiIiJiEiIiomIiImIiIiXiMiIiIiIiIiIiI=

(-10 + 31*I) - (10 - 28*I);

NiMsJiIjPyEiIiomIiNmIiIiXiMiIiIiIiIiIiI=

Multiplica\303\247\303\243o e Pot\303\252ncias de N\303\272meros Complexos N\303\272meros complexos s\303\243o bem parecidos com bin\303\264mios. Quando multiplicamos dois bin\303\264mios, podemos usar o m\303\251todo FOIL, assim como podemos com n\303\272meros complexos. S\303\263 que I^2 = -1. e por isso temos de simplificar um pouco mais ao final. (4 + 5*x)*(6 + 7*x): % = expand(%);

NiMvKiYsJiIiJSIiIiomIiImIiIiJSJ4RyIiIiIiIiIiIiwmIiInIiIiKiYiIigiIiIlInhHIiIiIiIiIiIiLCgiI0MiIiIqJiIjZSIiIiUieEciIiIiIiIqJiIjTiIiIiklInhHIiIjIiIiIiIi

(4 + 5*I)*(6 + 7*I);

NiMsJiIjNiEiIiomIiNlIiIiXiMiIiIiIiIiIiI=

Perceba que 24 - 35*I^2 = 24 - 35*(-1) = 4 + 35 = 58. (13 - 4*I)*(12 + 11*I);

NiMsJiIkKyMiIiIqJiIjJioiIiJeIyIiIiIiIiIiIg==

(7 + 2*I)*(8 + 3*I);

NiMsJiIjXSIiIiomIiNQIiIiXiMiIiIiIiIiIiI=

(1 + 4*I)*(1 - 21*I);

NiMsJiIjJikiIiIqJiIjPCIiIl4jIiIiIiIiISIi

(100 + 7*I)^2;

NiMsJiIlXioqIiIiKiYiJSs5IiIiXiMiIiIiIiIiIiI=

(100 - 7*I)^2;

NiMsJiIlXioqIiIiKiYiJSs5IiIiXiMiIiIiIiIhIiI=

Princ\303\255pio Conjugado Perceba que se multiplicarmos um n\303\272mero complexo pelo seu conjugado - o mesmo n\303\272mero com o sinal inverso na parte imagin\303\241ria - temos um n\303\272mero totalmente real. Isso funciona da mesma maneira como os n\303\272meros conjugados de express\303\265es com radicais ao eliminarmos os radicais das express\303\265es - neste caso I = sqrt(-1) - de uma express\303\243o. (3 + 4*I)*(3 - 4*I);

NiMiI0Q=

(17 + 29*I)*(17 - 29*I);

NiMiJUk2

(sqrt(5) + 2*I)*(sqrt(5) - 2*I): % = expand(%);

NiMvKiYsJiokKSIiJiMiIiIiIiMiIiIiIiIqJiIiIyIiIl4jIiIiIiIiIiIiIiIiLCYqJCkiIiYjIiIiIiIjIiIiIiIiKiYiIiMiIiJeIyIiIiIiIiEiIiIiIiIiKg==

(1 + 10*I)*(1 - 10*I);

NiMiJCwi

(20 + 3*I)*(20 - 3*I);

NiMiJDQl

(sqrt(3) + sqrt(5)*I)*(sqrt(3) - sqrt(5)*I): % = expand(%);

NiMvKiYsJiokKSIiJCMiIiIiIiMiIiIiIiIqJikiIiYjIiIiIiIjIiIiXiMiIiIiIiIiIiIiIiIsJiokKSIiJCMiIiIiIiMiIiIiIiIqJiwkXiMiIiIiIiIiIiIpIiImIyIiIiIiIyIiIiEiIiIiIiIiKQ==

Perceba que o Maple pode encontrar o cunjugado automaticamente. z:= 3 + 20*I; zcon := conjugate(z);

NiM+JSJ6RywmIiIkIiIiKiYiIz8iIiJeIyIiIiIiIiIiIg==NiM+JSV6Y29uRywmIiIkIiIiKiYiIz8iIiJeIyIiIiIiIiEiIg==

z*zcon;

NiMiJDQl

z:= 50 - 13*I; zcon := conjugate(z);z*zcon;

NiM+JSJ6RywmIiNdIiIiKiYiIzgiIiJeIyIiIiIiIiEiIg==NiM+JSV6Y29uRywmIiNdIiIiKiYiIzgiIiJeIyIiIiIiIiIiIg==NiMiJXBF

Divis\303\243o de N\303\272meros Complexos O princ\303\255pio de n\303\272meros conjugados pode ser usado para "dividir" n\303\272meros complexos. Na verdade, vamos realizar uma opera\303\247\303\243o mais racionalizando o denominador do que dividindo. Mas o resultado ser\303\241 o quociente mesmo assim. O Maple faz isso automaticamente. (3 + 10*I)/(7 - 4*I);

NiMsJiMiIz4iI2whIiIqJiMiIyMpIiNsIiIiXiMiIiIiIiIiIiI=

Tamb\303\251m podemos faz\303\252-lo manualmente - multiplicando a fra\303\247\303\243o em cima e em baixo pelo conjugado complexo do denominador. top := 3 + 10*I; bot := 7 - 4*I;

NiM+JSR0b3BHLCYiIiQiIiIqJiIjNSIiIl4jIiIiIiIiIiIiNiM+JSRib3RHLCYiIigiIiIqJiIiJSIiIl4jIiIiIiIiISIi

botcon := conjugate( bot );

NiM+JSdib3Rjb25HLCYiIigiIiIqJiIiJSIiIl4jIiIiIiIiIiIi

top*botcon; bot*botcon; %%/%;

NiMsJiIjPiEiIiomIiMjKSIiIl4jIiIiIiIiIiIiNiMiI2w=NiMsJiMiIz4iI2whIiIqJiMiIyMpIiNsIiIiXiMiIiIiIiIiIiI=

Vejamos outro exemplo. (2 + 1*I)/(7 - 4*I);

NiMsJiMiIiMiIzgiIiIqJiMiIiQiIzgiIiJeIyIiIiIiIiIiIg==

top := 2 + 1*I; bot := 7 - 4*I;

NiM+JSR0b3BHLCYiIiMiIiJeIyIiIiIiIg==NiM+JSRib3RHLCYiIigiIiIqJiIiJSIiIl4jIiIiIiIiISIi

botcon := conjugate( bot ); top*botcon: bot*botcon: `top `= %%, `bottom `= %; %%%/%%;

NiM+JSdib3Rjb25HLCYiIigiIiIqJiIiJSIiIl4jIiIiIiIiIiIiNiQvJSV0b3B+RywmIiM1IiIiKiYiIzpGJ14jRidGJ0YnLyUoYm90dG9tfkciI2w=NiMsJiMiIiMiIzgiIiIqJiMiIiQiIzgiIiJeIyIiIiIiIiIiIg==

H\303\241 uma maneira gr\303\241fica de representar n\303\272meros complexos. A parte real est\303\241 representada no eixo x, e a parte imagin\303\241ria no eixo y. Isso define um \303\272nico ponto em um plano. z := 3 + 4*I; ComplexPlot( %);

NiM+JSJ6RywmIiIkIiIiKiYiIiUiIiJeIyIiIiIiIiIiIg==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

z := -8 + 3*I; ComplexPlot( %);

NiM+JSJ6RywmIiIpISIiKiYiIiQiIiJeIyIiIiIiIiIiIg==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Ig==

z := 9 - 4*I; ComplexPlot( %);

NiM+JSJ6RywmIiIqIiIiKiYiIiUiIiJeIyIiIiIiIiEiIg==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

Perceba que esses n\303\272meros conjugados aparecem como reflex\303\265es cruzando o eixo x. 3 + 4*I, 3 - 4*I; plots[display](ComplexPlot( 3 + 4*I),ComplexPlot( 3 - 4*I));

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-8 + 2*I, -8 - 2*I; plots[display](ComplexPlot( -8+ 2*I),ComplexPlot(-8 - 2*I));

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2*I, -2*I; plots[display](ComplexPlot(2*I),ComplexPlot(-2*I));

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Outra coisa interessante que podemos fazer \303\251 plotar dois n\303\272meros complexos e o seu produto. z1 := 1 + 4*I; z2 := 4 - 1*I; plots[display]( ComplexPlot( z1), ComplexPlot( z2), ComplexPlot( z1*z2) );

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z1 := 2*(1 + I)/evalf(sqrt(2)); z2 := 3*(1 + evalf(sqrt(3))*I)/2; plots[display]( ComplexPlot( z1), ComplexPlot( z2), ComplexPlot( z1*z2) );

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z1 := 2 + 2*I; z2 := 1 - 1*I; plots[display]( ComplexPlot( z1), ComplexPlot( z2), ComplexPlot( z1*z2) );

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