Operações com Números Complexos
No estudo dos números reais vimos como realizar as operações de adição, subtração, multiplicação e divisão.
Agora vamos tratar cada uma destas operações trabalhando com os números complexos na forma algébrica.
Adição de Números Complexos
No artigo sobre polinômios vimos como realizar a redução de termos semelhantes.
Tratando os números complexos como binômios, podemos realizar a sua soma reduzindo os termos semelhantes como no exemplo abaixo:
![](MEx.ashx?KFxxdWFkOFxxdWFkK1xxdWFkNGlccXVhZClccXVhZCtccXVhZChccXVhZDJccXVhZCtccXVhZDVpXHF1YWQpXHF1YWQ9XHF1YWQ4XHF1YWQrXHF1YWQyXHF1YWQrXHF1YWQ0aVxxdWFkK1xxdWFkNWlccXVhZD1ccXVhZDEwXHF1YWQrXHF1YWQ5aQ==)
Como você pode perceber, isto é equivalente a somarmos separadamente as suas partes reais e imaginárias.
Exemplos da Adição de Números Complexos
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkN1xxdWFkK1xxdWFkOGlccXVhZClccXVhZCtccXVhZChccXVhZDJccXVhZCtccXVhZDZpXHF1YWQpXHF1YWQ9XHF1YWQoN1xxdWFkK1xxdWFkMilccXVhZCtccXVhZCg4aVxxdWFkK1xxdWFkNmkpXHF1YWQ9XHF1YWQ5XHF1YWQrXHF1YWQxNGk=)
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkM1xxdWFkK1xxdWFkMmlccXVhZClccXVhZCtccXVhZChccXVhZDFccXVhZCtccXVhZDRpXHF1YWQpXHF1YWQ9XHF1YWQoM1xxdWFkK1xxdWFkMSlccXVhZCtccXVhZCgyaVxxdWFkK1xxdWFkNGkpXHF1YWQ9XHF1YWQ0XHF1YWQrXHF1YWQ2aQ==)
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkMTZccXVhZCtccXVhZDI1aVxxdWFkKVxxdWFkK1xxdWFkKFxxdWFkMzRccXVhZCtccXVhZDE1aVxxdWFkKVxxdWFkPVxxdWFkKDE2XHF1YWQrXHF1YWQzNClccXVhZCtccXVhZCgyNWlccXVhZCtccXVhZDE1aSlccXVhZD1ccXVhZDUwXHF1YWQrXHF1YWQ0MGk=)
Subtração de Números Complexos
A subtração é realizada tal qual a adição, através da redução dos termos semelhantes, ou ainda subtraindo separadamente as partes reais e as partes imaginárias.
Exemplos da Subtração de Números Complexos
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkN1xxdWFkK1xxdWFkOGlccXVhZClccXVhZC1ccXVhZChccXVhZDJccXVhZCtccXVhZDZpXHF1YWQpXHF1YWQ9XHF1YWQoN1xxdWFkLVxxdWFkMilccXVhZCtccXVhZCg4aVxxdWFkLVxxdWFkNmkpXHF1YWQ9XHF1YWQ1XHF1YWQrXHF1YWQyaQ==)
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkM1xxdWFkK1xxdWFkMmlccXVhZClccXVhZC1ccXVhZChccXVhZDFccXVhZCtccXVhZDRpXHF1YWQpXHF1YWQ9XHF1YWQoM1xxdWFkLVxxdWFkMSlccXVhZCtccXVhZCgyaVxxdWFkLVxxdWFkNGkpXHF1YWQ9XHF1YWQyXHF1YWQtXHF1YWQyaQ==)
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkMTZccXVhZCtccXVhZDI1aVxxdWFkKVxxdWFkLVxxdWFkKFxxdWFkMzRccXVhZCtccXVhZDE1aVxxdWFkKVxxdWFkPVxxdWFkKDE2XHF1YWQtXHF1YWQzNClccXVhZCtccXVhZCgyNWlccXVhZC1ccXVhZDE1aSlccXVhZD1ccXVhZC0xOFxxdWFkK1xxdWFkMTBp)
Multiplicação de Números Complexos
Realizamos a multiplicação de números complexos tratando-os como binômios e os multiplicando como tal, ou seja, multiplicando cada termo do primeiro binômio por cada termo do segundo:
![](MEx.ashx?KFxxdWFkM1xxdWFkK1xxdWFkMmlccXVhZClccXVhZFxjZG90XHF1YWQoXHF1YWQxXHF1YWQrXHF1YWQ1aVxxdWFkKVxxdWFkPVxxdWFkKDNccXVhZFxjZG90XHF1YWQxKVxxdWFkK1xxdWFkKDNccXVhZFxjZG90XHF1YWQ1aSlccXVhZCtccXVhZCgyaVxxdWFkXGNkb3RccXVhZDEpXHF1YWQrXHF1YWQoMmlccXVhZFxjZG90XHF1YWQ1aSlccXVhZD1ccXVhZDNccXVhZCtccXVhZDE1aVxxdWFkK1xxdWFkMmlccXVhZCtccXVhZDEwaV4yXHF1YWQ9)
![](MEx.ashx?PVxxdWFkM1xxdWFkK1xxdWFkMTdpXHF1YWQrXHF1YWQxMGleMg==)
Note que o último termo é 10i2 e visto que
, logo i2 = -1, o que nos permite continuar os cálculos substituindo i2 por -1:
![](MEx.ashx?M1xxdWFkK1xxdWFkMTdpXHF1YWQrXHF1YWQxMGleMlxxdWFkPVxxdWFkM1xxdWFkK1xxdWFkMTdpXHF1YWQrXHF1YWQxMFxxdWFkXGNkb3RccXVhZC0xXHF1YWQ9XHF1YWQtN1xxdWFkK1xxdWFkMTdp)
Portanto:
![](MEx.ashx?KFxxdWFkM1xxdWFkK1xxdWFkMmlccXVhZClccXVhZFxjZG90XHF1YWQoXHF1YWQxXHF1YWQrXHF1YWQ1aVxxdWFkKVxxdWFkPVxxdWFkLTdccXVhZCtccXVhZDE3aQ==)
Exemplos da Multiplicação de Números Complexos
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkNFxxdWFkLVxxdWFkIGlccXVhZClccXVhZFxjZG90XHF1YWQoXHF1YWQtMlxxdWFkK1xxdWFkNiBpXHF1YWQpXHF1YWQ9XHF1YWQoNFxxdWFkXGNkb3RccXVhZC0yKVxxdWFkK1xxdWFkKDRccXVhZFxjZG90XHF1YWQ2IGkpXHF1YWQrXHF1YWQoLWlccXVhZFxjZG90XHF1YWQtMilccXVhZCtccXVhZCgtIGlccXVhZFxjZG90XHF1YWQ2IGkpXHF1YWQ9)
![](MEx.ashx?PVxxdWFkLThccXVhZCtccXVhZDI0IGlccXVhZCtccXVhZDJpXHF1YWQtXHF1YWQ2IGleMlxxdWFkPVxxdWFkLThccXVhZCtccXVhZDI2aVxxdWFkLVxxdWFkNlxxdWFkXGNkb3RccXVhZC0xXHF1YWQ9XHF1YWQtMlxxdWFkK1xxdWFkMjZp)
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkMTRccXVhZCtccXVhZDI1IGlccXVhZClccXVhZFxjZG90XHF1YWQoXHF1YWQzMVxxdWFkLVxxdWFkMTAgaVxxdWFkKVxxdWFkPVxxdWFkKDE0XHF1YWRcY2RvdFxxdWFkMzEpXHF1YWQrXHF1YWQoMTRccXVhZFxjZG90XHF1YWQtMTAgaSlccXVhZCtccXVhZCgyNSBpXHF1YWRcY2RvdFxxdWFkMzEpXHF1YWQrXHF1YWQoMjUgaVxxdWFkXGNkb3RccXVhZC0xMCBpKVxxdWFkPVxxdWFk)
![](MEx.ashx?PVxxdWFkNDM0XHF1YWQtXHF1YWQxNDBpXHF1YWQrXHF1YWQ3NzVpXHF1YWQtXHF1YWQyNTBpXjJccXVhZD1ccXVhZDQzNFxxdWFkK1xxdWFkNjM1aVxxdWFkLVxxdWFkMjUwXHF1YWRcY2RvdFxxdWFkLTFccXVhZD1ccXVhZDY4NFxxdWFkK1xxdWFkNjM1aQ==)
![](images/trianglered.gif)
![](MEx.ashx?KFxxdWFkNVxxdWFkK1xxdWFkOGlccXVhZClccXVhZFxjZG90XHF1YWQoXHF1YWQ1XHF1YWQtXHF1YWQ4aVxxdWFkKVxxdWFkPVxxdWFkKDVccXVhZFxjZG90XHF1YWQ1KVxxdWFkK1xxdWFkKDVccXVhZFxjZG90XHF1YWQtOGkpXHF1YWQrXHF1YWQoOGlccXVhZFxjZG90XHF1YWQ1KVxxdWFkK1xxdWFkKDhpXHF1YWRcY2RvdFxxdWFkLThpKVxxdWFkPQ==)
![](MEx.ashx?PVxxdWFkMjVccXVhZC1ccXVhZDQwaVxxdWFkK1xxdWFkNDBpXHF1YWQtXHF1YWQ2NGleMlxxdWFkPVxxdWFkMjVccXVhZC1ccXVhZDY0XHF1YWRcY2RvdFxxdWFkLTFccXVhZD1ccXVhZDg5)
Divisão de Números Complexos
A divisão de números complexos é realizada multiplicando o dividendo e o divisor pelo conjugado do divisor.
Observe no último exemplo de multiplicação acima que ao multiplicarmos o número imaginário 5 + 8i pelo seu conjugado 5 - 8i obtivemos como resultado o número real 89.
A multiplicação de um número imaginário pelo seu conjugado sempre resulta em um número real e isto pode ser utilizado para realizar a divisão de números complexos.
Agora vejamos este exemplo de divisão:
![](MEx.ashx?KDVccXVhZCtccXVhZDNpKVxxdWFkXGRpdlxxdWFkKDJccXVhZC1ccXVhZDdpKQ==)
Para começar vamos multiplicar o divisor e o dividendo pelo conjugado do divisor como explicado acima:
![](MEx.ashx?KDVccXVhZCtccXVhZDNpKVxxdWFkXGRpdlxxdWFkKDJccXVhZC1ccXVhZDdpKVxxdWFkPVxxdWFkXGZyYWN7KDVccXVhZCtccXVhZDNpKVxxdWFkXGNkb3RccXVhZCgyXHF1YWQrXHF1YWQ3aSl9eygyXHF1YWQtXHF1YWQ3aSlccXVhZFxjZG90XHF1YWQoMlxxdWFkK1xxdWFkN2kpfQ==)
Para realizar o produto no denominador vamos recorrer aos produtos notáveis, mais especificamente ao produto da soma pela diferença de dois termos, onde temos que:
![](MEx.ashx?KGFccXVhZCtccXVhZCBiKShhXHF1YWQtXHF1YWQgYilccXVhZD1ccXVhZCBhXjJccXVhZC1ccXVhZCBiXjI=)
Continuando o processo da divisão temos:
![](MEx.ashx?XGZyYWN7KDVccXVhZCtccXVhZDNpKVxxdWFkXGNkb3RccXVhZCgyXHF1YWQrXHF1YWQ3aSl9eygyXHF1YWQtXHF1YWQ3aSlccXVhZFxjZG90XHF1YWQoMlxxdWFkK1xxdWFkN2kpfVxxdWFkPVxxdWFkXGZyYWN7KDVccXVhZFxjZG90XHF1YWQyKVxxdWFkK1xxdWFkKDVccXVhZFxjZG90XHF1YWQ3aSlccXVhZCtccXVhZCgzaVxxdWFkXGNkb3RccXVhZDIpXHF1YWQrXHF1YWQoM2lccXVhZFxjZG90XHF1YWQ3aSl9ezJeMlxxdWFkLVxxdWFkKDdpKV4yfVxxdWFkPVxxdWFkXGZyYWN7MTBccXVhZCtccXVhZDM1aVxxdWFkK1xxdWFkNmlccXVhZCtccXVhZDIxaV4yfXs0XHF1YWQtXHF1YWQ0OWleMn09)
![](MEx.ashx?PVxxdWFkXGZyYWN7MTBccXVhZCtccXVhZDQxaVxxdWFkK1xxdWFkMjFccXVhZFxjZG90XHF1YWQtMX17NFxxdWFkLVxxdWFkNDlccXVhZFxjZG90XHF1YWQtMX1ccXVhZD1ccXVhZFxmcmFjey0xMVxxdWFkK1xxdWFkNDFpfXs1M31ccXVhZD1ccXVhZC1cZnJhY3sxMX17NTN9XHF1YWQrXHF1YWRcZnJhY3s0MX17NTN9aQ==)
Note que inicialmente tínhamos o divisor imaginário 2 - 7i e no final temos o divisor real 53. É por isto que utilizamos o conjugado como expediente para realizar a divisão, assim conseguimos transformar um divisor imaginário em um divisor real, o que facilita muito as coisas, como pudemos ver na passagem do penúltimo para o último passo.
Exemplos da Divisão de Números Complexos
![](images/trianglered.gif)
![](MEx.ashx?KDEyXHF1YWQrXHF1YWQyM2kpXHF1YWRcZGl2XHF1YWQoN1xxdWFkLVxxdWFkMThpKVxxdWFkPVxxdWFkXGZyYWN7KDEyXHF1YWQrXHF1YWQyM2kpXHF1YWRcY2RvdFxxdWFkKDdccXVhZCtccXVhZDE4aSl9eyg3XHF1YWQtXHF1YWQxOGkpXHF1YWRcY2RvdFxxdWFkKDdccXVhZCtccXVhZDE4aSl9XHF1YWQ9XHF1YWRcZnJhY3soMTJccXVhZFxjZG90XHF1YWQ3KVxxdWFkK1xxdWFkKDEyXHF1YWRcY2RvdFxxdWFkMThpKVxxdWFkK1xxdWFkKDIzaVxxdWFkXGNkb3RccXVhZDcpXHF1YWQrXHF1YWQoMjNpXHF1YWRcY2RvdFxxdWFkMThpKX17N14yXHF1YWQtXHF1YWQoMThpKV4yfVxxdWFkPQ==)
![](MEx.ashx?PVxxdWFkXGZyYWN7ODRccXVhZCtccXVhZDIxNmlccXVhZCtccXVhZDE2MWlccXVhZCtccXVhZDQxNGleMn17NDlccXVhZC1ccXVhZDMyNGleMn1ccXVhZD1ccXVhZFxmcmFjezg0XHF1YWQrXHF1YWQzNzdpXHF1YWQrXHF1YWQ0MTRccXVhZFxjZG90XHF1YWQtMX17NDlccXVhZC1ccXVhZDMyNFxxdWFkXGNkb3RccXVhZC0xfVxxdWFkPVxxdWFkXGZyYWN7LTMyM1xxdWFkK1xxdWFkMzc3aX17MzczfVxxdWFkPVxxdWFkLVxmcmFjezMyM317MzczfVxxdWFkK1xxdWFkXGZyYWN7Mzc3fXszNzN9aQ==)
![](images/trianglered.gif)
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![](images/trianglered.gif)
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![](MEx.ashx?PVxxdWFkXGZyYWN7MzIzXHF1YWQtXHF1YWQxMDc4aVxxdWFkK1xxdWFkODk5XHF1YWRcY2RvdFxxdWFkLTF9ezI4OVxxdWFkLVxxdWFkODQxXHF1YWRcY2RvdFxxdWFkLTF9XHF1YWQ9XHF1YWRcZnJhY3stNTc2XHF1YWQtXHF1YWQxMDc4aX17MTEzMH1ccXVhZD1ccXVhZC1cZnJhY3s1NzZ9ezExMzB9XHF1YWQtXHF1YWRcZnJhY3sxMDc4fXsxMTMwfWlccXVhZD1ccXVhZC1cZnJhY3syODh9ezU2NX1ccXVhZC1ccXVhZFxmcmFjezUzOX17NTY1fWk=)
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