Prove que:

a) overset(-)(z + w) = overset(-)(z) + overset(-)(w)

b) overset(-)(z w) = overset(-)(z) * overset(-)(w)

c) overset(-)(( (z)/(w) )) = (overset(-)(z))/(overset(-)(w))

d) overset(-)(overset(-)( z)) = z

Passo 1

Para fazermos essas questões vamos tomar

z = a + b i , overset(-)(z) = a - i b

w = c + d i , overset(-)(w) = c - i d

Letra a)

overset(-)((a + b i + c + d i)) = overset(-)(( a + c + i (b + d)) = a + c - i (b + d)

a - i b + c - d i = overset(-)(z) + overset(-)(w)

Letra b)

overset(-)(z w) = overset(-)(( a + b i ) ( c + d i )) = overset(-)(a c + a d i + b c i - b d)

= overset(-)(a c - b d + i ( a d + b c ))

= a c - b d - i (a d + b c) = overset(-)(z) * overset(-)(w)

Passo 2

Letra c)

overset(-)(( (z)/(w) )) = overset(-)(((a + b i)/(c + d i))) = overset(-)(((z overset(-)(w))/(w overset(-)(w)))) = (1)/(overset(-)(w overset(-)(w))) * overset(-)(z overset(-)(w)) = (1)/(overset(-)(w) * overset(̿)(w)) * overset(-)(z) * overset(̿)(w) = (overset(-)(z))/(overset(-)(w))

Passo 3

Letra d)

overset(-)(w) = c - i d

logo

overset(̿)(w) = overset(-)(c - i d) = c + i d = w

Resposta

Demonstração