Números complexos

Os números complexos são multiplicados com base na propriedade distributiva, sempre lembrando que um numeral complexo é formado por uma parte real e uma imaginária. Veja:

4 + 3i → Re(z) = 4 e Im(z) = 3
2 – 5i → Re(z) = 2 e Im(z) = –5
4 + 3i → Re(z) = 4 e Im(z) = 3
12 – 9i →Re(z) = 12 e Im(z) = –9

Multiplicando os complexos

Exemplos

a) (2 + 5i) * (1 – 2i)
2 – 4i + 5i – 5i² (lembrando que i² = – 1)
2 – 4i + 5i – 5 *(–1)
2 – 4i + 5i + 6
8 + i

b) (4 + 3i) * (2 + 6i)
8 + 24i + 6i + 18i² (lembrando que i² = – 1)
8 + 24i + 6i + 18 * (–1)
8 + 24i + 6i – 18
–10 + 30i

c) (6 – 3i) * (–3 + 7i)
–18 + 42i + 9i – 21i² (lembrando que i² = – 1)
–18 + 42i + 9i – 21 * (–1)
–18 + 42i + 9i + 21
3 + 51i

d) (10 + 10i) * (10 – 10i)
100 – 100i + 100i – 100i² (lembrando que i² = – 1)
100 – 100i + 100i – 100 * (–1)
100 + 100 + 0i
200 + 0i
200

e) 4 + 3i + (1 – 2i) * (3 + i)
4 + 3i + (3 + i – 6i – 2i²)
4 + 3i + 3 + i – 6i – 2i² (lembrando que i² = – 1)
4 + 3i + 3 + i – 6i – 2 * (–1)
4 + 3i + 3 + i – 6i + 2
9 – 2i

f) (2 – 3i) * (1 – 5i) – 4i – 8
2 – 10i – 3i + 15i² – 4i – 8 (lembrando que i² = – 1)
2 – 10i – 3i + 15 * (–1) – 4i – 8
2 – 10i – 3i – 15 – 4i – 8
2 – 15 – 8 – 10i – 3i – 4i
–21 – 17i


g) (–12 – 5i) * (5 + 5i) – 4i + 7
–60 – 60i – 25i – 25i² – 4i + 7 (lembrando que i² = – 1)
–60 – 60i – 25i – 25 * (–1) – 4i + 7
–60 – 60i – 25i + 25 – 4i + 7
–60 + 25 + 7 – 60i – 25i – 4i
–60 + 32 – 89i
–28 + 89i


h) (4 + 3i) * (2 – 5i) + (4 – 3i) * (2 + 5i)
8 – 20i + 6i – 15i² + (8 + 20i – 6i + 15i²)
8 – 20i + 6i – 15i² + 8 + 20i – 6i + 15i²
8 + 8 – 20i + 20i + 6i – 6i – 15i² + 15i²
16


i) (3 + 30i) * (2 – 3i) + 4 – 5i
6 – 9i + 60i – 90i² + 4 – 5i (lembrando que i² = – 1)
6 – 9i + 60i – 90 * (–1) + 4 – 5i
6 – 9i + 60i + 90 + 4 – 5i
6 + 90 + 4 – 9i + 60i – 5i
100 + 46i


j) (20 – 4i) * (2 + 5i) + (8 + 9i) * (7 – 10i) + 4 + 6i
40 + 100i – 8 – 20i² + (56 – 80i + 63i – 90i²) + 4 + 6i
40 + 100i – 8 – 20i² + 56 – 80i + 63i – 90i² + 4 + 6i (lembrando que i² = – 1)
40 + 100i – 8 – 20 * (–1) + 56 – 80i + 63i – 90 * (–1) + 4 + 6i
40 + 100i – 8 + 20 + 56 – 80i + 63i + 90 + 4 + 6i
40 – 8 + 20 + 56 + 90 + 4 + 100i – 80i + 63i + 6i
202 + 89i