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URL: https://nakafa.com/id/materi/matematika/bilangan-kompleks/perkalian-bilangan-kompleks
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/complex-number/multiplication-complex-numbers/id.mdx

Kalikan bilangan kompleks dengan sifat distributif dan rumus (x₁x₂-y₁y₂)+i(x₁y₂+x₂y₁). Pelajari ekspansi binomial dengan contoh i²=-1.

---

## Mengalikan Dua Bilangan Kompleks

Mengalikan dua bilangan kompleks mirip seperti mengalikan dua bentuk aljabar binomial. Kita bisa menggunakan sifat distributif perkalian terhadap penjumlahan.

Mari kita lihat bagaimana cara mengalikan $$z_1 = x_1 + iy_1$$ dengan $$z_2 = x_2 + iy_2$$.

Visible text: Mari kita lihat bagaimana cara mengalikan dengan .

Component: MathContainer
Children:

```math
z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2)
```

```math
= x_1(x_2 + iy_2) + iy_1(x_2 + iy_2)
```

```math
= (x_1x_2 + ix_1y_2) + (ix_2y_1 + i^2y_1y_2)
```

Ingat bahwa $$i^2 = -1$$, jadi kita bisa substitusi:

Visible text: Ingat bahwa , jadi kita bisa substitusi:

Component: MathContainer
Children:

```math
= (x_1x_2 + ix_1y_2) + (ix_2y_1 + (-1)y_1y_2)
```

```math
= x_1x_2 + ix_1y_2 + ix_2y_1 - y_1y_2
```

Sekarang, kita kelompokkan bagian real dan bagian imajiner:

Component: MathContainer
Children:

```math
= (x_1x_2 - y_1y_2) + (ix_1y_2 + ix_2y_1)
```

```math
= (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

Jadi, rumus umum untuk perkalian bilangan kompleks adalah:

```math
z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

## Contoh Perhitungan

Misalkan $$z_1 = 2+i$$ dan $$z_2 = 1-2i$$. Tentukan $$z_1 \times z_2$$.

Visible text: Misalkan dan . Tentukan .

**Penyelesaian:**

Menggunakan sifat distributif:

Component: MathContainer
Children:

```math
z_1 \times z_2 = (2+i)(1-2i)
```

```math
= 2(1-2i) + i(1-2i)
```

```math
= (2 - 4i) + (i - 2i^2)
```

```math
= (2 - 4i) + (i - 2(-1)) \quad \text{(karena } i^2 = -1)
```

```math
= (2 - 4i) + (i + 2)
```

```math
= (2+2) + (-4i + i)
```

```math
= 4 - 3i
```

Atau menggunakan rumus umum dengan $$x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2$$:

Visible text: Atau menggunakan rumus umum dengan :

Component: MathContainer
Children:

```math
z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

```math
= (2)(1) - (1)(-2) + i((2)(-2) + (1)(1))
```

```math
= (2 - (-2)) + i(-4 + 1)
```

```math
= (2+2) + i(-3)
```

```math
= 4 - 3i
```

Hasilnya sama!

Component: LineEquation
Props:
- title: Visualisasi Perkalian Bilangan Kompleks
- description: Visualisasi dari $$z_1 = 2+i$$,{" "}
$$z_2 = 1-2i$$, dan hasil perkaliannya{" "}
$$z_1 \times z_2 = 4-3i$$.
  Visible text: Visualisasi dari ,{" "}
, dan hasil perkaliannya{" "}
.
- cameraPosition: [0, 0, 15]
- showZAxis: false
- data: [
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2, y: 1, z: 0 },
],
color: getColor("SKY"),
labels: [{ text: "z₁ = 2+i", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1, y: -2, z: 0 },
],
color: getColor("EMERALD"),
labels: [{ text: "z₂ = 1-2i", at: 1, offset: [0.5, -0.5, 0] }],
cone: { position: "end" },
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4, y: -3, z: 0 },
],
color: getColor("ROSE"),
labels: [{ text: "z₁ × z₂ = 4-3i", at: 1, offset: [0.5, -0.5, 0] }],
cone: { position: "end" },
},
]

## Latihan

Misalkan $$z_1 = 1+i$$ dan $$z_2 = \frac{1}{2} - 2i$$. Tentukan $$z_1 \times z_2$$.

Visible text: Misalkan dan . Tentukan .

### Kunci Jawaban

Menggunakan sifat distributif:

Component: MathContainer
Children:

```math
z_1 \times z_2 = (1+i)(\frac{1}{2} - 2i)
```

```math
= 1(\frac{1}{2} - 2i) + i(\frac{1}{2} - 2i)
```

```math
= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2i^2)
```

```math
= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2(-1))
```

```math
= (\frac{1}{2} - 2i) + (\frac{1}{2}i + 2)
```

```math
= (\frac{1}{2} + 2) + (-2i + \frac{1}{2}i)
```

```math
= (\frac{1}{2} + \frac{4}{2}) + (-\frac{4}{2}i + \frac{1}{2}i)
```

```math
= \frac{5}{2} - \frac{3}{2}i
```

Menggunakan rumus umum dengan $$x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2$$:

Visible text: Menggunakan rumus umum dengan :

Component: MathContainer
Children:

```math
z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
```

```math
= ((1)(\frac{1}{2}) - (1)(-2)) + i((1)(-2) + (\frac{1}{2})(1))
```

```math
= (\frac{1}{2} - (-2)) + i(-2 + \frac{1}{2})
```

```math
= (\frac{1}{2} + 2) + i(-\frac{4}{2} + \frac{1}{2})
```

```math
= (\frac{1}{2} + \frac{4}{2}) + i(-\frac{3}{2})
```

```math
= \frac{5}{2} - \frac{3}{2}i
```