# Nakafa Framework: LLM

URL: /id/subject/high-school/11/mathematics/complex-number/multiplication-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/complex-number/multiplication-complex-numbers/id.mdx

Output docs content for large language models.

---

import { LineEquation } from "@repo/design-system/components/contents/line-equation";
import { getColor } from "@repo/design-system/lib/color";

export const metadata = {
  title: "Perkalian Bilangan Kompleks",
  description: "Kalikan bilangan kompleks dengan sifat distributif dan rumus (x₁x₂-y₁y₂)+i(x₁y₂+x₂y₁). Kuasai ekspansi binomial dengan contoh i²=-1.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/01/2025",
  subject: "Bilangan Kompleks",
};

## 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 <InlineMath math="z_1 = x_1 + iy_1" /> dengan <InlineMath math="z_2 = x_2 + iy_2" />.

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2)" />
  <BlockMath math="= x_1(x_2 + iy_2) + iy_1(x_2 + iy_2)" />
  <BlockMath math="= (x_1x_2 + ix_1y_2) + (ix_2y_1 + i^2y_1y_2)" />
</div>

Ingat bahwa <InlineMath math="i^2 = -1" />, jadi kita bisa substitusi:

<div className="flex flex-col gap-4">
  <BlockMath math="= (x_1x_2 + ix_1y_2) + (ix_2y_1 + (-1)y_1y_2)" />
  <BlockMath math="= x_1x_2 + ix_1y_2 + ix_2y_1 - y_1y_2" />
</div>

Sekarang, kita kelompokkan bagian real dan bagian imajiner:

<div className="flex flex-col gap-4">
  <BlockMath math="= (x_1x_2 - y_1y_2) + (ix_1y_2 + ix_2y_1)" />
  <BlockMath math="= (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />
</div>

Jadi, rumus umum untuk perkalian bilangan kompleks adalah:

<BlockMath math="z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />

## Contoh Perhitungan

Misalkan <InlineMath math="z_1 = 2+i" /> dan <InlineMath math="z_2 = 1-2i" />. Tentukan <InlineMath math="z_1 \times z_2" />.

**Penyelesaian:**

Menggunakan sifat distributif:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (2+i)(1-2i)" />
  <BlockMath math="= 2(1-2i) + i(1-2i)" />
  <BlockMath math="= (2 - 4i) + (i - 2i^2)" />
  <BlockMath math="= (2 - 4i) + (i - 2(-1)) \quad \text{(karena } i^2 = -1)" />
  <BlockMath math="= (2 - 4i) + (i + 2)" />
  <BlockMath math="= (2+2) + (-4i + i)" />
  <BlockMath math="= 4 - 3i" />
</div>

Atau menggunakan rumus umum dengan <InlineMath math="x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2" />:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />
  <BlockMath math="= (2)(1) - (1)(-2) + i((2)(-2) + (1)(1))" />
  <BlockMath math="= (2 - (-2)) + i(-4 + 1)" />
  <BlockMath math="= (2+2) + i(-3)" />
  <BlockMath math="= 4 - 3i" />
</div>

Hasilnya sama!

<LineEquation
  title="Visualisasi Perkalian Bilangan Kompleks"
  description={
    <>
      Visualisasi dari <InlineMath math="z_1 = 2+i" />,{" "}
      <InlineMath math="z_2 = 1-2i" />, dan hasil perkaliannya{" "}
      <InlineMath math="z_1 \times z_2 = 4-3i" />.
    </>
  }
  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 <InlineMath math="z_1 = 1+i" /> dan <InlineMath math="z_2 = \frac{1}{2} - 2i" />. Tentukan <InlineMath math="z_1 \times z_2" />.

### Kunci Jawaban

Menggunakan sifat distributif:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (1+i)(\frac{1}{2} - 2i)" />
  <BlockMath math="= 1(\frac{1}{2} - 2i) + i(\frac{1}{2} - 2i)" />
  <BlockMath math="= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2i^2)" />
  <BlockMath math="= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2(-1))" />
  <BlockMath math="= (\frac{1}{2} - 2i) + (\frac{1}{2}i + 2)" />
  <BlockMath math="= (\frac{1}{2} + 2) + (-2i + \frac{1}{2}i)" />
  <BlockMath math="= (\frac{1}{2} + \frac{4}{2}) + (-\frac{4}{2}i + \frac{1}{2}i)" />
  <BlockMath math="= \frac{5}{2} - \frac{3}{2}i" />
</div>

Menggunakan rumus umum dengan <InlineMath math="x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2" />:

<div className="flex flex-col gap-4">
  <BlockMath math="z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)" />
  <BlockMath math="= ((1)(\frac{1}{2}) - (1)(-2)) + i((1)(-2) + (\frac{1}{2})(1))" />
  <BlockMath math="= (\frac{1}{2} - (-2)) + i(-2 + \frac{1}{2})" />
  <BlockMath math="= (\frac{1}{2} + 2) + i(-\frac{4}{2} + \frac{1}{2})" />
  <BlockMath math="= (\frac{1}{2} + \frac{4}{2}) + i(-\frac{3}{2})" />
  <BlockMath math="= \frac{5}{2} - \frac{3}{2}i" />
</div>