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URL: https://nakafa.com/en/subjects/mathematics/complex-number/addition-complex-numbers
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/complex-number/addition-complex-numbers/en.mdx

Learn how to add complex numbers one step at a time with geometric visualization. Learn real and imaginary parts addition using parallelogram rule and examples.

---

## Addition of Two Complex Numbers

How do you add two complex numbers?

Suppose we have two complex numbers:

Component: MathContainer
Children:

```math
z_1 = x_1 + iy_1
```

```math
z_2 = x_2 + iy_2
```

To add them ($$z_1 + z_2$$), simply add the real parts together and the imaginary parts together.

Visible text: To add them (), simply add the real parts together and the imaginary parts together.

```math
z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)
```

### Addition Example

Let $$z_1 = 2 + 3i$$ and $$z_2 = 1 - i$$.

Visible text: Let and .

- The real part of $$z_1$$ is $$2$$, the real part of $$z_2$$ is $$1$$.
- The imaginary part of $$z_1$$ is $$3$$, the imaginary part of $$z_2$$ is $$-1$$.

Visible text: - The real part of is , the real part of is .
- The imaginary part of is , the imaginary part of is .

Then their sum is:

```math
z_1 + z_2 = (2 + 1) + i(3 + (-1)) = 3 + i(2) = 3 + 2i
```

### Visualization of Addition

Using the parallelogram rule, the addition of complex numbers can be viewed geometrically on the complex plane. If we represent $$z_1$$ and $$z_2$$ as vectors (arrows) from the origin $$(0, 0)$$, then their sum, $$z_1 + z_2$$, is the diagonal vector of the parallelogram formed by $$z_1$$ and $$z_2$$.

Visible text: Using the parallelogram rule, the addition of complex numbers can be viewed geometrically on the complex plane. If we represent and as vectors (arrows) from the origin , then their sum, , is the diagonal vector of the parallelogram formed by and .

Component: LineEquation
Props:
- title: Geometric Addition of Complex Numbers
- description: Visualization of the sum $$z_1 = 2+3i$$ and{" "}
$$z_2 = 1-i$$ using the parallelogram rule.
  Visible text: Visualization of the sum and{" "}
 using the parallelogram rule.
- cameraPosition: [0, 0, 12]
- showZAxis: false
- data: [
// Vector z1
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 2, y: 3, z: 0 },
],
color: getColor("SKY"),
labels: [{ text: "z₁ = 2 + 3i", at: 1, offset: [0.5, 0.5, 0] }],
cone: { position: "end" },
},
// Vector z2
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 1, y: -1, z: 0 },
],
color: getColor("EMERALD"),
labels: [{ text: "z₂ = 1 - i", at: 1, offset: [0.5, -0.5, 0] }],
cone: { position: "end" },
},
// Resultant vector z1 + z2
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 2, z: 0 },
],
color: getColor("ROSE"),
labels: [{ text: "z₁ + z₂ = 3 + 2i", at: 1, offset: [2, -1, 0] }],
cone: { position: "end" },
},
// Parallelogram helper line (from end of z1 to end of z1+z2)
{
points: [
{ x: 2, y: 3, z: 0 },
{ x: 3, y: 2, z: 0 },
],
color: getColor("EMERALD"),
},
// Parallelogram helper line (from end of z2 to end of z1+z2)
{
points: [
{ x: 1, y: -1, z: 0 },
{ x: 3, y: 2, z: 0 },
],
color: getColor("SKY"),
},
]

## Related Operations

Besides addition, other operations work similarly:

### Scalar Multiplication

Multiplying a complex number $$z = x + iy$$ by a real number (scalar) $$c$$ is straightforward. Just multiply $$c$$ into both the real and imaginary parts.

Visible text: Multiplying a complex number by a real number (scalar) is straightforward. Just multiply into both the real and imaginary parts.

```math
cz = c(x + iy) = cx + i(cy)
```

Geometrically, this scales the vector $$z$$ by a factor of $$c$$. If $$c$$ is negative, the vector's direction is reversed.

Visible text: Geometrically, this scales the vector by a factor of . If is negative, the vector's direction is reversed.

### Negative of a Complex Number

The negative of $$z = x + iy$$ is $$-z$$. This is the same as scalar multiplication by $$c = -1$$.

Visible text: The negative of is . This is the same as scalar multiplication by .

```math
-z = -(x + iy) = -x + i(-y) = -x - iy
```

Geometrically, $$-z$$ is a vector with the same length as $$z$$ but pointing in the opposite direction ($$180^\circ$$ rotation).

Visible text: Geometrically, is a vector with the same length as but pointing in the opposite direction ( rotation).

### Subtraction of Two Complex Numbers

Subtracting $$z_2$$ from $$z_1$$ ($$z_1 - z_2$$) is the same as adding $$z_1$$ to the negative of $$z_2$$ ($$z_1 + (-z_2)$$).

Visible text: Subtracting from () is the same as adding to the negative of ().

```math
z_1 - z_2 = z_1 + (-z_2) = (x_1 + (-x_2)) + i(y_1 + (-y_2)) = (x_1 - x_2) + i(y_1 - y_2)
```

So, subtract the real parts and subtract the imaginary parts.

Geometrically, $$z_1 - z_2$$ is the vector from the tip of $$z_2$$ to the tip of $$z_1$$.

Visible text: Geometrically, is the vector from the tip of to the tip of .

### Example of Combined Operations

Suppose we have:

Component: MathContainer
Children:

```math
z_1 = 2 + \frac{1}{2}i
```

```math
z_2 = -3 + \sqrt{2}i
```

Let's calculate some operations:

1.  **$$2z_1$$ (Scalar Multiplication):**

    
    
    ```math
    2z_1 = 2(2 + \frac{1}{2}i) = 2(2) + i(2 \times \frac{1}{2}) = 4 + i
    ```

2.  **$$z_1 + 3z_2$$ (Addition and Scalar Multiplication):**

    <MathContainer>
      
    
    ```math
    z_1 + 3z_2 = (2 + \frac{1}{2}i) + 3(-3 + \sqrt{2}i)
    ```

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

      
    
    ```math
    = (2 + \frac{1}{2}i) + (-9 + 3\sqrt{2}i)
    ```

      
    
    ```math
    = (2 - 9) + i(\frac{1}{2} + 3\sqrt{2})
    ```

      
    
    ```math
    = -7 + i(\frac{1}{2} + 3\sqrt{2})
    ```

    </MathContainer>

3.  **$$2z_1 - z_2$$ (Subtraction and Scalar Multiplication):**

    <MathContainer>
      
    
    ```math
    2z_1 - z_2 = (4 + i) - (-3 + \sqrt{2}i)
    ```

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

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

      
    
    ```math
    = 7 + i(1 - \sqrt{2})
    ```

    </MathContainer>

Visible text: 1. ** (Scalar Multiplication):**

 
 

2. ** (Addition and Scalar Multiplication):**

 <MathContainer>
 
 

 
 

 
 

 
 

 
 

 </MathContainer>

3. ** (Subtraction and Scalar Multiplication):**

 <MathContainer>
 
 

 
 

 
 

 
 

 </MathContainer>

## Exercise

If $$z_1 = 1 + 2i$$ and $$z_2 = 3 - i$$. Determine:

Visible text: If and . Determine:

1.  $$z_1 + z_2$$
2.  $$z_1 - z_2$$
3.  If $$z_3 = z_1 + z_2$$, draw $$z_1$$, $$z_2$$, and $$z_3$$ on the complex plane.

Visible text: 1. 
2. 
3. If , draw , , and on the complex plane.

### Answer Key

1.  $$z_1 + z_2 = (1+3) + i(2+(-1)) = 4 + i$$
2.  $$z_1 - z_2 = (1-3) + i(2-(-1)) = -2 + i(3) = -2 + 3i$$
3.  Visualization of $$z_1$$, $$z_2$$, and $$z_3 = z_1 + z_2$$ on the complex plane using the parallelogram rule:

    <LineEquation
      title="Addition of Complex Numbers"
      description={
        <>
          Visualization of $$z_1$$, $$z_2$$,
          and $$z_3 = z_1 + z_2$$ on the complex plane using
          the parallelogram rule.
        </>
      }
      cameraPosition={[0, 0, 12]}
      showZAxis={false}
      data={[
        {
          points: [
            { x: 0, y: 0, z: 0 },
            { x: 1, y: 2, z: 0 },
          ],
          color: getColor("SKY"),
          labels: [{ text: "z₁ = 1 + 2i", at: 1, offset: [0.5, 0.5, 0] }],
          cone: { position: "end" },
        },
        {
          points: [
            { x: 0, y: 0, z: 0 },
            { x: 3, y: -1, z: 0 },
          ],
          color: getColor("EMERALD"),
          labels: [{ text: "z₂ = 3 - i", at: 1, offset: [0.5, -0.5, 0] }],
          cone: { position: "end" },
        },
        {
          points: [
            { x: 0, y: 0, z: 0 },
            { x: 4, y: 1, z: 0 }, // z3 = z1 + z2
          ],
          color: getColor("ROSE"),
          labels: [{ text: "z₃ = 4 + i", at: 1, offset: [0.5, 0.5, 0] }],
          cone: { position: "end" },
        },
        // Parallelogram helper lines
        {
          points: [
            { x: 1, y: 2, z: 0 }, // end of z1
            { x: 4, y: 1, z: 0 }, // end of z3
          ],
          color: getColor("EMERALD"),
        },
        {
          points: [
            { x: 3, y: -1, z: 0 }, // end of z2
            { x: 4, y: 1, z: 0 }, // end of z3
          ],
          color: getColor("SKY"),
        },
      ]}
    />

Visible text: 1. 
2. 
3. Visualization of , , and on the complex plane using the parallelogram rule:

 <LineEquation
 title="Addition of Complex Numbers"
 description={
 <>
 Visualization of , ,
 and on the complex plane using
 the parallelogram rule.
 </>
 }
 cameraPosition={[0, 0, 12]}
 showZAxis={false}
 data={[
 {
 points: [
 { x: 0, y: 0, z: 0 },
 { x: 1, y: 2, z: 0 },
 ],
 color: getColor("SKY"),
 labels: [{ text: "z₁ = 1 + 2i", at: 1, offset: [0.5, 0.5, 0] }],
 cone: { position: "end" },
 },
 {
 points: [
 { x: 0, y: 0, z: 0 },
 { x: 3, y: -1, z: 0 },
 ],
 color: getColor("EMERALD"),
 labels: [{ text: "z₂ = 3 - i", at: 1, offset: [0.5, -0.5, 0] }],
 cone: { position: "end" },
 },
 {
 points: [
 { x: 0, y: 0, z: 0 },
 { x: 4, y: 1, z: 0 }, // z3 = z1 + z2
 ],
 color: getColor("ROSE"),
 labels: [{ text: "z₃ = 4 + i", at: 1, offset: [0.5, 0.5, 0] }],
 cone: { position: "end" },
 },
 // Parallelogram helper lines
 {
 points: [
 { x: 1, y: 2, z: 0 }, // end of z1
 { x: 4, y: 1, z: 0 }, // end of z3
 ],
 color: getColor("EMERALD"),
 },
 {
 points: [
 { x: 3, y: -1, z: 0 }, // end of z2
 { x: 4, y: 1, z: 0 }, // end of z3
 ],
 color: getColor("SKY"),
 },
 ]}
 />