explain me dividing and multiplicating complex numbers

### Multiplying and Dividing Complex Numbers

#### 1. **Complex Numbers Recap:**
A complex number is written in the form \( z = a + bi \), where:
- \( a \) is the real part
- \( b \) is the imaginary part
- \( i \) is the imaginary unit, defined by \( i^2 = -1 \)

#### 2. **Multiplying Complex Numbers:**
When multiplying two complex numbers, we apply the distributive property (FOIL method, like multiplying binomials) and simplify the terms.

Let’s multiply two complex numbers: \( z_1 = a + bi \) and \( z_2 = c + di \).

\[
z_1 \cdot z_2 = (a + bi)(c + di)
\]

Using the distributive property:

\[
z_1 \cdot z_2 = a(c + di) + bi(c + di)
\]

Expanding each term:

\[
= ac + adi + bci + bdi^2
\]

Since \( i^2 = -1 \):

\[
= ac + adi + bci - bd
\]

Now group real and imaginary parts:

\[
= (ac - bd) + (ad + bc)i
\]

So, the product of two complex numbers \( z_1 \) and \( z_2 \) is:

\[
z_1 \cdot z_2 = (ac - bd) + (ad + bc)i
\]

#### 3. **Dividing Complex Numbers:**
Dividing complex numbers involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

Let’s divide \( z_1 = a + bi \) by \( z_2 = c + di \).

\[
\frac{z_1}{z_2} = \frac{a + bi}{c + di}
\]

Step 1: Multiply the numerator and denominator by the conjugate of \( z_2 \), which is \( c - di \).

\[
\frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}
\]

Step 2: Multiply both the numerator and denominator.

For the denominator:
\[
(c + di)(c - di) = c^2 - (di)^2 = c^2 - d^2 i^2 = c^2 + d^2
\]

For the numerator:
\[
(a + bi)(c - di) = ac - adi + bci - bdi^2 = (ac + bd) + (bc - ad)i
\]

So the division is:

\[
\frac{z_1}{z_2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}
\]

This gives the real and imaginary parts of the result.

#### 4. **Summary of Formulas:**
- **Multiplication**: \( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \)
- **Division**: \( \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \)

Would you like further details or examples on this topic?

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### 5 Relative Questions:
1. Can you explain the geometric interpretation of multiplying complex numbers?
2. How do polar forms of complex numbers simplify multiplication and division?
3. How does the magnitude of a complex number affect multiplication?
4. What is the result of dividing a complex number by a real number?
5. How does the conjugate of a complex number assist in division?

**Tip:** Multiplying complex numbers can rotate vectors in the complex plane, and division can reverse or adjust this rotation.

Understand the multiplication and division of complex numbers with clear, step-by-step instructions. Learn key algebraic concepts and formulas, including how to use the complex conjugate in division. Suitable for Grades 10-12.

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