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.