# Divide Complex Numbers Worksheet

## 6 problems

It involves multiplying the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part. For example, to divide \frac{a+bi}{c+di}, multiply both by the conjugate $$c-di$$, resulting in \frac{(a+bi)(c-di)}{(c+di)(c-di)}, which simplifies to \frac{(ac+bd) + (bc-ad)i}{c^2+d^2}. Separating the real and imaginary components, get \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i. To practice this process, using a divide complex numbers worksheet and engaging in dividing complex numbers practice can help reinforce the steps and improve proficiency.

Algebra 2
Real And Complex Numbers

## How Will This Worksheet on "Divide Complex Numbers" Benefit Your Student's Learning?

• Helps students understand algebra better, especially using conjugates and simplifying expressions.
• Sharpens problem-solving skills and logical thinking.
• Essential for advanced math classes like calculus and linear algebra.
• Improves accuracy and precision in calculations.
• Encourages both analytical and creative thinking, useful in many areas beyond math.

## How to Divide Complex Numbers?

• Suppose we need to divide \frac{a+bi}{c+di}, where $$a+bi$$ is the numerator and $$c+di$$ is the denominator.
• Multiply both the numerator and the denominator by the conjugate of the denominator, $$c-di$$, to get \frac{(a+bi)(c-di)}{(c+di)(c-di)}.
• The denominator $$(c+di)(c-di)$$ simplifies to $$c^2 + d^2$$, since $$c^2 - (di)^2 = c^2 + d^2$$.
• Expand the numerator to get $$(ac+bd) + (bc-ad)i$$, then divide each term by the simplified denominator $$c^2+d^2$$ to get \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i.

## Solved Example

Q. Simplify $\frac{{-3i}}{{-7i}}$$\newline$$\newline$Write your simplified answer in the form $a + bi$.
Solution:
1. Separate numbers and imaginary unit: Separate the numbers and imaginary unit.$\newline$ $\frac{-3i}{-7i}$ can be written as $\frac{-3}{-7} \times \frac{i}{i}$.
2. Simplify fractions:$\newline$ Simplify: $\frac{-3}{-7} \cdot \frac{i}{i}$$\newline$ $\frac{-3}{-7} \cdot \frac{i}{i} = \frac{3}{7} \cdot 1$
3. Final simplification: Simplify $\frac{3}{7} \times 1$.$\newline$ $\frac{3}{7} \times 1 = \frac{3}{7}$

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