Simplify The Following Expression: 1 + 4 I 4 − 2 I = □ \frac{1+4i}{4-2i} = \square 4 − 2 I 1 + 4 I = □ Enter The Answer As A Reduced Fraction, When Necessary.

Mar 11, 2025 by ADMIN 161 views

**Simplifying Complex Fractions: A Step-by-Step Guide**

Introduction

Complex fractions, also known as complex numbers, are a fundamental concept in mathematics. They are used to represent numbers that have both real and imaginary parts. In this article, we will focus on simplifying complex fractions, specifically the expression $\frac{1+4i}{4-2i}$ . We will break down the steps involved in simplifying this expression and provide a clear, step-by-step guide.

What are Complex Fractions?

Complex fractions are numbers that have both real and imaginary parts. They are typically represented in the form $a+bi$ , where $a$ is the real part and $b$ is the imaginary part. The imaginary part is denoted by the letter $i$ , which is defined as the square root of $-1$ . Complex fractions can be added, subtracted, multiplied, and divided, just like real numbers.

Simplifying Complex Fractions: A Step-by-Step Guide

To simplify the complex fraction $\frac{1+4i}{4-2i}$ , we will use the following steps:

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

The conjugate of a complex number is obtained by changing the sign of the imaginary part. In this case, the conjugate of $4-2i$ is $4+2i$ . We will multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

$\frac{1+4i}{4-2i} \cdot \frac{4+2i}{4+2i}$

Step 2: Expand the Numerator and Denominator

We will expand the numerator and denominator by multiplying the two complex numbers.

$(1+4i)(4+2i) = 4+2i+16i+8i^2$

$= 4+18i+8i^2$

Since $i^2 = -1$ , we can substitute this value into the expression.

$= 4+18i-8$

$= -4+18i$

$(4-2i)(4+2i) = 16+8i-8i-4i^2$

$= 16-4i^2$

Since $i^2 = -1$ , we can substitute this value into the expression.

$= 16+4$

$= 20$

Step 3: Simplify the Expression

We will simplify the expression by dividing the numerator by the denominator.

$\frac{-4+18i}{20}$

We can simplify this expression by dividing both the real and imaginary parts by $20$ .

$= \frac{-4}{20} + \frac{18i}{20}$

$= -\frac{1}{5} + \frac{9i}{10}$

Conclusion

Simplifying complex fractions requires a step-by-step approach. By multiplying the numerator and denominator by the conjugate of the denominator, expanding the numerator and denominator, and simplifying the expression, we can simplify complex fractions. In this article, we simplified the complex fraction $\frac{1+4i}{4-2i}$ and obtained the result $-\frac{1}{5} + \frac{9i}{10}$ . We hope this article has provided a clear, step-by-step guide to simplifying complex fractions.

Common Mistakes to Avoid

When simplifying complex fractions, there are several common mistakes to avoid:

Not multiplying the numerator and denominator by the conjugate of the denominator: This is the most common mistake when simplifying complex fractions. Make sure to multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.
Not expanding the numerator and denominator: Make sure to expand the numerator and denominator by multiplying the two complex numbers.
Not simplifying the expression: Make sure to simplify the expression by dividing the numerator by the denominator.

Real-World Applications

Complex fractions have several real-world applications, including:

Electrical Engineering: Complex fractions are used to represent impedance in electrical circuits.
Signal Processing: Complex fractions are used to represent filters in signal processing.
Control Systems: Complex fractions are used to represent transfer functions in control systems.

Conclusion

Introduction

In our previous article, we discussed the steps involved in simplifying complex fractions. In this article, we will provide a Q&A guide to help you understand the concept of simplifying complex fractions better.

Q: What is a complex fraction?

A: A complex fraction is a number that has both real and imaginary parts. It is typically represented in the form $a+bi$ , where $a$ is the real part and $b$ is the imaginary part.

Q: Why do we need to simplify complex fractions?

A: We need to simplify complex fractions to make them easier to work with. Simplifying complex fractions helps us to:

Eliminate the imaginary part from the denominator: By multiplying the numerator and denominator by the conjugate of the denominator, we can eliminate the imaginary part from the denominator.
Simplify the expression: By simplifying the expression, we can make it easier to work with and understand.

Q: How do we simplify complex fractions?

A: To simplify complex fractions, we follow these steps:

Multiply the numerator and denominator by the conjugate of the denominator: This helps us to eliminate the imaginary part from the denominator.
Expand the numerator and denominator: This helps us to simplify the expression.
Simplify the expression: This helps us to make the expression easier to work with and understand.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of $4-2i$ is $4+2i$ .

Q: Why do we multiply the numerator and denominator by the conjugate of the denominator?

A: We multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. This helps us to simplify the expression and make it easier to work with.

Q: Can we simplify complex fractions with zero imaginary part?

A: Yes, we can simplify complex fractions with zero imaginary part. In this case, the complex fraction is simply a real number.

Q: Can we simplify complex fractions with zero real part?

A: Yes, we can simplify complex fractions with zero real part. In this case, the complex fraction is simply an imaginary number.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

Not multiplying the numerator and denominator by the conjugate of the denominator: This is the most common mistake when simplifying complex fractions.
Not expanding the numerator and denominator: Make sure to expand the numerator and denominator by multiplying the two complex numbers.
Not simplifying the expression: Make sure to simplify the expression by dividing the numerator by the denominator.

Q: What are some real-world applications of complex fractions?

A: Complex fractions have several real-world applications, including:

Electrical Engineering: Complex fractions are used to represent impedance in electrical circuits.
Signal Processing: Complex fractions are used to represent filters in signal processing.
Control Systems: Complex fractions are used to represent transfer functions in control systems.

Conclusion

Simplifying complex fractions is a fundamental concept in mathematics. By following the steps outlined in this article and avoiding common mistakes, we can simplify complex fractions and obtain the result in the form of a reduced fraction. We hope this article has provided a clear, step-by-step guide to simplifying complex fractions.