Simplify: $\frac{2-1 I}{-2-1 I}$

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Introduction

In mathematics, simplifying complex fractions is an essential skill that can be applied to various fields, including algebra, geometry, and calculus. A complex fraction is a fraction that contains one or more complex numbers, which are numbers that have both real and imaginary parts. In this article, we will simplify the complex fraction $\frac{2-1 i}{-2-1 i}$ using various techniques and methods.

Understanding Complex Numbers

Before we can simplify the complex fraction, we need to understand what complex numbers are. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. The real part of a complex number is the part that is not multiplied by $i$, while the imaginary part is the part that is multiplied by $i$.

Simplifying Complex Fractions

To simplify a complex fraction, we can use various techniques, including multiplying by the conjugate of the denominator, dividing the numerator and denominator by the conjugate of the denominator, and using the fact that $i^2 = -1$. In this article, we will use the first two techniques to simplify the complex fraction $\frac{2-1 i}{-2-1 i}$.

Multiplying by the Conjugate of the Denominator

One way to simplify a complex fraction is to 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 the denominator $-2-1 i$ is $-2+1 i$. Multiplying the numerator and denominator by the conjugate of the denominator, we get:

$$\frac{2-1 i}{-2-1 i} \cdot \frac{-2+1 i}{-2+1 i} = \frac{(2-1 i)(-2+1 i)}{(-2-1 i)(-2+1 i)}$$

Expanding the Numerator and Denominator

To simplify the expression, we need to expand the numerator and denominator. Expanding the numerator, we get:

$$(2-1 i)(-2+1 i) = -4 + 2 i + 2 i - i^2$$

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

$$-4 + 2 i + 2 i - (-1) = -4 + 4 i + 1 = -3 + 4 i$$

Expanding the denominator, we get:

$$(-2-1 i)(-2+1 i) = 4 - 2 i - 2 i - i^2$$

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

$$4 - 2 i - 2 i - (-1) = 4 - 4 i + 1 = 5 - 4 i$$

Simplifying the Expression

Now that we have expanded the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator:

$$\frac{-3 + 4 i}{5 - 4 i}$$

To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator, which is $5 + 4 i$. Multiplying the numerator and denominator by the conjugate of the denominator, we get:

$$\frac{-3 + 4 i}{5 - 4 i} \cdot \frac{5 + 4 i}{5 + 4 i} = \frac{(-3 + 4 i)(5 + 4 i)}{(5 - 4 i)(5 + 4 i)}$$

Expanding the Numerator and Denominator

To simplify the expression, we need to expand the numerator and denominator. Expanding the numerator, we get:

$$( -3 + 4 i)(5 + 4 i) = -15 - 12 i + 20 i + 16 i^2$$

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

$$-15 - 12 i + 20 i + 16 (-1) = -15 + 8 i - 16 = -31 + 8 i$$

Expanding the denominator, we get:

$$(5 - 4 i)(5 + 4 i) = 25 + 20 i - 20 i - 16 i^2$$

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

$$25 + 20 i - 20 i - 16 (-1) = 25 + 16 = 41$$

Simplifying the Expression

Now that we have expanded the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator:

$$\frac{-31 + 8 i}{41}$$

This is the simplified form of the complex fraction $\frac{2-1 i}{-2-1 i}$.

Conclusion

In this article, we simplified the complex fraction $\frac{2-1 i}{-2-1 i}$ using various techniques and methods. We multiplied the numerator and denominator by the conjugate of the denominator, expanded the numerator and denominator, and simplified the expression by dividing the numerator by the denominator. The simplified form of the complex fraction is $\frac{-31 + 8 i}{41}$. This article demonstrates the importance of simplifying complex fractions in mathematics and provides a step-by-step guide on how to simplify complex fractions using various techniques and methods.

Further Reading

For further reading on simplifying complex fractions, we recommend the following resources:

• Simplifying Complex Fractions
• Complex Fractions
• Simplifying Complex Fractions

References

• Complex Numbers
• Simplifying Complex Fractions
• Complex Fractions
  

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Introduction

In our previous article, we simplified the complex fraction $\frac{2-1 i}{-2-1 i}$ using various techniques and methods. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more complex numbers, which are numbers that have both real and imaginary parts.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you can use various techniques, including multiplying by the conjugate of the denominator, dividing the numerator and denominator by the conjugate of the denominator, and using the fact that $i^2 = -1$.

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 $2-1 i$ is $2+1 i$.

Q: How do I multiply a complex number by its conjugate?

A: To multiply a complex number by its conjugate, you can use the formula $(a+bi)(a-bi) = a^2 + b^2$, where $a$ and $b$ are the real and imaginary parts of the complex number.

Q: How do I divide a complex number by its conjugate?

A: To divide a complex number by its conjugate, you can use the formula $\frac{a+bi}{a-bi} = \frac{a^2 + b^2}{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number.

Q: What is the difference between multiplying and dividing a complex number by its conjugate?

A: Multiplying a complex number by its conjugate results in a real number, while dividing a complex number by its conjugate results in a complex number.

Q: Can I simplify a complex fraction using other methods?

A: Yes, you can simplify a complex fraction using other methods, such as using the fact that $i^2 = -1$ or using the formula for the sum and difference of complex numbers.

Q: How do I simplify a complex fraction with multiple complex numbers in the numerator and denominator?

A: To simplify a complex fraction with multiple complex numbers in the numerator and denominator, you can use the distributive property and simplify each complex number separately.

Q: Can I simplify a complex fraction with a complex number in the denominator?

A: Yes, you can simplify a complex fraction with a complex number in the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: How do I check if a complex fraction is simplified?

A: To check if a complex fraction is simplified, you can multiply the numerator and denominator by the conjugate of the denominator and see if the resulting expression is a real number.

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying complex fractions. We covered topics such as what a complex fraction is, how to simplify a complex fraction, and how to multiply and divide a complex number by its conjugate. We also discussed other methods for simplifying complex fractions and how to check if a complex fraction is simplified.

Further Reading

For further reading on simplifying complex fractions, we recommend the following resources:

• Simplifying Complex Fractions
• Complex Fractions
• Simplifying Complex Fractions

References

• Complex Numbers
• Simplifying Complex Fractions
• Complex Fractions