The Complex Conjugate Of $\omega$ Is Denoted By $\bar{\omega}$. Given That $\omega = 1 + 2i$, Find $z$ Where $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$.

Introduction

In mathematics, the complex conjugate of a complex number is a fundamental concept that plays a crucial role in various mathematical operations and applications. The complex conjugate of a complex number $\omega$ is denoted by $\bar{\omega}$. In this article, we will explore the concept of the complex conjugate of $\omega$ and its application in finding the value of $z$ where $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$.

The Complex Conjugate of Omega

The complex conjugate of a complex number $\omega = a + bi$ is denoted by $\bar{\omega} = a - bi$, where $a$ and $b$ are real numbers. In the given problem, $\omega = 1 + 2i$, so the complex conjugate of $\omega$ is $\bar{\omega} = 1 - 2i$.

Finding the Value of Omega Squared

To find the value of $z$, we need to calculate the value of $\omega^2$. Using the given value of $\omega = 1 + 2i$, we can calculate $\omega^2$ as follows:

$\omega^2 = (1 + 2i)^2$ $= 1^2 + 2 \cdot 1 \cdot 2i + (2i)^2$ $= 1 + 4i - 4$ $= -3 + 4i$

Finding the Value of Z

Now that we have the value of $\omega^2$, we can proceed to find the value of $z$. We are given that $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$. Substituting the values of $\omega$, $\bar{\omega}$, and $\omega^2$, we get:

$z = (1 + 2i) - \frac{25(1 - 2i)}{-3 + 4i}$

Simplifying the Expression

To simplify the expression, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $-3 + 4i$ is $-3 - 4i$. Multiplying the numerator and denominator by $-3 - 4i$, we get:

$z = (1 + 2i) - \frac{25(1 - 2i)(-3 - 4i)}{(-3 + 4i)(-3 - 4i)}$

Expanding the Numerator

Expanding the numerator, we get:

$z = (1 + 2i) - \frac{25(-3 - 4i + 6i + 8)}{(-3 + 4i)(-3 - 4i)}$

Simplifying the Numerator

Simplifying the numerator, we get:

$z = (1 + 2i) - \frac{25(-3 + 2i + 8)}{(-3 + 4i)(-3 - 4i)}$

Further Simplification

Further simplifying the numerator, we get:

$z = (1 + 2i) - \frac{25(5 + 2i)}{(-3 + 4i)(-3 - 4i)}$

Calculating the Denominator

To calculate the denominator, we need to multiply the two complex numbers:

$(-3 + 4i)(-3 - 4i) = (-3)^2 - (4i)^2$ $= 9 - 16i^2$ $= 9 + 16$ $= 25$

Substituting the Value of the Denominator

Substituting the value of the denominator, we get:

$z = (1 + 2i) - \frac{25(5 + 2i)}{25}$

Simplifying the Expression

Simplifying the expression, we get:

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

Final Calculation

Finally, we can calculate the value of $z$:

$z = 1 + 2i - 5 - 2i$ $= -4$

Conclusion

In this article, we have explored the concept of the complex conjugate of $\omega$ and its application in finding the value of $z$ where $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$. We have calculated the value of $\omega^2$ and used it to find the value of $z$. The final value of $z$ is $-4$.

Introduction

In our previous article, we explored the concept of the complex conjugate of $\omega$ and its application in finding the value of $z$ where $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$. In this article, we will answer some frequently asked questions related to the complex conjugate of $\omega$ and its application in mathematics.

Q&A

Q: What is the complex conjugate of a complex number?

A: The complex conjugate of a complex number $\omega = a + bi$ is denoted by $\bar{\omega} = a - bi$, where $a$ and $b$ are real numbers.

Q: How do you find the complex conjugate of a complex number?

A: To find the complex conjugate of a complex number, you need to change the sign of the imaginary part. For example, if $\omega = 1 + 2i$, then the complex conjugate of $\omega$ is $\bar{\omega} = 1 - 2i$.

Q: What is the significance of the complex conjugate in mathematics?

A: The complex conjugate plays a crucial role in various mathematical operations and applications, such as finding the value of $z$ where $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$.

Q: How do you calculate the value of $\omega^2$?

A: To calculate the value of $\omega^2$, you need to multiply the complex number $\omega$ by itself. For example, if $\omega = 1 + 2i$, then $\omega^2 = (1 + 2i)^2 = -3 + 4i$.

Q: What is the formula for finding the value of $z$?

A: The formula for finding the value of $z$ is $z = \omega - \frac{25 \bar{\omega}}{\omega^2}$.

Q: How do you simplify the expression for $z$?

A: To simplify the expression for $z$, you need to multiply the numerator and denominator by the conjugate of the denominator. For example, if $z = (1 + 2i) - \frac{25(1 - 2i)}{-3 + 4i}$, then you need to multiply the numerator and denominator by $-3 - 4i$.

Q: What is the final value of $z$?

A: The final value of $z$ is $-4$.

Conclusion

In this article, we have answered some frequently asked questions related to the complex conjugate of $\omega$ and its application in mathematics. We hope that this article has provided you with a better understanding of the complex conjugate of $\omega$ and its significance in mathematics.

Additional Resources

• Complex Conjugate: A Comprehensive Guide
• Complex Numbers: A Beginner's Guide
• Mathematics: A Subject of Beauty and Wonder

Related Articles

• The Complex Conjugate of Omega and Its Application in Mathematics
• The Significance of the Complex Conjugate in Mathematics
• The Complex Conjugate of Omega: A Mathematical Marvel

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