Calculate The Product: $(-11 - 7i)(2 + 6i$\]

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Introduction

Complex numbers are mathematical expressions that consist of a real number and an imaginary number. They are used to represent points in a two-dimensional plane and are essential in various fields, including mathematics, physics, and engineering. In this article, we will learn how to calculate the product of two complex numbers, specifically the product of $(-11 - 7i)$ and $(2 + 6i)$.

What are Complex Numbers?

Complex numbers are of the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary part is denoted by $i$, which is the square root of $-1$. Complex numbers can be represented graphically on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Multiplication of Complex Numbers

To multiply two complex numbers, we use the distributive property and the fact that $i^2 = -1$. The multiplication of two complex numbers is given by:

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Calculating the Product of $(-11 - 7i)$ and $(2 + 6i)$

Using the formula for multiplication of complex numbers, we can calculate the product of $(-11 - 7i)$ and $(2 + 6i)$ as follows:

$(-11 - 7i)(2 + 6i) = (-11 \cdot 2 - (-7) \cdot 6) + (-11 \cdot 6 + (-7) \cdot 2)i$

$= (-22 + 42) + (-66 - 14)i$

$= 20 - 80i$

Simplifying the Result

The result of the multiplication is a complex number in the form $a + bi$. In this case, the real part is $20$ and the imaginary part is $-80$. Therefore, the product of $(-11 - 7i)$ and $(2 + 6i)$ is $20 - 80i$.

Conclusion

In this article, we learned how to calculate the product of two complex numbers, specifically the product of $(-11 - 7i)$ and $(2 + 6i)$. We used the distributive property and the fact that $i^2 = -1$ to simplify the expression and obtain the final result. The product of $(-11 - 7i)$ and $(2 + 6i)$ is $20 - 80i$.

Examples and Applications

Complex numbers have numerous applications in various fields, including mathematics, physics, and engineering. Some examples of applications of complex numbers include:

• Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
• Control Systems: Complex numbers are used to analyze the stability of control systems.
• Navigation: Complex numbers are used to represent the position and velocity of objects in navigation systems.

Tips and Tricks

• Use the Distributive Property: When multiplying complex numbers, use the distributive property to simplify the expression.
• Use the Fact that $i^2 = -1$: When simplifying complex numbers, use the fact that $i^2 = -1$ to eliminate the imaginary part.
• Check Your Work: When calculating the product of complex numbers, check your work by plugging in the values and simplifying the expression.

Final Thoughts

In conclusion, complex numbers are a powerful tool in mathematics and have numerous applications in various fields. Calculating the product of complex numbers requires attention to detail and a thorough understanding of the distributive property and the fact that $i^2 = -1$. By following the steps outlined in this article, you can calculate the product of complex numbers with ease and apply this knowledge to real-world problems.

Introduction

Complex numbers are a fundamental concept in mathematics, and understanding them is crucial for various fields, including mathematics, physics, and engineering. In this article, we will answer some frequently asked questions about complex numbers, covering topics such as their definition, properties, and applications.

Q: What is a complex number?

A: A complex number is a mathematical expression that consists of a real number and an imaginary number. It is of the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Q: What is the imaginary part of a complex number?

A: The imaginary part of a complex number is denoted by $i$, which is the square root of $-1$. It is used to represent the magnitude and direction of a quantity that cannot be expressed by a single real number.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their sum is $(a + c) + (b + d)i$.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their difference is $(a - c) + (b - d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that $i^2 = -1$. The multiplication of two complex numbers is given by:

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Q: How do I divide complex numbers?

A: To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$. For example, if you have two complex numbers $a + bi$ and $c + di$, their quotient is:

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$. It is used to simplify complex expressions and to eliminate the imaginary part.

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a + bi$ is the distance from the origin to the point $(a, b)$ in the complex plane. It is given by:

$|a + bi| = \sqrt{a^2 + b^2}$

Q: What is the argument of a complex number?

A: The argument of a complex number $a + bi$ is the angle between the positive real axis and the line segment joining the origin to the point $(a, b)$ in the complex plane. It is given by:

$\arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right)$

Q: What are some common applications of complex numbers?

A: Complex numbers have numerous applications in various fields, including:

• Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
• Control Systems: Complex numbers are used to analyze the stability of control systems.
• Navigation: Complex numbers are used to represent the position and velocity of objects in navigation systems.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

• Forgetting to use the distributive property: When multiplying complex numbers, use the distributive property to simplify the expression.
• Forgetting to use the fact that $i^2 = -1$: When simplifying complex numbers, use the fact that $i^2 = -1$ to eliminate the imaginary part.
• Not checking your work: When calculating the product or quotient of complex numbers, check your work by plugging in the values and simplifying the expression.

Conclusion

In this article, we have answered some frequently asked questions about complex numbers, covering topics such as their definition, properties, and applications. We hope that this guide has been helpful in understanding complex numbers and their uses in various fields.