What Is The Modulus For $\left(\frac{z_1}{z_2}\right$\] When $z_1=27\left(\cos \left(55^{\circ}\right)+i \sin \left(55^{\circ}\right)\right$\] And $z_2=9\left(\cos \left(11^{\circ}\right)+i \sin

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Introduction

In mathematics, the modulus of a complex number is a measure of its magnitude or size. It is an essential concept in complex analysis and is used to describe the distance of a complex number from the origin in the complex plane. When dealing with complex numbers in the form of $z = r(\cos \theta + i \sin \theta)$, the modulus is simply the value of $r$. However, when we are given two complex numbers and asked to find the modulus of their ratio, things become more complicated.

Understanding the Modulus of a Complex Number

The modulus of a complex number $z = r(\cos \theta + i \sin \theta)$ is given by $|z| = r$. This is because the modulus is essentially the distance of the complex number from the origin, and in the complex plane, this distance is given by the value of $r$. For example, if we have a complex number $z = 3(\cos 45^{\circ} + i \sin 45^{\circ})$, the modulus of $z$ is simply $|z| = 3$.

Modulus of a Complex Number in Polar Form

When a complex number is given in polar form as $z = r(\cos \theta + i \sin \theta)$, the modulus is simply the value of $r$. This is because the polar form of a complex number already represents the distance of the complex number from the origin. For example, if we have a complex number $z = 4(\cos 30^{\circ} + i \sin 30^{\circ})$, the modulus of $z$ is simply $|z| = 4$.

Modulus of a Complex Number in Rectangular Form

When a complex number is given in rectangular form as $z = a + bi$, the modulus can be found using the formula $|z| = \sqrt{a^2 + b^2}$. This is because the modulus is essentially the distance of the complex number from the origin, and in the rectangular form, this distance is given by the square root of the sum of the squares of the real and imaginary parts.

Modulus of the Ratio of Two Complex Numbers

Now, let's consider the problem of finding the modulus of the ratio of two complex numbers. Given two complex numbers $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, the modulus of their ratio is given by $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}$.

Applying the Modulus Formula to the Given Problem

Now, let's apply the modulus formula to the given problem. We are given two complex numbers $z_1 = 27(\cos 55^{\circ} + i \sin 55^{\circ})$ and $z_2 = 9(\cos 11^{\circ} + i \sin 11^{\circ})$. To find the modulus of their ratio, we can use the formula $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}$.

Calculating the Modulus of the Ratio

The modulus of $z_1$ is $|z_1| = 27$, and the modulus of $z_2$ is $|z_2| = 9$. Therefore, the modulus of their ratio is $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} = \frac{27}{9} = 3$.

Conclusion

In conclusion, the modulus of the ratio of two complex numbers is given by the formula $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}$. By applying this formula to the given problem, we can find the modulus of the ratio of the two complex numbers $z_1 = 27(\cos 55^{\circ} + i \sin 55^{\circ})$ and $z_2 = 9(\cos 11^{\circ} + i \sin 11^{\circ})$. The modulus of their ratio is simply the ratio of their moduli, which is $3$.

Frequently Asked Questions

• What is the modulus of a complex number? The modulus of a complex number is a measure of its magnitude or size.
• How do you find the modulus of a complex number in polar form? The modulus of a complex number in polar form is simply the value of $r$.
• How do you find the modulus of a complex number in rectangular form? The modulus of a complex number in rectangular form can be found using the formula $|z| = \sqrt{a^2 + b^2}$.
• What is the modulus of the ratio of two complex numbers? The modulus of the ratio of two complex numbers is given by the formula $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}$.

References

• [1] Complex Analysis by Serge Lang
• [2] Complex Numbers by Michael Spivak
• [3] Calculus by Michael Spivak
  

Q: What is the modulus of a complex number?

A: The modulus of a complex number is a measure of its magnitude or size. It is an essential concept in complex analysis and is used to describe the distance of a complex number from the origin in the complex plane.

Q: How do you find the modulus of a complex number in polar form?

A: The modulus of a complex number in polar form is simply the value of $r$. This is because the polar form of a complex number already represents the distance of the complex number from the origin.

Q: How do you find the modulus of a complex number in rectangular form?

A: The modulus of a complex number in rectangular form can be found using the formula $|z| = \sqrt{a^2 + b^2}$. This is because the modulus is essentially the distance of the complex number from the origin, and in the rectangular form, this distance is given by the square root of the sum of the squares of the real and imaginary parts.

Q: What is the modulus of the ratio of two complex numbers?

A: The modulus of the ratio of two complex numbers is given by the formula $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}$. This means that the modulus of the ratio of two complex numbers is simply the ratio of their moduli.

Q: How do you apply the modulus formula to the given problem?

A: To apply the modulus formula to the given problem, we need to find the moduli of the two complex numbers $z_1 = 27(\cos 55^{\circ} + i \sin 55^{\circ})$ and $z_2 = 9(\cos 11^{\circ} + i \sin 11^{\circ})$. The modulus of $z_1$ is $|z_1| = 27$, and the modulus of $z_2$ is $|z_2| = 9$.

Q: What is the modulus of the ratio of the two complex numbers $z_1 = 27(\cos 55^{\circ} + i \sin 55^{\circ})$ and $z_2 = 9(\cos 11^{\circ} + i \sin 11^{\circ})$?

A: The modulus of the ratio of the two complex numbers $z_1 = 27(\cos 55^{\circ} + i \sin 55^{\circ})$ and $z_2 = 9(\cos 11^{\circ} + i \sin 11^{\circ})$ is simply the ratio of their moduli, which is $3$.

Q: What are some common applications of the modulus of complex numbers?

A: The modulus of complex numbers has many applications in various fields, including engineering, physics, and computer science. Some common applications include:

• Signal processing: The modulus of complex numbers is used to analyze and process signals in signal processing.
• Control systems: The modulus of complex numbers is used to analyze and design control systems.
• Image processing: The modulus of complex numbers is used to analyze and process images in image processing.

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

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

• Confusing the modulus with the argument of a complex number.
• Not using the correct formula for the modulus of a complex number.
• Not checking the units of the modulus of a complex number.

Q: How do you check the units of the modulus of a complex number?

A: To check the units of the modulus of a complex number, you need to check the units of the real and imaginary parts of the complex number. The units of the modulus of a complex number are typically the same as the units of the real and imaginary parts.

Q: What are some common tools and software used to work with the modulus of complex numbers?

A: Some common tools and software used to work with the modulus of complex numbers include:

• MATLAB: MATLAB is a popular software used to work with complex numbers and their moduli.
• Mathematica: Mathematica is a popular software used to work with complex numbers and their moduli.
• Python: Python is a popular programming language used to work with complex numbers and their moduli.

Q: How do you use MATLAB to work with the modulus of complex numbers?

A: To use MATLAB to work with the modulus of complex numbers, you need to first create a complex number using the complex function. Then, you can use the abs function to find the modulus of the complex number.

Q: How do you use Mathematica to work with the modulus of complex numbers?

A: To use Mathematica to work with the modulus of complex numbers, you need to first create a complex number using the Complex function. Then, you can use the Abs function to find the modulus of the complex number.

Q: How do you use Python to work with the modulus of complex numbers?

A: To use Python to work with the modulus of complex numbers, you need to first create a complex number using the complex function. Then, you can use the abs function to find the modulus of the complex number.

Q: What are some common applications of the modulus of complex numbers in engineering?

A: The modulus of complex numbers has many applications in engineering, including:

• Signal processing: The modulus of complex numbers is used to analyze and process signals in signal processing.
• Control systems: The modulus of complex numbers is used to analyze and design control systems.
• Image processing: The modulus of complex numbers is used to analyze and process images in image processing.

Q: What are some common applications of the modulus of complex numbers in physics?

A: The modulus of complex numbers has many applications in physics, including:

• Quantum mechanics: The modulus of complex numbers is used to describe the behavior of particles in quantum mechanics.
• Electromagnetism: The modulus of complex numbers is used to describe the behavior of electromagnetic waves.
• Optics: The modulus of complex numbers is used to describe the behavior of light in optics.

Q: What are some common applications of the modulus of complex numbers in computer science?

A: The modulus of complex numbers has many applications in computer science, including:

• Signal processing: The modulus of complex numbers is used to analyze and process signals in signal processing.
• Image processing: The modulus of complex numbers is used to analyze and process images in image processing.
• Machine learning: The modulus of complex numbers is used to analyze and process data in machine learning.

Q: What are some common challenges when working with the modulus of complex numbers?

A: Some common challenges when working with the modulus of complex numbers include:

• Confusing the modulus with the argument of a complex number.
• Not using the correct formula for the modulus of a complex number.
• Not checking the units of the modulus of a complex number.

Q: How do you overcome these challenges when working with the modulus of complex numbers?

A: To overcome these challenges when working with the modulus of complex numbers, you need to:

• Carefully check the units of the modulus of a complex number.
• Use the correct formula for the modulus of a complex number.
• Avoid confusing the modulus with the argument of a complex number.

Q: What are some common tools and software used to overcome these challenges when working with the modulus of complex numbers?

A: Some common tools and software used to overcome these challenges when working with the modulus of complex numbers include:

• MATLAB: MATLAB is a popular software used to work with complex numbers and their moduli.
• Mathematica: Mathematica is a popular software used to work with complex numbers and their moduli.
• Python: Python is a popular programming language used to work with complex numbers and their moduli.