Which Complex Number Has A Distance Of $\sqrt{17}$ From The Origin On The Complex Plane?A. $2 + 15i$ B. $17 + 1i$ C. $20 - 3i$ D. $4 - I$

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Introduction

In the complex plane, the distance of a complex number from the origin is given by the modulus or absolute value of the complex number. This distance is calculated using the formula ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}, where z=a+biz = a + bi is the complex number. In this article, we will explore which complex number has a distance of 17\sqrt{17} from the origin on the complex plane.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form z=a+biz = a + bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=βˆ’1i^2 = -1. The real part of the complex number is aa, and the imaginary part is bb. The modulus or absolute value of a complex number is given by ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}.

Calculating the Distance

To find the complex number with a distance of 17\sqrt{17} from the origin, we need to calculate the modulus of each option and compare it with 17\sqrt{17}. Let's start by calculating the modulus of each option.

Option A: 2+15i2 + 15i

The modulus of 2+15i2 + 15i is given by ∣2+15i∣=22+152=4+225=229|2 + 15i| = \sqrt{2^2 + 15^2} = \sqrt{4 + 225} = \sqrt{229}.

Option B: 17+1i17 + 1i

The modulus of 17+1i17 + 1i is given by ∣17+1i∣=172+12=289+1=290|17 + 1i| = \sqrt{17^2 + 1^2} = \sqrt{289 + 1} = \sqrt{290}.

Option C: 20βˆ’3i20 - 3i

The modulus of 20βˆ’3i20 - 3i is given by ∣20βˆ’3i∣=202+(βˆ’3)2=400+9=409|20 - 3i| = \sqrt{20^2 + (-3)^2} = \sqrt{400 + 9} = \sqrt{409}.

Option D: 4βˆ’i4 - i

The modulus of 4βˆ’i4 - i is given by ∣4βˆ’i∣=42+(βˆ’1)2=16+1=17|4 - i| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}.

Conclusion

From the calculations above, we can see that only option D has a modulus of 17\sqrt{17}, which means that the complex number 4βˆ’i4 - i has a distance of 17\sqrt{17} from the origin on the complex plane.

Final Answer

The final answer is D\boxed{D}.

Additional Information

In this article, we have explored how to calculate the distance of a complex number from the origin on the complex plane. We have also seen how to use the modulus formula to compare the distances of different complex numbers. This knowledge can be applied to a wide range of mathematical problems and is an essential tool for anyone studying complex analysis.

Real-World Applications

The concept of complex numbers and their distances from the origin has many real-world applications. For example, in electrical engineering, complex numbers are used to represent AC circuits and analyze their behavior. In signal processing, complex numbers are used to represent signals and filter them. In computer graphics, complex numbers are used to represent 3D objects and perform transformations on them.

Future Research Directions

There are many areas of research that involve complex numbers and their distances from the origin. Some potential areas of research include:

  • Complex Analysis: This is a branch of mathematics that deals with the study of complex functions and their properties.
  • Numerical Analysis: This is a branch of mathematics that deals with the study of numerical methods for solving mathematical problems.
  • Signal Processing: This is a branch of engineering that deals with the analysis and processing of signals.
  • Computer Graphics: This is a branch of computer science that deals with the creation and manipulation of 3D objects.

Conclusion

In conclusion, the complex number 4βˆ’i4 - i has a distance of 17\sqrt{17} from the origin on the complex plane. This knowledge can be applied to a wide range of mathematical problems and has many real-world applications. There are many areas of research that involve complex numbers and their distances from the origin, and this knowledge can be used to make new discoveries and advancements in these fields.

Introduction

In our previous article, we explored the concept of complex numbers and their distances from the origin on the complex plane. We also calculated the modulus of each option and compared it with 17\sqrt{17} to find the complex number with a distance of 17\sqrt{17} from the origin. In this article, we will answer some frequently asked questions (FAQs) about complex numbers and their distances from the origin.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form z=a+biz = a + bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=βˆ’1i^2 = -1.

Q: What is the modulus of a complex number?

A: The modulus of a complex number is given by ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}, where z=a+biz = a + bi is the complex number.

Q: How do you calculate the distance of a complex number from the origin?

A: To calculate the distance of a complex number from the origin, you need to calculate the modulus of the complex number using the formula ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}.

Q: What is the difference between the real and imaginary parts of a complex number?

A: The real part of a complex number is the part that is not multiplied by ii, while the imaginary part is the part that is multiplied by ii.

Q: Can you give an example of a complex number with a distance of 17\sqrt{17} from the origin?

A: Yes, the complex number 4βˆ’i4 - i has a distance of 17\sqrt{17} from the origin.

Q: How do you compare the distances of different complex numbers?

A: You can compare the distances of different complex numbers by calculating their moduli using the formula ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}.

Q: What are some real-world applications of complex numbers and their distances from the origin?

A: Complex numbers and their distances from the origin have many real-world applications, including electrical engineering, signal processing, and computer graphics.

Q: Can you give some examples of complex numbers and their distances from the origin?

A: Yes, here are some examples:

  • The complex number 2+3i2 + 3i has a distance of 13\sqrt{13} from the origin.
  • The complex number 5βˆ’2i5 - 2i has a distance of 29\sqrt{29} from the origin.
  • The complex number 1+4i1 + 4i has a distance of 17\sqrt{17} from the origin.

Q: How do you represent complex numbers in the complex plane?

A: Complex numbers can be represented in the complex plane using the x-axis and y-axis, where the x-axis represents the real part of the complex number and the y-axis represents the imaginary part.

Q: What is the significance of the modulus of a complex number?

A: The modulus of a complex number represents the distance of the complex number from the origin in the complex plane.

Q: Can you give some examples of complex numbers with different moduli?

A: Yes, here are some examples:

  • The complex number 3+4i3 + 4i has a modulus of 25\sqrt{25}.
  • The complex number 2βˆ’3i2 - 3i has a modulus of 13\sqrt{13}.
  • The complex number 1+2i1 + 2i has a modulus of 5\sqrt{5}.

Conclusion

In conclusion, complex numbers and their distances from the origin have many real-world applications and are an essential tool for anyone studying complex analysis. We hope that this article has answered some of the frequently asked questions about complex numbers and their distances from the origin.

Final Answer

The final answer is D\boxed{D}.

Additional Information

In this article, we have explored some frequently asked questions about complex numbers and their distances from the origin. We have also provided some examples of complex numbers and their distances from the origin. This knowledge can be applied to a wide range of mathematical problems and has many real-world applications.

Real-World Applications

The concept of complex numbers and their distances from the origin has many real-world applications, 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 filter them.
  • Computer Graphics: Complex numbers are used to represent 3D objects and perform transformations on them.

Future Research Directions

There are many areas of research that involve complex numbers and their distances from the origin. Some potential areas of research include:

  • Complex Analysis: This is a branch of mathematics that deals with the study of complex functions and their properties.
  • Numerical Analysis: This is a branch of mathematics that deals with the study of numerical methods for solving mathematical problems.
  • Signal Processing: This is a branch of engineering that deals with the analysis and processing of signals.
  • Computer Graphics: This is a branch of computer science that deals with the creation and manipulation of 3D objects.

Conclusion

In conclusion, complex numbers and their distances from the origin have many real-world applications and are an essential tool for anyone studying complex analysis. We hope that this article has answered some of the frequently asked questions about complex numbers and their distances from the origin.