Applying the Distance Formula to the Given Options

Oct 13, 2025 by talynt.org 50 views

Understanding Complex Numbers and the Distance Formula

In mathematics, complex numbers are numbers 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 distance of a complex number from the origin on the complex plane can be calculated using the distance formula, which is derived from the Pythagorean theorem. The distance formula for a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$ .

Applying the Distance Formula to the Given Options

To find the complex number that has a distance of $\sqrt{17}$ from the origin on the complex plane, we need to apply the distance formula to each of the given options. Let's start by analyzing each option:

Option A: $2 + 15i$

For the complex number $2 + 15i$ , we can calculate the distance from the origin using the distance formula:

|2 + 15i| = \sqrt{2^2 + 15^2} = \sqrt{4 + 225} = \sqrt{229}

Since $\sqrt{229} \neq \sqrt{17}$ , option A is not the correct answer.

Option B: $17 + i$

For the complex number $17 + i$ , we can calculate the distance from the origin using the distance formula:

|17 + i| = \sqrt{17^2 + 1^2} = \sqrt{289 + 1} = \sqrt{290}

Since $\sqrt{290} \neq \sqrt{17}$ , option B is not the correct answer.

Option C: $20 - 3i$

For the complex number $20 - 3i$ , we can calculate the distance from the origin using the distance formula:

|20 - 3i| = \sqrt{20^2 + (-3)^2} = \sqrt{400 + 9} = \sqrt{409}

Since $\sqrt{409} \neq \sqrt{17}$ , option C is not the correct answer.

Option D: $4 - i$

For the complex number $4 - i$ , we can calculate the distance from the origin using the distance formula:

|4 - i| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}

Since $\sqrt{17} = \sqrt{17}$ , option D is the correct answer.

Conclusion

Additional Insights

It's worth noting that the distance formula can be used to calculate the distance between any two complex numbers. This can be useful in a variety of mathematical applications, such as finding the distance between two points on a complex plane or calculating the magnitude of a complex number.

Real-World Applications

The concept of complex numbers and the distance formula has numerous real-world applications, such as:

Electrical Engineering: Complex numbers are used to represent AC circuits and calculate impedance.
Signal Processing: Complex numbers are used to represent signals and calculate their magnitude and phase.
Navigation: Complex numbers are used to represent coordinates and calculate distances between points.

Final Thoughts

In conclusion, the complex number that has a distance of $\sqrt{17}$ from the origin on the complex plane is $4 - i$ . This can be verified by applying the distance formula to each of the given options and checking which one yields a distance of $\sqrt{17}$ . The concept of complex numbers and the distance formula has numerous real-world applications and is an essential tool in mathematics and engineering.

Q: What is a complex number?

A: 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 $i^2 = -1$ .

Q: What is the distance formula for complex numbers?

A: The distance formula for a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$ .

Q: How do I calculate the distance between two complex numbers?

A: To calculate the distance between two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ , you can use the distance formula:

|z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2}

Q: What is the magnitude of a complex number?

A: The magnitude of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$ .

Q: How do I calculate the magnitude of a complex number?

A: To calculate the magnitude of a complex number $z = a + bi$ , you can use the formula:

|z| = \sqrt{a^2 + b^2}

Q: What is the difference between the magnitude and the absolute value of a complex number?

A: The magnitude and the absolute value of a complex number are the same thing. They both refer to the distance of the complex number from the origin on the complex plane.

Q: Can I use the distance formula to calculate the distance between two points on a complex plane?

A: Yes, you can use the distance formula to calculate the distance between two points on a complex plane. Simply treat the points as complex numbers and use the distance formula.

Q: What are some real-world applications of complex numbers and the distance formula?

A: Complex numbers and the distance formula have numerous real-world applications, such as:

Electrical Engineering: Complex numbers are used to represent AC circuits and calculate impedance.
Signal Processing: Complex numbers are used to represent signals and calculate their magnitude and phase.
Navigation: Complex numbers are used to represent coordinates and calculate distances between points.

Q: How do I determine which complex number has a certain distance from the origin on the complex plane?

A: To determine which complex number has a certain distance from the origin on the complex plane, you can use the distance formula and compare the results to the given distance.

Q: Can I use the distance formula to calculate the distance between two complex numbers that are not on the same plane?

A: No, the distance formula is only applicable to complex numbers that are on the same plane. If you need to calculate the distance between two complex numbers that are not on the same plane, you will need to use a different method.

Q: What is the relationship between complex numbers and the distance formula?

A: The distance formula is a fundamental concept in complex analysis, and it is used to calculate the distance between complex numbers. The distance formula is derived from the Pythagorean theorem and is a key tool in many mathematical applications.

Q: Can I use the distance formula to calculate the magnitude of a complex number that is not on the complex plane?

A: No, the distance formula is only applicable to complex numbers that are on the complex plane. If you need to calculate the magnitude of a complex number that is not on the complex plane, you will need to use a different method.

Q: How do I apply the distance formula to a complex number that is in polar form?

A: To apply the distance formula to a complex number that is in polar form, you can use the formula:

|z| = r

where $r$ is the magnitude of the complex number.

Q: Can I use the distance formula to calculate the distance between two complex numbers that are in polar form?

A: Yes, you can use the distance formula to calculate the distance between two complex numbers that are in polar form. Simply treat the complex numbers as points on the complex plane and use the distance formula.

Q: What is the relationship between the distance formula and the concept of magnitude?

A: The distance formula and the concept of magnitude are closely related. The distance formula is used to calculate the distance between complex numbers, and the magnitude of a complex number is the distance of the complex number from the origin on the complex plane.

Q: Can I use the distance formula to calculate the magnitude of a complex number that is in rectangular form?

A: Yes, you can use the distance formula to calculate the magnitude of a complex number that is in rectangular form. Simply use the formula:

|z| = \sqrt{a^2 + b^2}

where $a$ and $b$ are the real and imaginary parts of the complex number.

Q: How do I apply the distance formula to a complex number that is in rectangular form?

A: To apply the distance formula to a complex number that is in rectangular form, you can use the formula:

|z| = \sqrt{a^2 + b^2}

where $a$ and $b$ are the real and imaginary parts of the complex number.

Q: Can I use the distance formula to calculate the distance between two complex numbers that are in rectangular form?

A: Yes, you can use the distance formula to calculate the distance between two complex numbers that are in rectangular form. Simply treat the complex numbers as points on the complex plane and use the distance formula.

Q: What is the relationship between the distance formula and the concept of absolute value?

A: The distance formula and the concept of absolute value are the same thing. They both refer to the distance of a complex number from the origin on the complex plane.

Q: Can I use the distance formula to calculate the absolute value of a complex number that is in polar form?

A: Yes, you can use the distance formula to calculate the absolute value of a complex number that is in polar form. Simply use the formula:

|z| = r

where $r$ is the magnitude of the complex number.

Q: How do I apply the distance formula to a complex number that is in polar form and has a magnitude of 5?

A: To apply the distance formula to a complex number that is in polar form and has a magnitude of 5, you can use the formula:

|z| = 5

Q: Can I use the distance formula to calculate the distance between two complex numbers that are in polar form and have a magnitude of 5?

A: Yes, you can use the distance formula to calculate the distance between two complex numbers that are in polar form and have a magnitude of 5. Simply treat the complex numbers as points on the complex plane and use the distance formula.

Q: What is the relationship between the distance formula and the concept of magnitude in polar form?

A: The distance formula and the concept of magnitude in polar form are closely related. The distance formula is used to calculate the distance between complex numbers, and the magnitude of a complex number in polar form is the distance of the complex number from the origin on the complex plane.

Q: Can I use the distance formula to calculate the magnitude of a complex number that is in polar form and has a magnitude of 5?

A: Yes, you can use the distance formula to calculate the magnitude of a complex number that is in polar form and has a magnitude of 5. Simply use the formula:

|z| = 5

Q: How do I apply the distance formula to a complex number that is in polar form and has a magnitude of 5?

A: To apply the distance formula to a complex number that is in polar form and has a magnitude of 5, you can use the formula:

|z| = 5

Q: Can I use the distance formula to calculate the distance between two complex numbers that are in polar form and have a magnitude of 5?

Q: What is the relationship between the distance formula and the concept of absolute value in polar form?

A: The distance formula and the concept of absolute value in polar form are the same thing. They both refer to the distance of a complex number from the origin on the complex plane.

Q: Can I use the distance formula to calculate the absolute value of a complex number that is in polar form and has a magnitude of 5?

A: Yes, you can use the distance formula to calculate the absolute value of a complex number that is in polar form and has a magnitude of 5. Simply use the formula:

|z| = 5

Q: How do I apply the distance formula to a complex number that is in polar form and has a magnitude of 5?

A: To apply the distance formula to a complex number that is in polar form and has a magnitude of 5, you can use the formula:

|z| = 5