Find The Difference Of The Complex Numbers:$(6 + 2i) - (4 - 8i$\]A. $10 + 10i$ B. $2 + 10i$ C. $10 - \sqrt{\text{th}}$ D. $2 - 6i$

by ADMIN 134 views

===========================================================

Complex numbers are a fundamental concept in mathematics, and understanding how to perform operations with them is crucial for solving various mathematical problems. In this article, we will focus on finding the difference of complex numbers, which is a basic operation that involves subtracting one complex number from another.

What are Complex Numbers?

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. Complex numbers can be represented graphically on a complex plane, with the real part of the number on the x-axis and the imaginary part on the y-axis.

The Difference of Complex Numbers

The difference of complex numbers is a basic operation that involves subtracting one complex number from another. To find the difference of two complex numbers, we simply subtract the real parts and the imaginary parts separately.

Example: (6+2i)−(4−8i)(6 + 2i) - (4 - 8i)

Let's consider the example given in the problem: (6+2i)−(4−8i)(6 + 2i) - (4 - 8i). To find the difference, we will subtract the real parts and the imaginary parts separately.

Step 1: Subtract the Real Parts

The real part of the first complex number is 6, and the real part of the second complex number is 4. To find the difference of the real parts, we simply subtract 4 from 6.

real_part_difference = 6 - 4
print(real_part_difference)

Step 2: Subtract the Imaginary Parts

The imaginary part of the first complex number is 2i, and the imaginary part of the second complex number is -8i. To find the difference of the imaginary parts, we simply subtract -8i from 2i.

imaginary_part_difference = 2i - (-8i)
print(imaginary_part_difference)

Step 3: Combine the Real and Imaginary Parts

Now that we have found the difference of the real and imaginary parts, we can combine them to get the final result.

final_result = real_part_difference + imaginary_part_difference
print(final_result)

The Final Answer

After performing the calculations, we get the final result:

(6+2i)−(4−8i)=2+10i(6 + 2i) - (4 - 8i) = 2 + 10i

Therefore, the correct answer is:

A. 2+10i2 + 10i

Conclusion

In this article, we have learned how to find the difference of complex numbers by subtracting the real parts and the imaginary parts separately. We have also seen an example of how to perform this operation using Python code. By following these steps, you can easily find the difference of complex numbers and solve various mathematical problems.

Frequently Asked Questions

Q: What is the difference between complex numbers and real numbers?

A: 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. Real numbers, on the other hand, are numbers that can be expressed in the form a, where a is a real number.

Q: How do I represent complex numbers graphically on a complex plane?

A: To represent complex numbers graphically on a complex plane, you can plot the real part of the number on the x-axis and the imaginary part on the y-axis.

Q: What is the imaginary unit i?

A: The imaginary unit i is a number that satisfies the equation i^2 = -1. It is used to represent the imaginary part of complex numbers.

References

Note: The references provided are for informational purposes only and are not directly related to the content of this article.

===========================================================

Complex numbers are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will provide answers to frequently asked questions about complex numbers, covering topics such as their definition, representation, and operations.

Q: What is the definition of 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: How do I represent complex numbers graphically on a complex plane?

A: To represent complex numbers graphically on a complex plane, you can plot the real part of the number on the x-axis and the imaginary part on the y-axis. This is known as the Cartesian representation of complex numbers.

Q: What is the imaginary unit i?

A: The imaginary unit i is a number that satisfies the equation i^2 = -1. It is used to represent the imaginary part of complex numbers.

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, (a + bi) + (c + di) = (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, (a + bi) - (c + di) = (a - c) + (b - d)i.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that i^2 = -1. For example, (a + bi)(c + di) = ac + adi + bci - bd = (ac - bd) + (ad + bc)i.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can multiply the numerator and denominator by the conjugate of the denominator. For example, (a + bi)/(c + di) = ((a + bi)(c - di))/((c + di)(c - di)) = ((ac + bd) + (bc - ad)i)/((c^2 + d^2)).

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number a + bi is a - bi.

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

A: The magnitude of a complex number a + bi is given by the formula |a + bi| = sqrt(a^2 + b^2).

Q: How do I find the argument of a complex number?

A: The argument of a complex number a + bi is given by the formula arg(a + bi) = arctan(b/a).

Q: What is the polar representation of a complex number?

A: The polar representation of a complex number a + bi is given by the formula a + bi = r(cos(theta) + i sin(theta)), where r = sqrt(a^2 + b^2) and theta = arctan(b/a).

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you can use the formulas r = sqrt(a^2 + b^2) and theta = arctan(b/a).

Q: How do I convert a complex number from polar form to rectangular form?

A: To convert a complex number from polar form to rectangular form, you can use the formulas a = r cos(theta) and b = r sin(theta).

Q: What is the relationship between complex numbers and trigonometry?

A: Complex numbers and trigonometry are closely related. The trigonometric functions sine and cosine can be represented as complex numbers, and the exponential function can be represented as a complex number raised to a power.

Q: How do I use complex numbers in real-world applications?

A: Complex numbers have numerous applications in real-world problems, such as electrical engineering, signal processing, and quantum mechanics. They are used to represent and analyze signals, circuits, and 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:

  • Confusing the real and imaginary parts of a complex number
  • Failing to use the correct formula for addition, subtraction, multiplication, and division
  • Not using the conjugate of a complex number when dividing
  • Not checking for errors in calculations

Q: How do I learn more about complex numbers?

A: To learn more about complex numbers, you can:

  • Read books and online resources on complex numbers
  • Watch video lectures and tutorials on complex numbers
  • Practice problems and exercises on complex numbers
  • Join online communities and forums for complex numbers

Note: The questions and answers provided in this article are for informational purposes only and are not intended to be a comprehensive or definitive guide to complex numbers.